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					Casao                                                                                Montwood High School
                                                   Physics Equations

1. Vector Resolution:
     a. Perpendicular Vectors:
             Pythagorean Theorem for Magnitude: a 2 + b2 = c2
             Trig Functions for Direction:
                             opposite
                    sin θ 
                            hypotenuse
                                             adjacent
                              cos θ 
                                            hypotenuse
                             opposite
                              tan θ 
                             adjacent
        b. Non perpendicular Vectors:
                  Law of C osines for Magnitude: c2  a2  b2  2  a  b  cos C
                                                         sin A sin B sin C
                  Law of Sines for Direction:                      
                                                           a     b     c

                       x
2. Velocity: v                         v = velocity, x = displacement or distance, t = time
                        t

3. Velocity and Acceleration:
      vf  vi  a  t                                    vi = initial velocity, v f = final velocity
                          
        Δx  v i  t  0.5  a  t    2
                                                          x = displacement or distance, t = time
        v f 2  v i 2  2  a  Δx                       a = acceleration

4. Newton’s Second Law:                 F = m·a            F = force, m = mass, a = acceleration

5. Weight:       Fw = mg               Fw = Force (weight), m = mass, g = gravity (9.8 m/s 2 on Earth)

6. Tension:
     Up as positive: T = m(g + a)                                          T = tension, m = mass, g = gravity
     Down as positive: T = m(g – a)                                        a = acceleration

 7. Acceleration of masses attached to a string, one accelerating horizontally and one
    accelerating vertically.
     m hg = (m h + m s)a        m h = hanging mass, ms = sliding mass, a = acceleration
                                  g = gravity

8. Atwood’s machine (two vertically suspended masses over a pulley):
                                         m 1 and m 2 = hanging masses, g = gravity
         2  m1  m2  g
      T                                 T = tension, a = acceleration
            m1  m2

                                                       G  m1  m 2
9. Law of Universal Gravitation:                  F
                                                           d2
                                                                                                         N  m2
    F = gravitational force of attraction, G = gravitational constant = 6.672 x 10 -11
                                                                                                          kg 2
                m 1 and m 2 = masses, d = distance between the centers of the masses
                                                                                              G  m1  m 2
10. For an object located some distance above the surface of a body:                     F
                                                                                                r  d2
                  r = radius of body, d = distance above the surface of the body

                                                    G  Mb
11. Surface gravity (acceleration):         g              g = surface gravity (acceleration)
                                            d2
                        Mb = mass of body [Earth’s mass = 5.98 x 10 24 kg]
                           d = distance b/w the centers of the masses

                                                                                   2
                                                                   g1 d 2
12. Proportional relati onship between gravity and distance:           
                                                                   g2    d1 2
                 For objects orbiting the Earth: g 2 = 9.8 m/s 2 , d1 = 6.37 x 106 m
                        d2 = 6.37 x 106 m + distance above Earth’s surface

                                                                            1  1    
13. Work required to separate two masses:                W  G  m1  m 2   
                                                                            r
                                                                                     
                                                                                     
                                                                             1 r2   
                                                                         N  m2
      W = work, G = gravitational constant = 6.672 x 10 -11           , m 1 and m 2 = masses
                                                                 kg 2
       r1 = initial distance b/w centers of mass; r 2 = final distance b/w centers of mass

                                                                                                    G  m1  m 2
14. Gravitational potential energy of two masses separated by a distance: U  
                                                                                                         d
                                                                              N  m2
U = potential energy, G = gravitational constant = 6.672 x 10 -11                        , m 1 and m 2 = masses
                                                                    kg 2
                               d = distance b/w centers of the masses

                                                2GM
15. Esca pe velocity of Earth:        v esc                           vesc = escape velocity
                                                  d
                         N  m2
    G = 6.672 x 10-11              , M = 5.98 x 1024 kg, d = distance b/w centers of the masses
                            kg 2

                                            GM
16. S peed in a circular orbit:       v                     v = tangential speed of object in circular orbit
                                             d
                    N  m2
G = 6.672 x 10-11            , M = mass of body being orbited, d = distance b/w centers of masses
                     kg 2

                                      4  π2  r3
17. Kepler’s Third Law:        T2                             T = period of orbit, r = radius of orbit
                                         GM
                                           N  m2
                    G = 6.672 x 10-11                 , M = mass of body being orbited
                                             kg 2
            r3
       k              k = 3.35 x 1018 m 3 /s2 , T = period of orbit, r = ra dius of orbit
            T2
                                                         Ta 2       ra 3
18. Ratio of orbital period to orbital radius:                  =
                                                   Tb 2 rb 3
       T a and T b = period of orbit for the bodies, r a and rb = radii of orbits for the bodies

 19. Resolving the weight of an object on an incline into parallel and perpendicular
     components:
             a. Parallel (Fp ): Fx  Fw  sinθ  m  g  sinθ
             b. Perpendicular (Fn):    Fy  Fw  cos θ  m  g  cos θ

                       Ff
20. Friction:    μ                      Ff  μ  Fn  μ  m  g
                       Fn
   = coefficient of friction; Ff = force of friction; Fn = normal force (the force pressing the two
                                           surfaces together)

       a. Horizontal Surfaces:        Fn = Fw on a horizontal surface

       b. Constant S peed (a = 0 m/s 2 ) or With Acceleration:
                                Fx  Ff  m  a or Fx  μ  m  g   m  a

       c. Pulling an Object on a Horizontal Surface:         constant speed (a = 0 m/s 2 )
                   Fx                                                            
                                          μ  m  g   m  a   μ  Fapplied  cos θ  m  a 
             μ            Fapplied 
                 Fw  Fy                   cos θ  μ  sin θ                            
                                                                             m  g   Fapplied  sin θ   
       d. Pushing an Object on a Horizontal Surface: constant speed (a = 0 m/s 2 )
               Fx                                                           
                                     μ  m  g   m  a   μ  Fapplied  cos θ  m  a 
         μ             Fapplied 
             Fw  Fy                  cos θ  μ  sin θ                             
                                                                        m  g   Fapplied  sin θ    
       e. For an Object Skidding to a Stop (when friction is the decelerating force acting upon
                                                                     a
          an object moving along a horizontal surface):           μ
                                                                      g
       f. Down an Incline with Constant Speed:                  μ  tanθ

       g. Down an Incline with an Acceleration:
                  Fx  Ff  m  a  m  g  sinθ  μ  m  g  cos θ  m  a
                      a  g  sinθ  μ  g  cos θ
                      Fnet  Fx  Ff

       h. Up an Incline with Constant Speed [a = 0 m/s 2 ]:
                         Ftotal  Fx  Ff  m  g  sinθ  μ  m  g  cos θ

       i. Up an Incline with Acceleration:         Ftotal  Fx  Ff  m  a
                Fp  Fx  Ff  m  a          Fp  m  g  sinθ  μ  m  g  cos θ  m  a
Fp = the force applied parallel to the surface up the incline (could be a push or a pull – no angle
                                 is involved with the push or pull)

21. Torque:    T  F  r
F = the perpendicular force; r = the perpendicular distance from the pivot point to the point
                                   of a pplication of the force
              T  F  r  sinθ              = the angle between the force and r 
                     For equilibrium problems:              ΣTclockw ise  ΣTcounterclo ise
                                                                                      ckw


22. Projectile Motion:
      a. Vertical Velocity (v y):      v y  v o  sinθ                    v y  v yo  g  t   v o  sin θ  g  t 
      vo = initial projectile velocity;  = angle of projection; v y = vertical velocity at time t
                                     vyo = initial vertical velocity

                                                                       x
       b. Horizontal Velocity (v x):      v x  v o  cos θ                vx 
                                                                       t
          vo = initial projectile velocity;  = angle of projection; x = horizontal distance

                                                                  v o 2  sin2  θ v o 2  2  sin θ  cos θ
       c. Range (horizontal distance):           x  vx  t           R           
                                                                           g                      g
          vo = initial projectile velocity;  = angle of projection; v x = horizontal velocity

      d. Horizontal Position Component at Time t: x  v x  t  v o  cos θ  t
 x = horizontal distance; v o = initial projectile velocity;  = angle of projection; v x = horizontal
                                               velocity

                                                    v o 2  sin θ2
       e. Maximum Vertical Height (h):             h
                                                          2g
                            vo = initial projectile velocity;  = angle of projection

       f. Vertical Position Component (y):                                                               
                                                     y  v yo  t  0.5  g  t 2  v o  sin θ  t   0.5  g  t 2        
       y = vertical distance; v yo = initial vertical velocity; v o = initial projectile velocity;  = angle
                                              of projection

                                                                                                  x2  g          
       g. Vertical Position at any Horizontal Position:                 y  x  tan θ                          
                                                                                           2  v 2  cos θ2     
                                                                                                o                 

       h. Horizontal and Vertical Motion Equations: y-axis direction (upward) as positive with
          a x = 0 m/s2 , a y = -9.8 m/s2
                                 horizontal position: x  x o  v x  t
                           vertical position:                              
                                                    y  y o  v yo  t  0.5  g  t 2     
                                                          v y  v yo  g  t 
                               vertical velocity:
                                                          v y 2  v yo 2  2  g  y 


       i. Initial Velocity from Component Velocities:                vo  v x 2  v y 2

                                              vy
       j. Angle of Projection:      tan θ 
                                              vx

                                             v2
23. Centripetal Acceleration (a c):       ac    r  ω2
                                              r
        a c = centripetal acceleration; v = linear velocity; r = radius;  = angular velocity
                                   m  v2
24. Centripetal Force (Fc):        Fc     m  r  ω2
                                     r
      Fc = centripetal force, m = mass, v = linear velocity, r = radius,  = angular velocity

                                  v2
25. Banking Angles:        tan θ 
                                  rg
                     = banking angle; v = linear velocity , r = ra dius of curvature

26. Centripetal Force in a Vertical Circle:
                                                                       m  v2 
       a. Top of Vertical Circle:         Fc  Ft  Fw Ft  Fc  Fw  
                                                                                m  g 
                                                                       r 
                                                                              
   Ft = tension, Fw = weight, Fc = centripetal force, m = mass, v = linear velocity, r = radius

                                                                          m  v2 
       b. Bottom of Vertical Circle:                      Ft  Fc  Fw  
                                               Fc  Ft  Fw                        m  g 
                                                                          r 
                                                                                 
   Ft = tension, Fw = weight, Fc = centripetal force, m = mass, v = linear velocity, r = radius

27. Critical Velocity (v min)for an Orbiting Object:  v min  r  g
                                r = distance between centers of mass

                                           ω
28. Angular Acceleration ():        =              = angular velocity, t = time
                                           t
               Δω
      avg =            avg = average angular acceleration;  = change in angular velocity
               Δt

                                  Δθ
29. Angular Velocity ():       ω
                                  Δt
                      = change in angular displacement, t = change in time

30. Arc Length (s):      s  rθ            = angular displacement, r = radius

31. Rotary Motion Equations:
       = o + (t)            = angular velocity; o = initial angular velocity;
       = (ot) + (0.5t2 )  = angular displacement;  = angular acceleration
       = o2 + (2)
      vt  r  ω       vt = tangential velocity; r = radius;  = angular velocity
      a t = r                a t = tangential acceleration; r = radius;  = angular acceleration

32. Torque for Rotary Motion:      T  F  r  I  α
       T = torque; F = force; r = ra dius; I = rotational inertia;  = angular acceleration

33. Common Rotational Inertia Equations: m = mass; r = ra dius
            2                                                 1
      ball:    m  r2           ring: m  r 2       disk:       m  r2
            5                                                 2

34. Relationship between Angular Velocity and Linear Velocity: v  ω  r
                     v = linear velocity;  = angular velocity; r = radius
                                   event
35. Frequency (f):           f                  event = swing, pulse, wave, oscillation, etc.
                                   time

 36. Simple Harmonic Motion (for a physical system consisting of a mass connected to
     an oscillator):
                                                 2πt 
       a. Displacement:               x  A  sin        2πf t
                                                  T 
                                                          1             2π           ω
       x  A  sinω  t              x  A  cosω  t            f 
                                                                     T          f 
                                                          T              ω           2π
          x = displacement; A = amplitude; T = period; f = frequency;  = angular velocity
Note: the sin equation is used if the motion starts with zero displacement. The cos equation is
used if the motion starts with the displacement at a maximum. If the motion begins at any
point between zero and the maximum amplitude A, these equations can be used:
  x  a  sinω  t  θ       x  a  cosω  t  θ       determines the starting position


                                                      x  k  e ct  sinω  t 
       b. Damped Harmonic Motion:                                                          c>0
                                                      x  k  e ct  cosω  t 
        c = damping constant; k = original displacement of object; amplitude = k  e ct

                                                 F
       c. S pring Constant (k):             k              F = distorting force; d = change in length
                                                 Δd

                              1          2π                     m
       d. Period (T):          T  T           T  2π
                              f           ω                      k
                  f = frequency; m = mass of oscillating object; k = spring constant

                                                k
       e. Angular Velocity ():             ω
                                                m
                                    m = mass of oscillating object; k = spring constant

                                        ω       1     k
       f. Angular Frequency (f):             f   
                                       2π 2π        m
              = angular velocity; m = mass of oscillating object; k = spring constant

       g. Restoring Force:              F  k  x          k = spring constant; x = displacement

                                                2π A
       h. Maximum Speed (v max) :                        ωA
                                                 v max  
                                                   T
                           A = amplitude; T = period;  = angular velocity
Note: the sign (+ or -) depends on the direction of the displacement from the equilibrium
position; positive if the mass is moving towards the equilibrium position; negative if the mass is
moving away from the equilibrium position.

