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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 = mg 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 hg = (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 d2 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 2GM 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 GM 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 GM 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 sin2 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 θ2 e. Maximum Vertical Height (h): h 2g 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 θ2 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 rg = 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 rθ = angular displacement, r = radius 31. Rotary Motion Equations: = o + (t) = angular velocity; o = initial angular velocity; = (ot) + (0.5t2 ) = 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): 2πt a. Displacement: x A sin 2πf t T 1 2π ω x A sinω t x A cosω t f T f T ω 2π 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 sinω t θ x a cosω t θ determines the starting position x k e ct sinω t b. Damped Harmonic Motion: c>0 x k e ct cosω t c = damping constant; k = original displacement of object; amplitude = k e ct F c. S pring Constant (k): k F = distorting force; d = change in length Δd 1 2π m d. Period (T): T T T 2π 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 2π 2π m = angular velocity; m = mass of oscillating object; k = spring constant g. Restoring Force: F k x k = spring constant; x = displacement 2π A h. Maximum Speed (v max) : ωA 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): dv v dt ω A sinω 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 dv k. Acceleration (a): a ω 2 A cosω 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 sinω t θ2 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 cosω t θ2 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 mg d b. Angular Velocity for a Physical Pendulum ( ): ω I d = distance from center of gravity to pivot; I = rotational inertia l 2π c. Period (T): T 2π l = length; = angular velocity g ω 2π I d. Period for a Physical Pendulum (T): T 2π ω mg 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 TI k b. Angular Velocity (): ω θ l T = torque; k = torsional constant; = displacement angle; I = rotational inertia I c. Period (T): T 2π 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 Fv 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 2g h or h 2g 52. Impulse (J) and Momentum: J F t F t mv F = force; t = time; m = mass; v = velocity 53. Momentum (p): p mv 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 Fl 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 Fl 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 rDg 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 rDg h = capillary rise; = coefficient of surface tension; r = radius of ca pillary tube; D = density Fl 71. Viscosity (): η Av 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 Av 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 8lη 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 + (ma) 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 KAt 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 QW b. Work Done On System: U QW 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 λf = 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 Tm 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 2π 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 Nm Tm 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 cosk x ω t E = electric field strength; = angular velocity ( ω 2 π f );k = angular wave number 2π k ; x = position of wave with respect to x-a xis λ d. Sinusoidal Magnetic Field equation: B Bmax cosk x ω t B = magnetic field strength; = angular velocity ( ω 2 π f );k = angular wave number 2π 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 EB 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; Tm 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 Tm 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 Nm B2 h. Instantaneous Energy Density (associated with the magnetic field): uB 2 μo remember that u E = u B Tm 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 Nm Tm 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 Tm 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 2S 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 4l 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 T0 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 vv 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 vv 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 hf E = energy; h = Planck’s constant = 6.63 x 10 -34 Js; f = frequency 123. Work Function: = hf = work function; h = Planck’s constant = 6.63 x 10 -34 Js; 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 Js; 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 Js; f = frequency h 126. deBroglie Wavelength: λ mv - wavelength; h = Planck’s constant = 6.63 x 10 -34 Js; 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 4πI = 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 nλ 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: 2x 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. dq a. Volume Charge Density: ρ dV = volume charge density, d(q) = charge differential, d(V) = volume differential dq b. Surface Charge Density: σ dA = surface charge density; d(q) = charge differential; d( A ) = area differential dq c. Linear Charge Density: λ dl = 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 dq 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 EQ 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 dA 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 Qr 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 2k 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 ds 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 ds 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 ds E ds 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 ab 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 2C 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 dQ b. Instantaneous Electric Current: i dt 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 τE d. Drift Velocity: v D q n q nA 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 σE 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 σA 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) ρl 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 RC Q 1 e RC 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: It e RC 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 RC 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 RC t dq t t Q e RC It I e RC I e τ dt dt 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 It 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: mv 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 2π r 2π 2π 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 2m 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 4π r 2 B 4π r 2 Tm 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 4πa Tm 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 Iθ 208. Magnetic Field Due to a Curved Wire Segment: B o 4πr 7 Tm 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 Tm 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 2πr r Tm 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 2πr Tm 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 Tm 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 2πr Tm B = magnetic field strength, o = 4 π x 10 7 , I = total enclosed current A r = distance from the center of the wire μ Ir b. Within the Radius of the Wire: B o 2 π R2 Tm 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 Tm 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 dΦ E 214. Displacement Current (Capacitor): ID ε o dt ΦE E d A C2 I D = displacement current, o = 8.854 x 10-12 , E = electric flux, E = electric field 2 