                                                                   4  π2  A
       i. Maximum Acceleration (a max):                 a max             2
                                                                                 ω2  A
                                                                T
                                   A = amplitude; T = peri od;  = angular velocity
      j.   Velocity (v):
              dv 
             v
               dt
                     ω  A  sinω  t   θ  v  
                                                         k
                                                         m
                                                              
                                                            A2  x2   
                                                                      v  ω  A 2  x 2
   A = amplitude;  = angular velocity; k = spring constant; x = displacement; m = mass;
                                   determines the starting point

                                  dv 
      k. Acceleration (a):       a      ω 2  A  cosω  t   θ
                                   dt
              A = amplitude;  = angular velocity;  determines the starting point

      l. Given Initial Conditions:      xo an d vo
                                                                                            2
                                                               vo                  v 
       x o  A  cos θ  v o  ω  A  sin θ  tan θ                A  x o2   o 
                                                                                     ω 
                                                              ω  xo                   
      xo = initial displacement; v o = initial velocity; A = amplitude;  = angular velocity

     m. Kinetic Energy (K):     K  0.5  m  v 2  0.5  m  ω 2  A 2  sinω  t   θ2
 m = mass; v = velocity;  = angular velocity; A = amplitude;  determines the starting point

      n. Potential Energy (U):   U  0.5  k  x 2  0.5  k  A 2  cosω  t   θ2
         k = spring constant; x = displacement;  = angular velocity; A = amplitude;
                                determines the starting point

      o. Total Energy (E):       E  0.5  k  A 2
                                 k = spring constant; A = am plitude

37. Pendulum:
                                           g
      a. Angular Velocity ():        ω                l = length
                                           l

                                                                      mg d
      b. Angular Velocity for a Physical Pendulum ( ):           ω
                                                                         I
                  d = distance from center of gravity to pivot; I = rotational inertia

                                      l   2π
      c. Period (T):       T  2π                    l = length;  = angular velocity
                                      g    ω


                                                          2π          I
      d. Period for a Physical Pendulum (T):         T        2π
                                                           ω         mg d
    I = rotational inertia; d = distance from center of gravity to pivot;  = angular velocity

                                               g  sin θ
      e. Pendulum Acceleration (a):        a
                                                    l
                               determines the starting point; l = length

      f. Restoring Force:       F  m  g  sin θ
                              m = mass;  determines the starting point
38. Torsional Pendulum:
                                                d 2 θ
       a. Hooke’s Law:       T  k  θ  I 
                                         dt 2
        T = torque; k = torsional constant;  = displacement angle; I = rotational inertia

                                        TI     k
       b. Angular Velocity ():       ω     
                                         θ       l
        T = torque; k = torsional constant;  = displacement angle; I = rotational inertia

                                      I
       c. Period (T):     T  2π                    I = rotational inertia; k = torsional constant
                                      k

39. Work (W):      W  Fapplied  d
            Fapplied = applied force; d = distance moved in direction of a pplied force
           Work (W) when applied force is at an angle  with the direction of motion:
                                           W  Fapplied  cos θ  d

40. Work (W) in Rotary Motion:    W  T  θ  F  r  θ
   T = torque;  = angular displacement (in radians); F = the perpendicular force; r = the
      perpendicular distance from the pivot point to the point of a pplication of the force

                          work output               Fr  d r 
41. Efficiency (e):   e work input   100 %   F  d   100 %
                                                              
                                                    e e
Fr = the resistance force; Fe = the effort force; dr = the distance the resistance force moves; de =
                                the distance the effort force moves
                     Fw  h 
       Lifting: e                  100 %
                     Fapplied  d 
                                  
      Fw = weight; Fapplied = the force applied to the object; h = height the weight is lifted
                                d = distance the applied force moves
                        Note: efficiencies can be reported as a decimal.

                                                      Fr
42. Mechanical Advantage (MA or AMA):                 MA 
                                                      Fe
                            Fr = the resistance force; Fe = the effort force

                                                de
43. Ideal Mechanical Advantage (IMA):              IMA 
                                                dr
     dr = the distance the resistance force moves; de = the distance the effort force moves

                            MA 
       Efficiency (e):  e        100 %
                            IMA 
                  MA = mechanical advantage; IMA = ideal mechanical advantage

                         W F d
44. Power (P):     P           Fv                 W = work; F = force; d = distance; v = velocity
                         t   t
                                                         F  r  θ
45. Power (P) in Rotary Motion:       P  T  ω  F  r  ω 
                                                              t
T = torque;  = angular velocity;  = angular dis placement (in radians); F = the perpendicular
 force; r = the perpendicular distance from the pivot point to the point of a pplication of the
                                              force

46. Gravitational Potential Energy (Ep or U):          U  m g  h                m = mass; h = height

                                                             F
47. Elastic Force Constant or S pring Constant (k):                k
                                                             x
                           F = force (can be Fw); x = distance of stretch

48. Elastic Potential Energy (Ep or U):   U  0.5  k  x 2
             k = elastic force constant or spring constant; x = distance of stretch

49. Kinetic Energy (Ek or K):      K  0.5  m  v 2                        m = mass; v = velocity

50. Kinetic Energy (Ek or K) in Rotary Motion:     K  0.5  I  ω2
                          I = rotational inertia;  = angular velocity

51. Law of Conservation of Energy: many pro blems require setting U = K, commonly
                                                                        v2
     resulting in the equation:       v  2g h         or        h
                                                                        2g

52. Impulse (J) and Momentum:     J  F t  F t  mv
                        F = force; t = time; m = mass; v = velocity

53. Momentum (p):       p  mv                   m = mass; v = velocity

54. Law of Conservation of Momentum: many problems will involve setting the momentum
    before a collision to the momentum after the collision.
      a. Momentum of two objects that collide elastically:
                               m1  v1  m 2  v 2  m1  v1'  m 2  v 2 '
m 1 and m 2 = masses before and after the collision; v 1 and v2 = velocity before the collision; v 1 ’
                             and v2 ’ = velocity after the collision

      b. Relative velocity of the two objects before the collision (v 1 – v2 ) equals the negative of
                                                                                    
          the relative velocity of the two objects after the collision  1  v1'  v 2 '        
                                        v 1  v 2  v 2 '  v 1'
                                        v 1  v 2   v 1'  v 2 '
                                                          
                                        v1  v 2  1  v1'  v 2 '     
      c. Momentum of two objects that collide inelastically:
                                   m1  v1  m2  v 2  m1  m2   v '
  m 1 and m 2 = masses before and after the collision; v 1 and v2 = velocity before the collision
                                v’ = velocity after the collision
       d. Momentum for a particle that divides:         m1  v1  m2  v 2  m3  v 3
 m 1 = mass of original particle; v 1 = velocity of original particle; m 2 = mass of original particle
remaining; v 2 = velocity of this particle; m3 = mass of particle removed or ejected; v 3 = velocity
                                            of this particle

                      F1     x
55. Hooke’s Law:           1
                      F2     x2
              F1 and F2 = a pplied forces; x1 and x2 = proportional distance of stretch

                                                F
56. Tensile Strength:        tensile strength  
                                                A
                              F = a pplied force; A = cross-sectional area

                         F                                   
57. Stress:     stress                 F = a pplied force; A = cross-sectional area
                         A

                       Δl x d  x n
58. Strain:     strain  
                       l       xn
      l = change in length; l = original length; x d = deformed length; xn = normal length

59. Modulus Equations:
                                         stress     Fl
      a. Young’s Modulus (Y):         Y              
                                         strain Δl  A
                                                               
   F = force; l = original length; l = change in length; A = cross-sectional area (generally
                                       
                                      A  π  r 2  0.25  π  d 2 )

                                                       stress
      b. Elastic Modulus:         elastic mod ulus 
                                                       strain

                                          shear stress     Fl     F              Δl
      c. Shear Modulus (G):         G                                  φ
                                          shear strain A  Δl A  φ               l
                                         = angle of displacement

                                    V  ΔP                                  1    ΔV
      d. Bulk Modulus (B):         B               Compressibility (k): k    
                                      ΔV                                     B V  ΔP
              V = original volume; V = change in volume; P = change in pressure

                          F
60. Pressure (P):     P 
                          A
                                                            
              F = force; A = cross-sectional area (generally A  π  r 2  0.25  π  d 2 )

61. Columnar Fluid Pressure (P): P  h D g
                        D = density; h = depth (columnar height)

62. Fluid Pressure at Depth h: PT  1 atm  h  D  g 
  PT = total pressure (atmospheric pressure + columnar fluid pressure); h = depth (columnar
                         height); D = density; 1 atm = 1.013 x 10 5 N/m 2
                                                                            F1  F
63. Pascal’s Principle (Transmission of Pressure):            P1  P2        2
                                                                            A1 A 2
                                                                                 
P1 = input pressure; P2 = output pressure; F1 = input force; F2 = output force; A 1 = input area
                                                               
        A 2 = output area; A = cross-sectional area (generally A  π  r 2  0.25  π  d 2 )
                                                  
                                             F2   A2
64. Hydraulic Force Multiplication:              
                                             F1   A1
                                                                          
  F1 = input force; F2 = output force; A 1 = input area; A 2 = output area; A = cross-sectional
                                               
                             area (generally A  π  r 2  0.25  π  d 2 )

                                                         F2
65. Actual Mechanical Advantage (AMA):              AMA 
                                                         F1
                                  F1 = input force; F2 = output force

                                                     BF  mass in air  mass in fluid  g
66. Archimede’s Principle (Buoyant Force):           BF  true weight  apparent weight
                                                 BF  Fw displaced fluid  D  V  g
                         BF = buoyant force; D = density of fluid, V = volume

                                       Dx   Fw x
67. S pecific Gravity:     sp. gr.       
                                       Dw   Fw w
Dx = density of unknown; Dw = density of water; Fwx = weight of unknown; Fww = weight of water

                                                 F
68. Coefficient of Surface Tension ():        γ
                                                 L
                         F = force required to increase the surface area by L

                                                      2 γ
69. Capillary Rise (0, perfectly wetting):         h
                                                    rDg
  h = capillary rise;  = coefficient of surface tension; r = radius of ca pillary tube; D = density

                                                          2  γ  cos θ
70. Capillary Rise (>0, not perfectly wetting):         h
                                                            rDg
  h = capillary rise;  = coefficient of surface tension; r = radius of ca pillary tube; D = density

                         Fl
71. Viscosity ():  η 
                         Av
    F = force needed to produce velocity v; v = velocity difference of two layers separated by
                                                   
                                     distance l; A = area

                                                 V 
72. Volumetric Flow Rate or Discharge (Q):   Q      Av
                                                  t
                                       
                 V = volume; t = time; A = cross-sectional area; v = velocity
                                                                                             
73. Equation of C ontinuity (Fluid Flow Through a Pipe): Q1  Q2  A1  v1  A 2  v 2
                                                        
  Q1 an d Q2 = volumetric flow rates; A 1 = input area; A 2 = output area; v 1 = input velocity
                                      v2 = output velocity
                                    π  r 4  ΔP
74. Poisseuille’s Equation:         Q
                                      8lη
  Q = volumetri c flow rate; r = radius of pi pe; l = length of pi pe;  = viscosity; P = pressure
                          difference between the two ends of the pipe

75. Work Done by Piston Moving Fluid Against Opposing Pressure: W  P  ΔV
                     W = work; P = pressure; V = change in volume

76. Bernoulli’s Equation:                                                            
                            P1  0.5  D  v12  h1  D  g   P2  0.5  D  v 2 2  h 2  D  g 
 P1 = input pressure; P2 = output pressure; D = flui d density; v 1 = input velocity; v 2 = output
                        velocity; h 1 = input height; h 2 = output height

77. Lift (FL):
            
                                
                                    
                 FL  0.5  p  A  v t 2  v u 2   
            A    = area of wing; p = density of air (1.3 kg/m 3 ); vt = airflow over top of wing
                                       vu = airflow un der the wing
                                                
78. Drag (Fdrag):   Fdrag  0.5  C  p  v 2  A
                          
   C = drag coefficient; A = cross-sectional area; p = densi ty of air (1.3 kg/m 3 ); v = velocity

       a. Terminal Velocity:         Fdrag = Fw

       b. Prior to Terminal Velocity:           Fw = Fdrag + (ma)

79. Torricelli’s Theorem:        v  2 g  h                    v = velocity; h = height above hole

80. Linear Expansion (Solid):    Δl  α  l  ΔT  α  l  Tf  Ti 
 l = change in length;  = coefficient of linear expansion; l = original length; T = change in
                                             temperature

                    To determine the new length:                new length = old length + l
                                                                   
81. Area Expansion (Solid):    ΔΑ  2  α  Α  ΔT  2  α  Α  Tf  Ti 
                       
   A = change in area; A = original area;  = coefficient of linear expansion; T = change in
                                            temperature
                                                                                        
                      To determine the new area:                new area = old area +  A

82. Volume Expansion (Solid):  ΔV  3  α  V  ΔT  3  α  V  Tf  Ti 
       V = change in volume; V = original volume;  = coefficient of linear expansion
                               T = change in temperature

       a. To determine the new volume: new volume = old volume + V

       b. To determine the new reading on a solid object containing a liquid:
                          new reading  old reading  ΔV flask  ΔV liquid

                      V   P
83. Boyles Law:         
                     V  P
         V = original volume; V’ = final volume; P = original pressure; P ’ = final pressure
                     V     T
84. Charles Law:         K'
                     V  TK
   V = original volume; V’ = final volume; T K = original Kelvin temperature; T K’ = final Kelvin
                                          temperature

                             P    T
85. Gay-Lussac’s Law:            K
                            P  TK '
  P = original pressure; P’ = final pressure; T K = original Kelvin temperature; T K’ = final Kelvin
                                            temperature

                             V P   TK
86. Com bined Gas Law:          
                             V' P  TK '
 V = original volume; V’ = final volume; P = original pressure; P ’ = final pressure; T K = original
                      Kelvin temperature; T K’ = final Kelvin temperature

87. Universal Gas Law:   P  V  n  R  TK
P = pressure (in atmospheres); V = volume (in liters); n = moles; R = Universal Gas Constant =
                                   L  atm
                         0.0821             ; T K = Kelvin temperature
                                   mol  K

                                     D    P
88. Gas Density and Pressure:           
                                     D P 
D = original gas density; D’ = final gas density; P = original gas pressure; P’ = final gas pressure

89. Heat (Q):  Q  m  c  ΔT  m  c  Tf  Ti 
    m = mass; c = specific heat; T = change in temperature (often high temperature – low
                                             temperature)

90. Electric Energy to Heat Energy:     P  t  m  c  ΔT
    P = power (in Watts); t = time; m = mass; c = specific heat; T = change in temperature

91. Gravitational Potential Energy to Heat Energy:     m  g  h  m  c  ΔT
             m = mass; h = height; c = specific heat; T = change in temperature
                  Note: be very careful with the units on the mass (kg or g)

92. Kinetic Energy to Heat Energy:     0.5  m  v 2  m  c  ΔT
             m = mass; v = velocity; c = specific heat; T = change in temperature
                  Note: be very careful with the units on the mass (kg or g)

93. Law of Heat Exchange:       Qlost = Qgained

94. Heat and Changes of Phase:
      a. Fusion:  Q  m Lf             Q = heat; m = mass; L f = heat of fusion

       b. Va porization:    Q  m Lv             Q = heat; m = mass; L v = heat of vaporization

       c. Com bustion:     Q  m  Hc             Q = heat; m = mass; Hc = heat of com bustion
                                    T  T1
95. Heat Con duction:    Q  KAt 2
                                         d
 K = thermal conductivity constant; T 2 -T1 = temperature difference between surfaces; t = time;
                              
                              A = area; d = thickness of material

                                            Q 
96. Rate of Ra diation (Stefan’s Law):         A  e  σ  TK 4
                                            t
   Q                            
     = radiation per unit time; A = area; e = emissivity; T K = Kelvin temperature;  = Stefan-
   t
                                                                  W
                          Boltzmann constant = 5.67 x 10 -8
                                                                 2
                                                               m  K4

                          ΔQ
97. Entropy (S):     ΔS 
                          TK
                     Q = change in heat energy; T K = Kelvin temperature
    Note: if more than one Kelvin temperature is given, use the average value to resolve the
                                          problem.

98. First Law of Thermodynamics: Q  W  U               Q = heat energy; W = work
 Note: work done on a system is the negative value of the work done by a system, therefore:

       a. Work Done By System:        U  QW

       b. Work Done On System:         U  QW

99. First Law of Thermodynamics (Heat Engine): Qh  Qc  W
           Qh = heat input due to fuel combustion; Qc = heat energy lost; W = work

100. Work Output for Com bustion Engine: W  Qh  Qc
          Qh = heat input due to fuel combustion; Qc = heat energy lost; W = work

                                                 W    Q  Qc
101. Efficiency of Com bustion Engine:       e      h
                                                Qh      Qh
     e = efficiency; Qh = heat input due to fuel com bustion; Qc = heat energy lost; W = work

                                                     1
102. Efficiency of Gasoline Engine:      e 1
                                                             γ 1
                                                   V1   
                                                        
                                                  V     
                                                   2    
  V1
     = com pression ratio; V1 = initial volume; V2 = final volume;  = ratio of the molar specific
  V2
                                                    Cp             
                                             heats 
                                                   C
                                                                    
                                                                    
                                                    V              

                                        Th  Tc
103. Carnot Efficiency (e max):   e max 
                                          Th
                        T h = input temperature; T c = output temperature
104. Work Done by Gas (Isothermal Expansion):     W  P  Vf  Vi 
               W = work; P = pressure; Vf = final volume; Vi = initial volume

105. Work (Adiabatic Process): W  Uf  Ui
               W = work; Uf = final internal energy; Ui = initial internal energy

106. Heat Pumps:
      a. First Law of Thermodynamics: Qh  Qc  Win
                   Qh = heat output; Qc = heat removed; Win = work input

                                                           Qh
       b. Coefficient of Performance (COP):        COP 
                                                           Win

                                       Qh
       c. heat pum p C OP:     COP 
                                       Win

                                    Qc
       d. refrigerator COP: COP 
                                    Win
                     Qh = heat output; Qc = heat removed; Win = work input

                                                       Qc       Tc       Qc
       e. Carnot Efficiency (refrigerator):   COP                  
                                                      Win    Th  Tc   Qh  Qc
            T h = output temperature (in Kelvins); T c = input temperature (in Kelvins)

                                                Tc
       f. Work to Run a Refrigerator:        W
                                              Th  Tc
            T h = output temperature (in Kelvins); T c = input temperature (in Kelvins)

107. Wave Velocity (v):    v  λf             = wavelength; f = frequency

108. Electromagnetic Waves:
                                                       k           1        N  m2
       a. S peed of Propagation (vacuum):         c      
                                                       k        ε o  μo       C2
                                              N  m2                        N                          C2
k = electrostatic constant = 8.98755 x 10 9            ; k’ = 1 x 10-7          ; o = 8.854 x 10-12        ;
                                                C2                 A2                     N  m2
                                               Tm                  N
                            o = 4  π x 10 7       1.257 x 10 6 2
                                                A                  A
             
      E max   E ω                                      
   c                 E = electric field strength; B = magnetic field strength;  = angular
      B max   B k
                                                                            2π
                   velocity ( ω  2  π  f ); k = angular wave number  k     
                                                                             λ 
                                                                                        
                                                                                      E2
       b. Magnetic Field-Electric Field Relationship:             B2  ε o  μ o  E 2  2
                                                                                        c
                      C2                                     
o = 8.854 x 10-12             ; E = electric field strength; B = magnetic field strength; c = speed of
                           2
                     Nm
                                                                 Tm                 N
                 light = 3 x 108 m/s; o = 4  π x 10 7              1.257 x 10 6 2
                                                                  A                 A
                                                            
      c. Sinusoidal Electric Field equation: E  E max  cosk  x   ω  t 
      
      E = electric field strength;  = angular velocity ( ω  2  π  f );k = angular wave number
                               2π
                           k      ; x = position of wave with respect to x-a xis
                                λ 
                                                           
       d. Sinusoidal Magnetic Field equation:             B  Bmax  cosk  x   ω  t 
     
     B = magnetic field strength;  = angular velocity ( ω  2  π  f );k = angular wave number
                          2π
                     k        ; x = position of wave with respect to x-a xis
                           λ 

   e. Energy Carried by Electromagnetic Waves (Power per Unit Area):
                 
      S
         P c  B2
            
         A 2  μo
                             
                                              
                      ε o  E 2   c  ε o  E 2 (equal contributions from both the magnetic field
                                 
                                
                                             and the electric field)
                                                                               
                 or:   S
                              1  
                             μo
                                       
                                  ExB       EB
                                                 μo
                                                       S
                                                                E2
                                                               μo  c
                                                                       S
                                                                              c  B2
                                                                               μo
         P                                                                              C2
     S   = power per unit area (also called a Poynting vector); o = 8.854 x 10-12         ;
        A                                                                           N  m2
   E = electric field strength; B = magnetic field strength; c = speed of light = 3 x 10 8 m/s;
                                               Tm                 N
                            o = 4  π x 10 7      1.257 x 10 6 2
                                                A                 A
                                                                          
                                           E max  B max     E max 2     c  B max
        f. Wave Intensity (I): I  S avg                             
                                               2  μo       2  μo  c    2  μo
                                                     2                                   
                                                   C
I = S avg = wave intensity; o = 8.854 x 10-12          ; E = electric field strength; B = magnetic
                                                N  m2
                                                                             Tm                   N
   field strength; c = speed of light = 3 x 10 8 m/s; o = 4  π x 10 7            1.257 x 10 6 2
                                                                               A                  A
                                                                                                                  
       g. Instantaneous Energy Density (associated with the electric field):                    u E  0.5  ε o  E 2
                                                                                                  


                                                                       C2         
   u E = instantaneous energy density; o = 8.854 x 10-12
     
                                                                            2
                                                                                ; E = electric field strength
                                                                     Nm
                                                                                                            
                                                                                                           B2
        h. Instantaneous Energy Density (associated with the magnetic field):                   uB 
                                                                                                 
                                                                                                          2  μo
                                       remember that u E = u B
                                                            

                                                                          Tm                 N
          u B = instantaneous energy density; o = 4  π x 10 7
                                                                              1.257 x 10 6 2
                                                                           A                 A
                                  
                                  B = magnetic field strength
                                                                              
                                                                          2 B2
        i. Total Instantaneous Energy Density:     u       
                                                           uE
                                                                  εo  E 
                                                                    uB
                                                                     
                                                                              μo
                                                                  2     
                                                                C
   uE
       = instantaneous energy density; o = 8.854 x 10-12            ; E = electric field strength
                                                                    2
                                                              Nm
                                                                   Tm                    N
           u B = instantaneous energy density; o = 4  π x 10 7
                                                                         1.257 x 10 6 2
                                                                     A                    A
                                   
                                   B = magnetic field strength

   j. Average Energy Density of an Electromagnetic Wave:
                                                      
                           2                     2  B max 2
             uavg  ε o  E avg  0.5  ε o  E max                    or       I  S avg  c  uavg
                                                      2  μo
                                                                C2         
      uavg = average energy density; o = 8.854 x 10-12                  ; E = electric field strength;
                                                             N  m2
                                  Tm                   N     
             o = 4  π x 10 7        1.257 x 10 6     2
                                                            ; B = magnetic field strength
                                   A                    A

                                                          U
        k. Momentum (Perfectly Absorbing Surface):           p
                                                          c
                p = momentum; U = total energy; c = speed of light = 3 x 10 8 m/s

                                                                     2S
        l. Ra diation Pressure (Perfectly Reflecting Surface):           P
                                                                      c
              P = pressure; S = rate of energy flow; c = speed of light = 3 x 10 8 m/s

109. Electromagnetic Wave Velocity (Transparent Medium): c  λ  f
              c = speed of light = 3 x 108 m/s; λ = wavelength; f = frequency

                                                                         F
110. S peed of Transverse Waves (Stretched Spring):          vw 
                                                                         m
                                                                           l
              vw = wave velocity; F = tension in spring; m           = mass per unit length
                                                                l

                                                                       E
111. S peed of Com pressional Waves (Liquid or Solid Rod):               vw 
                                                                       D
           vw = wave velocity; E = elastic or stretch modulus of rod; D = density of rod
112. Wavelength and Pipes:
                       λ  2 l
      a. Open Pipes:
                       λ  2  l  0.8  d
                   λ = wavelength; l = length of pipe; d = diameter of pipe

                              λ  4l
        b. Close d Pipes:
                              λ  4  l  0.4  d
                        λ   = wavelength; l = length of pipe; d = diameter of pipe

113. Velocity of Soun d in Air:    v = 331.5 m/s at 0 C
                                          m             
               T0     then    v  331 .5    0.6  T        T = Celsius temperature
                                          s             

                                                                            v  vL 
114. Doppler Equation (listener moving; source stationary):        f  f 
                                                                            v     
                                                                                   
f’ = frequency heard by listener; f = frequency of soun d; v = velocity of soun d in air; v L = velocity
                                              of listener
    Note: positive if listener is moving towards source; negative if listener is moving away from
                                               source.

                                                                            v 
115. Doppler Equation (listener stationary; source moving):        f  f 
                                                                           vv 
                                                                                 
                                                                              S 
f’ = frequency heard by listener; f = frequency of soun d; v = velocity of soun d in air; v S = velocity
                                             of source
    Note: positive if source is moving towards listener; negative if source is moving away from
                                              listener.

                                                                             v  vL 
116. Doppler Equation (source and listener both moving):            f  f 
                                                                            vv 
                                                                                     
                                                                                  S 
       f’ = frequency heard by listener; f = frequency of source; v = velocity of soun d in air
                          vL = velocity of listener; v S = velocity of source

                                          v Δf
117. Doppler Effect for Light Waves:        
                                          c    f
c = speed of light = 3 x 10 8 m/s; v = speed of moving source or moving observer; f = difference

            between observed frequency and actual frequency; f = frequency of light

118. Beat Frequency (fbeat): f beat  f1  f 2
                           f1 and f2 = frequencies of individual waves

                                                        I 
119. Relative Sound Intensity ():        β  10  log  
                                                       I 
                                                        o
                      I = intensity; I o = threshold intensity = 1 x 10 -16 W/cm 2

                                                             2        2
120. Soun d Intensity – Distance Relationship: I1  d1  I 2  d 2
 I 1 = initial sound intensity; d1 = initial distance; I 2 = final sound intensity; d2 = final distance
121. Energy – Mass Relationship:  E  m  c2
                   E = energy; m = mass; c = sped of light = 3 x 10 8 m/s

122. Energy of a Photon:   E  hf
              E = energy; h = Planck’s constant = 6.63 x 10 -34 Js; f = frequency

123. Work Function:  = hf
           = work function; h = Planck’s constant = 6.63 x 10 -34 Js; f = frequency

124. Kinetic Energy of Ph otoelectrons:    K  h  f  - 
  K = kinetic energy;  = work function; h = Planck’s constant = 6.63 x 10 -34 Js; f = freque ncy

125. Cutoff Potential:   q  Vo   h  f   
  q = electron charge = 1.602 x 10 -19 C; Vo = cutoff potential;  = work function; h = Planck’s
                            constant = 6.63 x 10-34 Js; f = frequency

                                     h
126. deBroglie Wavelength:      λ
                                   mv
          - wavelength; h = Planck’s constant = 6.63 x 10 -34 Js; m = mass; v = velocity

                                             f   L
127. Laws of Strings – Law of Lengths:         
                                            f L
                  f = frequency; f’ = new frequency; L = length; L’ = new length

                                              f   d
128. Laws of Strings – Law of Diameters:        
                                             f d
               f = frequency; f’ = new frequency; d = diameter; d’ = new diameter

                                              f    T
129. Laws of Strings – Law of Tensions:          
                                             f   T
                 f = frequency; f’ = new frequency; T = tension; T’ = new tension

                                             f   D
130. Law of Strings – Law of Densities:        
                                             f D
                 f = frequency; f’ = new frequency; D = density; D’ = new density

131. Luminous Flux:      Φ  4πI
                            = luminous flux; I = intensity of source

                         Φ     I
132. Illumination:   E   2
                         A r
                                            
        E = illuminance;  = luminous flux; A = area; I = intensity of source; r = radius

                                                            Φ  cos θ
If the light source is not perpendicular to the surface:   E        , where  is the angle the
                                                                A
 light beam makes with the normal to the illuminated surface or is the tilt angle of the surface
                  from its perpendicular position with respect to the beam.
                                             I1   r2
133.Intensity of Source (Photometer):            12
                                             I 2 r2
            I 1 and I 2 = intensities of sources; r 1 and r2 = distance of source to pa per

                                     R
134. Focal Length of Mirror:     f 
                                     2
                             f = focal length; R = ra dius of curvature

                                    1     1    1
135. Mirror or Lens Equation:              
                                    f do di
                     f = focal length; do = object distance; di = image distance

                                  hi    d
136. Images and Distances:             i
                                  ho    do
         h i = image height; h o = object height; do = object distance; di = image distance


137. For lens problems involving a ratio of distances for both objects and images, the focal
                                      1     1     1    1
     length is constant, therefore:               '
                                                       '
                                     do di       do    di
do = object distance; di = image distance; do’ = distance of 2nd object; di’ = distance of 2nd image

                                   1              1   1 
138. Lens-Maker’s Equation:           n  1  
                                                 R  R 
                                                          
                                   f              1    2 
 f = focal length; n = index of refraction of lens material; R 1 and R2 = ra dii of curvature of first
                                         and second surfaces
Note: R is positive if the surface is convex; negative if the surface is concave; and infinite is the
                                          surface is planar.

                             hi   d            25
139. Magnification:     M       i      M
                             ho  do             f
    M = magnification; h i = image height; h o = object height; do = object distance; di = image
                                    distance; f = focal length

140. Total Magnification of Two Lenses: mtotal  m1  m2
           m total = total magnification; m 1 and m 2 = magnification of the two lenses

141. Optical Instruments:
                                            angular size of image θ i
       a. Angular Magnification:       M                         
                                            angular size of object θ o

                                               angular size of image through magnifier
       b. Strength of a Magnifier:     M
                                             angular size of object at near po int ( 25 cm)

                                                           25 cm 
       c. Minimum Magnification (Magnifier):          M         
                                                           f     
                                            f = focal length
                                                              25 cm 
       d. Ma ximum Magnification (Magnifier):            M          1
                                                              f     
                                              f = focal length

                                                 d   25 cm 
       e. Magnification (Microscope):     M   i   
                                                d   f
                                                               
                                                               
                                                 o       e  
            do = object distance; di = image distance; fe = focal length of eyepiece lens
                         Note: negative sign indicates an inverted image

                                                                   fo
       f. Magnification (Astronomical Telescope):          M
                                                                   fe
                   fe = focal length of eyepiece lens; fo = focal length of objective lens
                             Note: negative sign indicates an inverted image

                                                            1      1    1
142. Effective Focal Length of Lenses in Contact:                   
                                                             f eff f1 f 2
           feff   = effective focal length; f1 and f2 = focal lengths of two le nses in contact

143. Index of Refraction:
                                            c
       a. Transparent Substance:            n
                                            v
n = index of refraction; c = speed of light = 3 x 10 8 m/s; v = speed of light through the medium

                               sin i   v
       b. Snell’s Law:        n      1
                               sin r v 2
 n = index of refraction; i = incident angle; r = refracted angle; v 1 and v2 = velocities in the two
                                 media of different optical densities

         c. Snell’s Law:    n1  sin θ1  n2  sin θ2
    n 1 = index of refraction in incident media; 1 = incident angle; n2 = index of refracti on in
                                  refracted media; 2 = refracted angle

                                        n2
       d. Critical Angle:       sin θ c 
                                         n1
        c = critical angle; n 1 = larger index of refraction; n 2 = smaller index of refraction

                                                           λ2  v
       e. Ratio of Wavelengths in Two Different Media:         2
                                                           λ1  v1
  1 = wavelength in media one; 2 = wavelength in media two; v 1 = velocity in media one; v 2 =
                                     velocity in media two

                                                                    v2   n
       f. Ratio of the Speeds of Light in Two Different Media:          1
                                                                    v1   n2
n 1 = index of refraction in media one; n2 = index of refraction in media two; v 1 = velocity in
media one; v 2 = velocity in media two
                                                                           λ2    n
       g. Relationship Between Wavelengths and Indices of Refraction:           1
                                                                           λ1    n2
 1 = wavelength in media one; 2 = wavelength in media two; n 1 = index of refraction in media
                          one; n 2 = index of refraction in media two

144. Wave Optics:
                                                                                     Δs
       a. Relationship Between Phase Difference and Path Difference:             2  π 
                                                                                     λ
                 = phase difference (in radians); s = path difference;  = wavelength

                                                          nλ
       b. Angular Posi tions of Bright Fringes:         sin θ 
                                                            a
        = angle between maximum intensity (n = 0) and position of bright fringe; a = distance
             between slits; n = fringe order (n = 0, 1, 2, 3, 4, …);  = wavelength

                                             d  sin θ
       c. Diffraction and Interference:         λ
                                                 n
               = wavelength; d = grating constant;  = angle; n = order of image

                                               1
       d. Grating Constant:       d
                                       number of lines/ cm

                                                        di
       e. Distance to a First Order Image:           tan θ 
                                                       ds
              di = distance to first order image; ds = distance to screen;  = angle

       f. Angular Positions of Minimum Intensities (Dark Fringes):      n  λ  w  sin θ
                n = fringe order (n = 1, 2, 3, …);  = wavelength; w = slit width

                                          2  Δx
       g. Michelson Interferometer:          λ
                                            n
             n = number of fringes the observer sees; x = distance mirror moves
            Note: 2x represents the path difference between the two light beams

                              k  q1  q 2
145. Coulom b’s La w:    F
                                  r2
                                                                                              N  m2
 F = electrostatic force; q1 and q2 = charges; k = electrostatic constant = 8.93 x 10 9                or
                                                                                               C2
                                       N  m2
                    8.98755 x 109               ; r = distance between the charges
                                         C2

146. Charge Density (Uniform Distribution of Charge):
                                       Q
      a. Linear Charge Density:    λ
                                       L
                       = linear charge density; Q = charge; L = length
                                          Q
       b. Surface Charge Density:     σ 
                                          A                     
                         = surface charge density; Q = charge; A = area

                                         Q
       c. Volume Charge Density:       ρ
                                         V
                        = volume charge density; Q = charge; V = volume

147. Charge Density (Nonuniform Distribution of Charge): remember to integrate over an
     interval small enough so that the charge is uniformly distributed.
                                          dq 
       a. Volume Charge Density: ρ 
                                         dV 
         = volume charge density, d(q) = charge differential, d(V) = volume differential


                                            dq 
       b. Surface Charge Density:      σ
                                             
                                            dA
                                              
                                                                    
          = surface charge density; d(q) = charge differential; d( A ) = area differential


                                          dq 
       c. Linear Charge Density:      λ
                                          dl
           = linear charge density; d(q) = charge differential; d(l) = length differential

                        F V
148. Electric Field: E       
                            Q d
       
      E = electric field, F = force, Q = charge, V = potential difference (voltage), d = distance

                                        k Q
149. Electric Field (Point Charge):    E 2
                                          r
                                                             N  m2                  N  m2
 E = electric field, k = electrostatic constant = 8.93 x 10 9        or 8.98755 x 109        ; Q=
                                                               C2                      C2
                                  charge, r = distance from charge

                                                                      dq 
150. Electric Field of a C ontinuous Charge Distribution:     E  k    r2
                                                                             r
                                                                              ˆ

                                                                N  m2                    N  m2
    E = electric field, k = electrostatic constant = 8.93 x 10 9     2
                                                                          or 8.98755 x 109        ;
                                                                  C                         C2
                                                                       ˆ
              d(q) = charge differential, r = distance from charge, r = unit vector r

                                                              σ
151. Electric Field (just outside a charged con ductor):    E
                                                               εo
                                                                                    C2
               E = electric field,  = surface charge density, o = 8.854 x 10-12
                                                                                    N  m2
                                                                          
                                                                          EQ
152. Acceleration of Charged Particles in a Uniform Electric Field:   a
                                                                           m
               a = acceleration, E = electric field, Q = charge, m = mass of particle

                                                                                   
153. Electric Flux: Φ E 
                
                             E  dA where Φ   E E  A (when E is perpendicular to A ) and
                                                                  
          ΦE  E  A  cos θ (when E is not perpendicular to A )
                                                                    
                             E = electric flux, E = electric field, A = area

                                                                                                    Q
154. Electric Flux (any closed surface surrounding a point charge):          ΦE  4  π  k  Q 
                                                                                                    εo
                                                                   N  m2                       N  m2
    E = electric flux, k = electrostatic constant = 8.93 x 10 9            or 8.98755 x 109             ;
                                                                    C2                              C2
                                                                                C2
                     Q = charge enclosed in surface, o = 8.854 x 10-12
                                                                           N  m2
         Note: net electric flux through a closed surface is zero if there is no charge inside


155. Gauss’ Law:            
                                 
                     ΦE  E  d A  q inside
                                          εo
                                                  
           E = electric flux, E = electric field, A = area, q = charge enclosed in surface,
                                                              C2
                                         o = 8.854 x 10-12
                                                         N  m2
          
    Note: E represents the total electric field, which includes contributions from charges both
                           inside and outside the Gaussian surface.

156. Typical Electric Field Calculations Using Gauss’s Law


     Charge Distribution                     Electric Field                          Location

                                                  k Q
Insulating sphere of radius R,                                                        r>R
                                                      r2
 uniform charge density and
        total charge Q                          k Qr
                                                                                      r<R
                                                      R3
                                                  k Q
Thin spherical shell of radius                                                        r>R
                                                      r2
    R an d total charge Q
                                                       N
                                                  0                                   r<R
                                                       C

Line charge of infinite length                  2k  λ
and charge per unit length                       r                            Outside the line

   Noncon ducting, infinite                         σ
charged plane having surface                      2  εo              Everywhere outside the plane
      charge density 
                                                   σ
                                                   εo                          Just outside the conductor
  Conductor having surface
      charge density                                 N                            Inside the conductor
                                                  0
                                                      C

                                                                     B    
157. Change in Electric Potential Energy:     ΔU  U B  U A  q  E  ds 
                                                                           
                                                                     A
                                                              
U = electric potential energy, q = charge, E = electric field, s = distance from point A to point B


158. Potential difference (Voltage):
                                                             ΔU W
       a. Electric Pote ntial at Any Point in a Field:     V    
                                                              Q    Q
V = potential difference (voltage), U = electric potential energy, Q = charge, W = work to move
                                             charge Q

                                                                                    k Q
       b. Potential Difference a Distance r from a Point Charge:               V
                                                                                      r
                                                                                            N  m2
       V = potential difference (voltage), k = electrostatic constant = 8.93 x 10 9                  or
                                                                                             C2
                                    N  m2
                  8.98755 x 109              ; Q = charge, r = distance from charge
                                       C2

                                                                             ΔU      B     
       c. Potential Difference Between Any Two Points:       ΔV  VB  VA         E  ds 
                                                                                             
                                                                             Q       A
                                                                           
      V = potential difference, U = electric potential energy, Q = charge, E = electric field,
                                 
                                 s = distance from point A to point B
       Note: negative sign indicates that point B is at a lower electric potential than point A

       d. Potential Difference in a Uniform Electric Field:
                                                 B                  B       
                               ΔV  VB  VA   E  ds   E 
                                                                      ds
                                                      A               A
                                                                           
      V = potential difference, U = electric potential energy, Q = charge, E = electric field,
                                 
                                 s = distance from point A to point B
       Note: negative sign indicates that point B is at a lower electric potential than point A

       e. Potential Difference Between Two Points a Distance r from Charge Q:
                                                          1    1
                                        VB  VA  k  Q        
                                                           rB rA 
                                                                                            N  m2
       V = potential difference (voltage), k = electrostatic constant = 8.93 x 109                   or
                                                                                             C2
                         N  m2
         8.98755 x 109            ; Q = charge, r = distance from charge to indicated point
                          C2

159. Work in Moving a Charge Across a Potential Difference: W  V  Q
                  W = work, V = potential difference (voltage), Q = charge
                          Q
160. Capacitance:      C
                          V
                  C = capacitance, Q = charge, V = potential difference (voltage)

                                  C    V
161. Dielectric Constant:      K     o
                                 Co    V
        K = dielectric constant, C = new capacitance, C o = old ca pacitance, V = new voltage
                                          Vo = old voltage

162. Permittivity of a New Dielectric:        ε  K  εo
                                                                                                        C2
                K = dielectric constant,  = new permittivity, o = 8.854 x 10-12
                                                                                                      N  m2

                                  V
163. Dielectric Strength:     Ed 
                                   d
             
             E d = electric field for dielectic, V = voltage, d = distance between the plates
                                                                
                                                       K  εo  A
164. Capacitance of a Parallel Plate Capacitor: C 
                                                           d
                                                                                      
     C = capacitance, K = dielectric constant, d = distance between capacitor plates, A = area
                                                                          C2
                                          o = 8.854 x 10-12
                                                                         N  m2

                                                                                                    ab
165. Capacitance of an Isolated S pherical Conductor:                     C  4  π  εo  R 
                                                                                                 k  b  a 
                                                   C2                                                                N  m2
    C = capacitance, , o = 8.854 x 10-12                      , k = electrostatic constant = 8.93 x 10 9
                                                  N  m2                                                              C2
                                    N  m2
               or 8.98755 x 109              ; R = ra dius, a = inner radius, b = outer radius
                                     C2

                                                                          L
166. Capacitance of a Cylindrical Capacitor:                   C
                                                                              b
                                                                    2  k  ln 
                                                                              a
                                                                                                       N  m2
           C = capacitance, L = length, k = electrostatic constant = 8.93 x 10 9                                or
                                                                                                          C2
                                          N  m2
                       8.98755 x 109                ; a = inner radius, b = outer radius
                                             C2

                                1    1   1   1
167. Capacitors in Series:                   
                                CT   C1 C 2 C 3

                         Recommended:                      
                                                C T  C1   C 2   C 3   
                                                                    1        1       1
                                                                                                 
                                                                                                 1

     CT = total ca pacitance; C1 , C2 , C 3 = individual ca pacitances; -1 = inverse key on calculator

168. Capacitors in Parallel:    C T  C1  C2  C3  
                   C T = total ca pacitance; C1 , C 2 , C 3 = individual ca pacitances
169. Work Done in Charging a Capacitor or Energy Stored in a Capacitor:
                                                                    Q2
                                   W or U  0.5  C  V 2  0.5  Q  V 
                                                                   2C
                    W or U = work (energy), C = ca pacitance, V = voltage, Q = charge
                                                                               
170. Energy Stored in a Parallel Plate Capacitor:   U  0.5  ε o  A  d  E 2
                                                                               
         U = energy, E = electric field, d = distance between capacitor plates, A = area
                                                                 C2
                                          o = 8.854 x 10-12
                                                               N  m2
                                                                        
171. Energy Density (Energy per Unit Volume):          uE  0.5  ε o  E 2
                                                                                       C2
               uE = energy density, E = electric field, A = area, o = 8.854 x 10-12
                                                                                        N  m2
                                                              
                                                          Eo
172. Electric Field in the Presence of a Dielectric: E 
                                                            K
    E = electric field with dielectric, E = electric field without dielectric, K = dielectric constant

                                                                K  1
173. Induced Charge Density on a Dielectric:       σ induced        σ
                                                                K 
 induced   = induced surface charge density,  = surface charge density, K = dielectric constant

174. Change in Charge if a Dielectric is Added While Voltage is Maintained by a Battery:
                                            Qnew  K  Q
               Qnew = new charge, K = dielectric constant, Q = original charge

175. Electric Current:
                                              Q
       a. Average Electric Current:         I
                                              t
                                    I = current, Q = charge, t = time

                                                     dQ
       b. Instantaneous Electric Current:           i
                                                     dt 
             I = instantane ous current, d(Q) = current differential, d(t) = time differential
                                                                        
      c. Charge Within a Volume of a Conductor:     ΔQ  n  A  Δx  q
                                             
 Q = charge, n = number of charge carriers, A = cross-sectional area, x = change in length
                                q = charge on each carrier
                                                         
                                  J         I       q τE
       d. Drift Velocity: v D                 
                                 q n q nA          me
                                                  
q = 1.602 x 10-19 C, n = number of atoms/m 3 , A = cross-sectional area,  = mean time between
                               
                   collisions, E = electric field strength, m e = 9.11 x 10-31 kg
                                                                  
                                                           ΔQ n  A  Δx  q
            e. Average Current in a Conductor:     I avg     
                                                           Δt       Δt
                                                                            
I avg   = average current, Q = charge, n = number of charge carriers, A = cross-sectional area, x
                   = change in length, q = charge on each carrier, t = change in time

                                  Δx                               
                                      v D , therefore, I avg  n  A  q  v D
                                   Δt
                                                                
 I avg   = average current, n = number of charge carriers, A = cross-sectional area, q = charge on
                                    each carrier, v D = drift velocity

                                        I
           f. Current Density:      J    n  q  vD
                                        A                                    
        J = current density, I = current, n = number of charge carriers, A = cross-sectional area
                               q = charge on each carrier, v D = drift velocity
                              
176. Ohm’s Law:         J  σE                                 
                        J = current density,  = con ductivity, E = electric field

                                                                              V
                             Most common expression for Ohm’s Law:                   I
                                                                              R
                      I = current, V = potential difference (voltage), R = resistance

                                                   ΔVl
177. Resistance of a Conductor:                 
                                               R
                                           σA  I
 R = resistance,  = con ductivity, l = length, A = cross-sectional area, V = potential difference
                                         (voltage), I = current

                             1
178. Resistivity:       R                      = resistivity,  = con ductivity
                             σ

179. Variation of Resistivity with Temperature:                                             
                                                                 ρ  ρo  1  α  T  To   ρo  ρo  α  T  To 
    = resistivity (new),  o = resistivity (old),  = temperature coefficient, T = temperature (new)
                                           T o = temperature (old)

                                                         ρl
180. Resistance of a Uniform Conductor:            R 
                                                          A     
                   R = resistance,  = resistivity, l = length, A = cross-sectional area

181. Variation of Resistance with Temperature:
                                                            
                                 R  R o  1  α  T  To   R o  R o  α  T  To 
  R = resistance (new), Ro = resistance (old),  = temperature coefficient, T = temperature (new)
                                      T o = temperature (old)

                                    R1     R         Rx    l
182. Wheatstone Bridge:                   3 or           3
                                    R2 R4            R2    l4
              R1 , R2 , R3 , R4 = ratio of resistances on both sides of the Wheatstone Bridge
            Rx = unknown resistance, R2 = known resistance, l 3 and l4 = equivalent lengths
                                          V2
183. Electric Power:      P  I  V  I2  R 
                                           R
             P = power, I = current, V = potential difference (voltage), R = resistance

                                                   V2
184. Power Delivered to a Resistor:          P  I2  R 
                                                    R
             P = power, I = current, V = potential difference (voltage), R = resistance

185. Terminal Voltage Across a Battery with Inte rnal Resistance:
                                                    RL                 r 
                            V  E  I  r   E                          
                                                    R  r   E  1  R  r 
                                                    L                 L    
     V = potential difference (voltage), E = electromotive force (emf), I = current, r = internal
                               resistance, RL = resistance of the load

                                                       E
186. Current Produced by an EMF Source:               I
                                                    RL  r
  E = electromotive force (emf), I = current, r = internal resistance, R L = resistance of the load

187. Power Delivered by an EMF Source:                     
                                            P  I2  R L  I2  r            
          P = power, I = current, r = internal resistance, RL = resistance of the load

188. Parallel Circuits:
        a. Voltage:     VT  V1  V2 V 3  
                         VT = total voltage; V1 , V2 , V3 = individual voltages

        b. Current divided among branches depending upon resistance:                        I T  I1  I 2  I 3  
                        I T = total current; I 1 , I 2 , I3 = individual currents

                            1    1   1   1
        c. Resistance:                   
                            RT   R1 R 2 R 3

                        Recommended:                   
                                              R T  R1   R 2   R 3   
                                                               1         1       1
                                                                                        
                                                                                        1

     RT = total resistance; R1 , R2 , R3 = individual resistances; -1 = inverse key on calculator

189. Series Circuit:
        a. Voltage:     VT  V1  V2 V 3  
                         VT = total voltage; V1 , V2 , V3 = individual voltages

        b. Current:      I T  I1  I 2  I 3  
                            I T = total current; I 1 , I 2 , I3 = individual currents

        c. Resistance: R T  R1  R 2  R 3  
     RT = total resistance; R1 , R2 , R3 = individual resistances; -1 = inverse key on calculator

190. Kirchhoff’s Junction Rule:     I in   I out an d for a closed loop: E  I  r  0
                   I = current, E = electromotive force (emf), R = resistance
191. RC (resistor – capacitor) Circuit Equations:
                                              Q
        a. Kirchhoff’s Rule (Charging): E   I  R   0
                                              C
     I = current, E = electromotive force (emf), R = resistance, Q = charge, C = capacitance

                                                         E
       b. Ma ximum Current (when t = 0 s):        I mas 
                                                         R
                   I max = current, E = electromotive force (emf), R = resistance

       c. Maximum Charge on Capacitor: Q  C  E
                Q = charge, C = ca pacitance, E = electromotive force (emf)

       d. Charge vs. Time for Capacitor Being Charged: where τ  R  C
                                                t               t             t 
                        q t   C  E  1  e RC   Q  1  e RC   Q  1  e τ 
                                                                                   
                                                                                   
   q(t) = charge at time t, C = capacitance, E = ele ctromotive force (emf), R = resistance, Q =
                  maximum charge, t = time, e = natural log,  = time constant

                                                                      t         t
                                                                   E         E
         e. Current vs. Time for a Charging Capacitor:      It    e RC   e τ
                                                                   R         R
  I(t) = current at time t, E = electromotive force (emf), R = resistance, t = time, e = natural log
                                           = time constant

       f. Energy Output of the Battery as Capacitor is Charged:    E  EMF  Q  C  EMF 2
           E = energy, EMF = electromotive force (emf), Q = charge, C = ca pacitance

                                              q
       g. Kirchhoff’s Rule (Discharging):       I  R   0
                                              C
                     C = capacitance, q = charge, I = current, R = resistance

                                                                                         t           t
       h. Charge vs. Time for Discharging Capacitor (where τ  R  C ):  q t   Q  e RC  Q  e   τ

    q(t) = charge at time t, C = capacitance, R = resistance, Q = maximum charge, t = time
                                 e = natural log,  = time constant

       i. Current vs. Time for Discharging Capacitor:
                                                t 
                                       d Q  e RC           t
                                                   
                               dq 
                                                                            t         t
                                                     Q  e RC
                       It                                     I  e RC  I  e τ
                               dt        dt         R C
I(t) = current at time t, d(q) = charge differential, d(t) = time differential, Q = maximum charge,
           R = resistance, C = capacitance, t = time, e = natural log,  = time constant
Note: negative sign indicates current direction as opposite the current direction when charging

                          I2  R  t
192. Joules Law:     Q
                              J
                                                                               J
                 Q = heat, I = current, R = resistance, t = time, J = 4.18
                                                                             g o C
                                                              gram atomic weight
193. Chemical Equivalent:      chem equivalent 
                                                                 ionic ch arg e

                                        chemical equivalent
194. Electrochemical equivalent:           z
                                              faraday
                  Z = electrochemical equivalent, faraday = 965000 coulom bs

195. Fara day’s Law of Electrolysis:  m  z It
                 M = mass, I = current, t = time, z = electrochemical equivalent

                                                                V
196. Ammeter (Galvanometer and Shunt in Parallel):                     V  I1  R g
                                                                    Rs               V I 2 R s
                                                                 I
       Rs = resistance of shunt, V = voltage, I = current, Rg = resistance of galvanometer

                                                 Im
197. Pointer Deflection of Galvanome ter:               d
                                                  k
 d = pointer deflection, I m = current in galvanometer coil, k = current sensitivity (A/division)

                                                             Im
198. Voltage Sensitivity of Galvanometer:                         Rm
                                                        VS  k  R m 
                                                             div
    VS = voltage sensitivity, k = current sensitivity (A/division), I m/div = current per scale
                        division, Rm = resistance in galvanometer meter

                                                                 V
199. Voltmeter (Galvanometer and Resistance in Series):      R     Rg
                                                                  I
   Rg = resistance of galvanometer, R = resistance of series resistor, I = current, V = voltage

                       Im  R m
200. Ammeter:     Rs 
                       I T  Im
      Rs = shunt resistance, I T = total current, I m = meter current, Rm = meter resistance

201. Magnitude of Magnetic Force on Charged Particle Moving in a Magnetic Field:

                              
                                             
                                                        
                                                        
                                 FB  q  v x B  q  v  B  sin θ
                                                                                         
   FB = magnetic force, v x B = cross product of velocity and magnetic field, q = charge, v =
                                                                              
                velocity, B = magnetic field strength,  = angle between v and B
   Note: direction of FB on a moving positive charge can be determined using the right-hand;
 direction of FB on a moving negative charge can be determined using the left hand; maximum
              FB occurs when the charge moves perpendicular to the magnetic field.

202. Force on a Wire Section in a Uniform Magnetic Field:
                                          
                                                        
       a. Straight Wire: FB  I  L x B  I  L  B  sin θ
                                                                                    
  FB = magnetic force, L x B = cross product of length and magnetic field, L = length, B =
                                                                     
                     magnetic field strength,  = angle between L and B


       b. Curved Wire:     FB  I    
                                      a
                                       b
                                        ds x B
                                           

                                        
  FB = magnetic force, I = current, ds x B = cross product of length differential and magnetic
                                   
   field, ds = length differential, B = magnetic field strength, a and b = endpoints of circular
                                  portion of wire in magnetic field
203. Maximum Torque on a Current Carrying Loop in a Uniform Magnetic Field:

                             
                                   
                                   
                                                                    
                          T  I  A x B  I  A  B  sin θ  N  I  A  B  sin θ
                                                                                       
   T = torque, I = current, A x B = cross product of area and magnetic field, A = area, B =
                                                                
      magnetic field strength,  = angle between A and B , N = num ber of turns or loops

                                                                            
       If magnetic moment  is provided, then:        μ  I  A and    T  μ x B  μ  B  sin θ
                                                                                       
  T = torque, I = current, μ x B = cross product of magnetic moment and magnetic field, μ =
                                                                                     
    magnetic moment, A = area, B = magnetic field strength,  = angle between μ and B

204. Motion of a Charged Particle in a Uniform Magnetic Field:
                           
                       mv
      a. Ra dius: r       
                        q B
                                                        
         r = radius, m = mass, q = charge, v = velocity, B = magnetic field strength

                                                               
                                                          v q B
       b. Cyclotron Frequency (angular velocity): ω  
                                                          r    m                 
    = cyclotron frequency (angular velocity), r = radius, m = mass, q = charge, v = velocity
                                   
                                   B = magnetic field strength

                        2π r 2π 2π m
       c. Period:        T                
                           v        ω      q B
                                                                  
                   T = period, r = radius, m = mass, q = charge, v = velocity
              
              B = magnetic field strength,  = cyclotron frequency (angular velocity)
                                                                                     
                                                                               q 2  B2  r 2
205. Kinetic Energy of a Charged Particle Upon Exit from a Cyclotron:      K
                                                                                   2m
                                       
       K = kinetic energy, q = charge, B = magnetic field strength, r = radius, m = mass
                                                                   
                                 μ o I  ds x r 
                                               ˆ          μ o  I ds x r 
                                                                        ˆ
206. Biot-Savart Law:     dB 
                                4π
                                     
                                           r 2
                                                     B
                                                          4π    
                                                                    r 2

                       Tm     
    o = 4  π x 10 7     , ds x B = cross product of length differential and magnetic field
                        A      
       
                                                                 ˆ
     ds = length differential, B = magnetic field strength, r = unit vector (magnitude = 1)
                                        r = distance (radius)

                                                                       μ  I  cos θ1  cos θ 2 
207. Magnetic Field Surroun ding a Thin, Straight Conductor:          B o
                                                                                  4πa
                   Tm                 
  o = 4  π x 10 7    , I = current, B = magnetic field strength, a = perpendicular distance
                     A                                                                         
from wire to point P, cos 1 and cos 2 are the angles the wire makes with the magnetic field B

                                                            μ Iθ
208. Magnetic Field Due to a Curved Wire Segment:          B o
                                                              4πr
                    7   Tm                
    o = 4  π x 10          , I = current, B = magnetic field strength, r = radius,  = angular
                          A
                                      measurement (from s  r  θ )
                                                                                       μ oI  r 2
209. Magnetic Field on the Axis of a Circular Current Loop:                 BX 
                                                                                                    
                                                                                                         3
                                                                                    2  x2  r2          2

                     Tm                  
   o = 4  π x 10 7      , I = current, B X = magnetic field strength directed along the x-axis
                       A
                   r = radius of loop, x = distance from center of loop to point P

                                                                              μ o  I1  I 2  L 2  k  I1  I 2  L
210. Magnetic Force Between Two Parallel Conductors:                   FB                      
                                                                                 2πr                    r
                                                          Tm
         FB = magnetic force, o = 4  π x 10 7              , I = current, L = length of con ductor
                                                           A
                                         N
                        k = 1 x 10-7         , r = distance between parallel conductors
                                        A2

                                                     FB     μ I I
                                                          o 1 2
                                                      L      2πr
                                                   Tm                              N
FB = magnetic force, o = 4  π x 10 7                 , I = current, k = 1 x 10-7    , r = distance between
                                                    A                               A2
                                                   parallel conductors

                                 
211. Am pere’s Law:
                               B  ds  μ o  I
                                                                                Tm
        B = magnetic field strength, ds = length differential, o = 4  π x 10 7     , I = total
                                                                                   A
                                         enclosed current

212. Magnetic Field Created by a Long, Current-Carrying Wire:
                                                μo  I
      a. Beyon d the Ra dius of the Wire: B 
                                                 2πr
                                                          Tm
           B = magnetic field strength, o = 4  π x 10 7       , I = total enclosed current
                                                            A
                                r = distance from the center of the wire

                                                         μ Ir
       b. Within the Radius of the Wire:                B o
                                                          2  π  R2
                                                             Tm
              B = magnetic field strength, o = 4  π x 10 7     , I = total enclosed current
                                                               A
                    r = distance from the center of the wire, R = ra dius of the wire

                                                   N                                   N
213. Magnetic Field Inside a Solenoid:     B  μo   I  μo  n  I          where n 
                                                    L                                   L
                                                  Tm
   B = magnetic field strength, o = 4  π x 10 7      , I = current, L = length, N = number of
                                                    A
                   turns or coils, n = num ber of turns (coils) per unit length
                                                                 dΦ E 
214. Displacement Current (Capacitor):              ID  ε o 
                                                                  dt 
                                                                                
                                                                                    
                                                                                      
                                                                           ΦE  E  d A

                                                   C2                          
    I D = displacement current, o = 8.854 x 10-12      ,  E = electric flux, E = electric field
                                                      2
                                                Nm
                                               A = area

                                                                                      dΦ E  
215. Am pere -Maxwell Law:      B  ds   μ o  I  I D   μ o  I   μ o  ε o 
                                                                                               
                                                                                         dt  
                                                    Tm
 B = magnetic field strength, o = 4  π x 10 7             , I D = displacement current, I = conduction
                                                        A
                                 C2                                                                
   current, o = 8.854 x 10-12          ,  E = electric flux, d(t) = time differential, ds = length
                                     2
                               Nm
                                              differential

216. Magnetic Flux:
                              
                                     
                                            
                       ΦB  B x A  B  A  cos θ
                                                                       
   B = magnetic flux, B x A = cross product of magnetic field and area, B = magnetic field
                                                                    
                       strength, A = area,  = angle between B and A


217. Fara day’s Law of In duction (Induced EMF):    E
                                                          dΦ B 
                                                                    N 
                                                                          dΦ B 
                                                                                   N 
                                                                                            
                                                                                            
                                                                                         d B  A  cos θ
                                                           dt            dt                dt 
                                                                                       
 E = electromotive force (emf), N = number of turns or coils, B = magnetic flux, B = magnetic
                                                                           
                     field strength, A = area,  = angle between B and A
                                                            dΦ B 
           Note: negative sign indicates that the EMF an d             oppose each other.
                                                              dt 
                                  
218. Motional EMF:    E  B  L  v
                                     
      E = electromotive force (emf), B = magnetic field strength, L = length, v = velocity

219. Generators and Motors:
        a. Magnetic Flux:
                                   
                                          
                                                        
                           ΦB  B x A  B  A  cos θ  B  A  cos ω  t 
                               where θ  ω  t an d ω  2  π  f
                                                                           
   B = magnetic flux, B x A = cross product of magnetic field and area, B = magnetic field
                                              
 strength, A = area,  = angle between B and A ,  = angular velocity, t = time, f = frequency

                           dΦ B           dcos w  t         
        b. EMF:     E  N         N  B  A               N  B  A  ω  sin ω  t 
                            dt                    dt 
                                                                                            
    B = magnetic flux, N = num ber of turns or coils, B = magnetic field strength, A = area
                                                 
                   = angle between B and A ,  = angular velocity, t = time
                                             
        c. Maximum EMF:         E max  N  B  A  ω
                                                                                          
 E = electromotive force (emf), N = number of turns or coils, B = magnetic field strength, A =
                                    area,  = angular velocity

220. Motor Effect:   T  FB  w  cos α
  T = torque, FB = magnetic force acting on conductor loo p, w = width of con ducting loop,  =
                      angle between conducting loop an d magnetic flux
221. Maxwell’s Equations:

                             E  dA   ε  4  π  k  Q
                                          Q enclosed
       a. Gauss’ Law:                                                     enclosed
                                                   o

                         dA  = area differential, Q
                          
   E = electric field,                                    = charged enclosed by Gaussian surface,
                                                               enclosed
                                        2
                                    C                                                        N  m2
             o = 8.854 x 10-12                 , k = electrostatic constant = 8.93 x 10 9            or
                                   N  m2                                                     C2
                                                                      N  m2
                                                 8.98755 x 109
                                                                          C2

       b. Gauss’ Law of Magnetism:
                                           
                                       Bd A  0         
                                                   
                    B = magnetic field strength, d( A ) = area differential


       c. Fara day’s Law:     ):    E
                                        dΦ B 
                                                 N 
                                                       dΦ B 
                                                                N 
                                                                         
                                                                      d B  A  cos θ   
                                         dt           dt                dt 
                                                                                    
 E = electromotive force (emf), N = number of turns or coils, B = magnetic flux, B = magnetic
                                                                          
                     field strength, A = area,  = angle between B and A
                                                            dΦ B 
           Note: negative sign indicates that the EMF an d            oppose each other.
                                                              dt 

                                                                                           dΦ E  
       d. Am pere -Maxwell Law:      B  ds   μ o  I  I D   μ o  I   μ o  ε o 
                                                                                                    
                                                                                              dt  
                                                 Tm
 B = magnetic field strength, o = 4  π x 10 7           , I D = displacement current, I = conduction
                                                     A
                                 C2                                                                     
   current, o = 8.854 x 10-12         ,  E = electric flux, d(t) = time differential, ds = length
                                    2
                               Nm
                                             differential

                                        dΦ B         dI
222. Self-Induced EMF:       E L  N           L 
                                         dt          dt 
    EL = EMF induced in the inductor, N = num ber of turns or coils, d(B) = magnetic flux
         differential, d(t) = time differential, L = inductance, d(I) = current differential
                                                               dΦ B 
          Note: negative sign indicates that the EMF an d              oppose each other.
                                                                dt 

                                            ΦB
223. Inductance of an N-turn Coil:               L  N
                                             I
          L = inductance, N = number of turns or coils, B = magnetic flux, I = current

                                                                 t 
                                                                      E          L
224. Current as Function of Time (RL Circuit):             1  e τ  where τ 
                                                                 I
                                                                    R          R
                                                                    
   I = current, E = electromotive force (emf) or voltage, R = resistance,  = time constant, L =
                              inductance, e = natural log, t = time

225. Energy Stored in an Inductor:  U  0.5  L  I 2
                           U = energy, L = inductance, I = current
                                                           
                                                     u     B2
226. Magnetic Energy Density (Solenoid):     uB       
                                                   A  L 2  μo
                                                                       
 UB = magnetic energy density, u = energy, B = magnetic field strength, A = area, L = length
                                                       Tm
                                    o = 4  π x 10 7
                                                        A

                                               Φ1on 2
227. Mutual In ductance:      M1on 2  N 2 
                                            I1
M1 on 2 = mutual inductance of inductor 1 on inductor 2, N = num ber of turns or coils, 1 on 2 =
  magnetic flux (produced by I 1 ) that passes through inductor 2, I 1 = current in inductor 1

                                                            dI 2 
228. Induced EMF (Mutual In ductance):            E1  M2 on1 
                                                             dt 
 E1 = induced emf, M2 on 1 = mutual inductance of inductor 2 on in ductor 1, d(I 2 ) = current 2
                              differential, d(t) = time differential
 Note: the induced EMF in one coil is always proportional to the rate at which the current in
   the second coil is changing; negative sign indicates that the induced EMF is opposite in
                                         direction to E2

229. Inductors in Series (No Mutual Inductance):        L T  L1  L 2  L 3  
                   LT = total inductance, L1 , L2 , L3 = individual inductances

                               1    1   1   1
230. Inductors in Parallel:                 
                               LT   L1 L 2 L 3
                       Recommended:                                         
                                          L T  L 1 1  L 2 1  L 3 1  
                    LT = total inductance, L 1 , L2 , L3 = individual inductances

231. LC Circuit Equations:
                                                                                          2
                                                                     Q2     L  I 2 Q max
       a. Total Energy Stored in an LC Circuit:          U  uC  uL             
                                                                    2C       2      2C
U = total energy, u C = energy in capacitor, u L = energy in inductor, Q = charge, C = ca pacitance,
                                          L = inductance

                                       1
       b. Angular Velocity:     ω
                                    L C
                       = angular velocity, L = inductance, C = capacitance

       c. Charge vs. Time for an Ideal LC Circuit: Q = Qmax + cos[(t) + ]
 Q = charge at time t, Qmax = maximum charge,  = angular velocity, t = time,  = phase angle

                                                        dQ
       d. Current vs. Time for an LC Circuit:           I    = -Qmax  sin[(t) + ]
                                                        dt 
     I = current, d(Q) = charge differential, d(t) = time differential, Qmax = maximum charge
                           = angular velocity, t = time,  = phase angle

232. Alternating Current (AC) Circuit Equations:
       a. Instantaneous Voltage of Source of EMF:  ΔV  Vmax  sin ω  t
     V = instantaneous voltage, V max = maximum voltage,  = angular velocity, t = time
        b. Instantaneous Voltage Across a Resistor:       ΔVR  I max  R  sin ω  t  Vmax  sin ω  t
       VR = instantaneous voltage across resistor, I max = maximum current, R = resistance
                     Vmax = maximum voltage,  = angular velocity, t = time

        c. Instantaneous Current in Resistor:    i  I max  sin ω  t
        I = instantaneous current, I max = maximum current,  = angular velocity, t = time

                                          I max
        d. RMS Current:         I rms         0.7071  I max
                                           2
                              I rms   = RMS current, I max = maximum current

                                          Vmax
        e. RMS Voltage:        Vrms           0.7071  Vmax
                                           2
                             Vrms     = RMS voltage, Vmax = maximum voltage

        f. Average Power Delivered to Resistor:
                              Pavg = I rms2 R = I rmsVrmscos  = 0.5VmaxI max cos 
Pavg   = average power, I rms = rms current, Vrms = rms voltage, I max = maximum current, Vmax =
                                  maximum voltage,  = phase angle

                                              2π
        g. Angular Velocity:          ω  2π f 
                                                T
                              = angular velocity, f = frequency, T = period

                                                              ω
        h. Frequency of Alternating Current:                   f 
                                                            2π
                                        = angular velocity, f = frequency

        Inductors:
                                                                VL
                 i. Inductive Reactance:               XL  ω  L  2  π  f  L 
                                                                 I
 XL = inductive reactance, L = inductance,  = angular velocity, f = frequency of AC source
                          VL = voltage across the coil, I = current

                                                       Vrms
                 j. RMS Current:          I rms 
                                                        XL
                   I rms = rms current, Vrms          = rms voltage, XL = inductive reactance

              k. Instantaneous Current in Inductor:
                               V                           V
                         i L  max  sin ω  t   dt   max  cos ω  t 
                                                  
                                 L                          ωL
i L = instantaneous current, Vrms = rms voltage,  = angular velocity, L = inductance, t = time

                                                                 Vmax  V
                 l. Maximum Current in Inductor:                       max
                                                                     I max 
                                                                  ωL   XL
  I max = maximum current, Vmax           = maximum voltage,  = angular velocity, L = inductance
                                           XL = inductive reactance
                         m. Potential Difference in Inductor:     VL  I max  X L  sin ω  t
     VL = potential difference (voltage) in inductor, I max = maximum current, XL = inductive
                             reactance,  = angular velocity, t = time

                                              XL
                n. Phase Angle:       = tan1
                                               R
                       = phase angle, XL = inductive reactance, R = resistance


                o. Im pedance:  Z  R2  XL 2
                     Z = impedance, XL = inductive reactance, R = resistance

        Capacitors:
                                                    1        1
                p. Ca pacitive Reactance:        XC   
                                                  ωC 2πf C
  XC = ca pacitive reactance, f = frequency of AC source,  = angular velocity, C = ca pacitance

                                                                             Vrms
                q. RMS Current:       I rms  Vrms  ω  C  Vrms  2  π  f  C 
                                                                              XC
  I rms = rms current, Vrms   = rms voltage, XC = capacitive reactance, f = frequency of AC source
                                  = angular velocity, C = capacitance

               r. Instantaneous Current in Capacitor:
                               dq 
                          iC         ω  C  Vmax  cos ω  t   2  π  f  C  Vmax  cos ω  t 
                               dt 
iC = instantaneous current, Vmax = maximum voltage, XC = capacitive reactance, f = frequency of
                    AC source,  = angular velocity, C = capacitance, t = time

                                                                                        Vmax
                s. Maximum Current in Capacitor:             I max  ω  C  Vmax  2  π  f  C  Vmax 
                                                                                         XC
 I max = maximum current, Vmax = maximum voltage, XC = capacitive reactance, f = frequency of
                      AC source,  = angular velocity, C = capacitance

              t. Potential Difference in Capacitor:    VC  Vmax  sin ω  t  I max  X C  sin ω  t
   VC = potential difference in capacitor, V max = maximum voltage, XC = ca pacitive reactance
                                   = angular velocity, t = time

              u. Charge in Capacitor: q  C  Vmax  sin ω  t
      q = charge, C = ca pacitance, Vmax = maximum voltage,  = angular velocity, t = time

233. RLC Series Circuit Equations:
        a. Instantaneous Potential Difference: v  Vmax  sin ω  t
 v = instantaneous potential difference, Vmax = maximum voltage,  = angular velocity, t = time

       b. Instantaneous Current: i = I maxsin [(t) + ]
       i = instantaneous current, I max = maximum current,  = angular velocity, t = time
                                           = phase angle

        c. Total Reactance:    X  XL  XC
             X = total reactance, XL = inductive reactance, XC = capacitive reactance
      d. Impe dance: Z  R 2  X L  X c 2
      Z = impedance, R = resistance, XL = inductive reactance, XC = capacitive reactance

                                      Vmax
       e. Maximum Current:          I max 
                                        Z
               I max = maximum current, Vmax = maximum voltage, Z = impedance

                                    X  XC 
       f. Phase Angle:    = tan1  L
                                           
                                            
                                      R    
       = phase angle, XL = inductive reactance, XC = capacitive reactance, R = resistance

                                      X  XC                 VR
       g. Power Factor:     = tan1  L
                                              
                                                or cos  = V
                                         R    
   = phase angle, XL = inductive reactance, XC = capacitive reactance, R = resistance, cos  =
                    power factor, V R = resistance voltage, V = circuit voltage

       h. Change in Potential Difference Across the Resistor:
                                  VR  I max  R  sin ω  t  VR  sin ω  t
VR = resistance voltage, I max = maximum current, R = resistance,  = angular velocity, t = time

       i. Change in Potential Difference Across the Inductor:
                                                                π
                                VL  I max  X L  sinω  t     VL  cos ω  t 
                                                                2
 VL = inductor voltage, I max = maximum current, XL = in ductive reactance,  = angular velocity
                                                 t = time

       j. Change in Potential Difference Across the Capacitor:
                                                        π
                        VC  I max  X C  sinω  t     VC  cos ω  t 
                                                        2
VC = ca pacitor voltage, I max = maximum current, XC = capacitive reactance,  = angular velocity
                                            t = time

                                                                           1
       k. Resonant Frequency (Series RLC Circuit):            f res 
                                                             2π L C
       Fres = resonant frequency (sometimes labeled o), L =inductance, C = capacitance

       l. Average Power Delivered to an RLC Circuit:           Pavg  I rms 2  R
                    Pavg = average power, I rms = rms current, R = resistance
                  Note: no power loss occurs in an ideal inductor or ca pacitor

       m. Power in AC Circuit Containing Resistance and Inductance: P = VIcos 
                    P = power, V = voltage, I = current,  = phase angle
                   n. Average Power as a Function of Frequency in an RLC Circuit:
                                                      Vrms 2  R  ω 2
                                 Pavg 
                                            R      
                                                               
                                             ω 2  L2  ω 2  ω o 2 
                                                 2
                                                    
                                                                     2
                                                                       
                                                                       
                                                                         
  Pavg = average power, Vrms   = rms voltage, R = resistance,  = angular velocity, o = resonant
                                     frequency, L = inductance

                                 VS  N     I
234. Transformer Equation:           S  P
                                 VP  NP    IS
    VS = secondary voltage, VP = primary voltage, N S = secondary turns, N P = primary turns
                         I P = primary current, I S = secondary current

                                     PS
235. Transformer Efficiency:        e
                                     PP
  e = efficiency, PS = power dissipated in secondary circuit, P P = power dissipated in primary
                                              circuit

236. Transistor Characteristics:
                                                                        ΔI C
       a. Current Amplification (Gain), Common-Base Circuit:                 =
                                                                        ΔI E
       = current gain, IC = change in collector current, I E = change in emitter current

                                                                            ΔI C
       b. Current Amplification (Gain), Common Emitter Circuit:               β
                                                                            ΔI B
         = current gain, I C = change in collector current, I B = change in base current

237. Relativity:
                                         Δt o
       a. Time Dilation:    Δt 
                                     v2 
                                 1  2 
                                    c 
                                        
 t = time interval measured by Earth boun d observers, to = time interval measured in space
                             v = velocity of craft, c = 3 x 10 8 m/s

                                               v2 
       b. Length Contraction:   L  Lo  1  2 
                                              c 
                                                  
L = length measured by Earth boun d observers, L o = length measured in space, v = velocity of
                                    craft, c = 3 x 108 m/s
Note: length contraction occurs only in that dimension of an object parallel to the direction of
                                            motion.

                                         mo
       c. Mass Increase:       m
                                       v2 
                                   1  2 
                                      c 
                                          
    m = relativistic mass, m o = rest mass, v = velocity of object or particle, c = 3 x 10 8 m/s

       d. Kinetic Energy:                       
                            K  m  c 2  mo  c 2         
          K = kinetic energy, m = relativistic mass, m o = rest mass, c = 3 x 108 m/s
239. Mass-Energy Relationship:   E  Δm  c 2
   E = energy, m = mass converted to energy or energy converted to mass, c = 3 x 10 8 m/s
                                                 
                                         e       E
240. Charge to Mass Ratio of Electron:       
                                         m B2  r
      e                                            
        = charge to mass ratio, E = electric field, B = magnetic field, r = ra dius of circle
      m

                                                             hc
241. Planck’s Equation (Quanta of Energy):        E  hf 
                                                               λ
        E = energy, h = 6.626 x 10-34   Js, f = frequency, c = 3 x 108 m/s,  = wavelength

                                  h
242. Momentum of a Photon:        p
                                  λ
             P = momentum, h = 6.626 x 10-34 Js, c = 3 x 108 m/s,  = wavelength

243. Kinetic Energy Transfe rred to an Electron (Com pton Effect):
                                    K = 0.5mv2 = (hfi) – (hfs)
K = kinetic energy, m = 9.11 x 10 -31 kg, v = velocity of electron, hfi = energy of incident photon,

                               hfs = energy of scattered photon

                                    h
244. de Broglie Wavelength:       λ
                                   mv
        = wavelength, h = 6.626 x 10-34 Js, m = mass of particle, v = velocity of particle

                                                   h
245. Heisenberg Uncertainty Principle:       Δx  Δp 
                                                  4π
          x = uncertainty in measurement of particle’s position, h = 6.626 x 10 -34 Js
               p = uncertainty in measurement of momentum in the x-direction

                       1      1    1 
246. Balmer Series:       R 2  2 
                                      
                       λ     2    m 
     = wavelength, R = Rydberg constant = 1.097 x 10 7 /m, m = any integer greater than 2
                           (represents energy level greater than 2)

                                               1       1    1 
247. Wavelengths in the Hydrogen Spectrum:        R 2  2 
                                               λ      n     m 
 = wavelength, R = Rydberg constant = 1.097 x 10 7 /m, m = any integer greater than n (n and
                                 m represent energy levels)

                                                              Z2 
248. Energy Level (Hydrogen-Like Atom):     E n  13.6 eV   2 
                                                             n 
                                                                 
                       Z = atomic number, n = principal quantum num ber

                                             n2  h2
249. Ra dius of Electron Orbit:    r
                                        4  π2  m  k  Z  e2
 r = radius of orbit, h = 6.626 x 10 -34 Js, n = principal quantum number, Z = atomic number
                              m = 9.11 x 10-31 kg, e = 1.602 x 10-19 C
                                             m  e4
250. Energy of an Electron:      E
                                        8  ε o 2  h2  n2
                                                                                           C2
         E = energy, m = 9.11 x 10-31 kg, e = 1.602 x 10-19 C, o = 8.854 x 10-12
                                                                                         N  m2
                      h = 6.626 x 10-34 Js, n = principal quantum number

                                                                                 1   1 
251. Energy of Photon Emitted by Hydrogen Atom:               h  f  13.6 eV   2  2 
                                                                                n   m 
           hf = energy, h = 6.626 x 10-34       Js, n and m represent energy levels (m > n)

                                                                                     1    1 
252. Energy of K X-Ray (n = 2 to n = 1 Transition):      K α  13.6 eV  Z  12   2  2 
                                                                                            
                                                                                      1  2 
                              K = kinetic energy, Z = atomic number

                                                                                     1    1 
253. Energy of K X-Ray (n = 3 to n = 1 Transition):      K β  13.6 eV  Z  12   2  2 
                                                                                     1   3 
                              K = kinetic energy, Z = atomic number

                                                        1
254. Ra dius of a Nucleus:     r  1.2 x 10 15 m  A 3                     A = mass number


                        number of protons x amu proton
                        number of neutrons x amu neutron

255. Mass Defect:      mass of individual nucleons                         amu = atomic mass unit
                       mass of nucleus
                       mass defect

256. Nuclear Binding Energy:
                                                              931 .5017   1.602189 x 10 13 J
            E = mass converted to energy (amu) x                        x
                                                               1 amu            1 MeV

                                         0.693
257. Disintegration Constant:      λ
                                          T1
                                             2
                              = disintegration constant, T 1 = half-life
                                                                   2


258. Ra dioactive Decay: N  No  e λt
     N = final number of nuclei, N o = initial number of nuclei,  = disintegration constant
                           e = natural log base (≈ 2.718), t = time

259. Activity of Ra dioactive Nuclide:    A  λN
                 A = activity,  = disintegration constant, N = number of nuclei
260. Activity as a Function of Time: A  A o  e λt
              A = activity at time t, A o = initial activity,  = disintegration constant
                               e = natural log base (≈ 2.718), t = time

261. Power of Nuclear Reactor: P  n  Δm  c 2
       P = power, n = number of reactions/second, m = loss of mass in each reaction
                                      c = 3 x 108 m/s

                                          A 
                                        ln o 
                                           A 
                                              
262. Dating by Ra dioisotope:        t
                                            λ
t = age, Ao = initial activity, A = current activity,  = disintegration constant (proba bility per unit
                                    time that the nucleus will decay)

                                                     1     N       
263. Age of Uranium Bearing Sample:            t       ln P b  1
                                                           N       
                                                    λU      U      
   t = age, U = disintegration constant, N Pb      = number of Pb atoms, N U = number of U atoms

                                    T1
                                      2
264. Mean Lifetime:      Tmean 
                                    ln 2
                                T mean = mean lifetime, T 1 = half-life
                                                              2


266. Disintegration Energy:
       a. Alpha Decay: Q = (m p – m d – m )·c2
  Q = energy, m p = mass of parent nuclei, m d = mass of daughter nuclei, m = mass of alpha
                                   particle, c = 3 x 108 m/s

        b. Beta Decay:                    
                           Q  m p  md  c 2
    Q = energy, m p = mass of parent nuclei, m d = mass of daughter nuclei, c = 3 x 10 8 m/s

        c. Positron Decay:                         
                               Q  mp  md  2  mβ  c 2
          Q = energy, m p = mass of parent nuclei, m d = mass of daughter nuclei, m  = mass of
                                 beta particle, c = 3 x 108 m/s

                                    m 
267. Threshold Energy:     E th  1   A   Q
                                           
                                    m B 
                                             
      Eth = threshold energy, m A = mass of incoming particle, m B = mass of target particle
                                     Q = disintegration energy

                                                 k  q1  q 2
268. Coulom b Potential Energy Barrier:         U
                                                      r
                                            N  m2
    U = potential energy, k = 8.9875 x 10 9         , q1 and q2 = nuclear charges, r = distance
                                             C2
                                    between centers of mass
269. Fusion Reactions:      K  kB  T
                                                                      J
 K = kinetic energy, k B = Boltzmann’s constant = 1.380662 x 10 -23     , T = Kelvin temperature
                                                                      K

                                            0.1 J                          J
                                     1 rad        0.01 gray, 1 gray  1    , 1 rem  0.01 Sv
                                              kg                          kg
270. Biological Radiation Doses:
                                            1 Sv 1 rad
                                     RBE        
                                            gray    rem
        Ra d = ra diation absorbed dose, rem = Roentgen equivalent man, 1 rad = 1 rem
                                                                                             C
  RBE = relative biological effectiveness, Roentgen (X-ray and  rays in air) = 2.58 x 10-4
                                                                                            kg
                                     E
       a. Absorbe d Dose:     AD 
                                     m
                            AD = a bsorbed dose, E = energy, m = mass

       b. Equivalent Dose: E D  RBE  A D or         ED Sv  Q  A D gray
 ED = equivalent dose, RBE = relative biological effectiveness, A D = absorbed dose, Q = quality
                                             factor

				
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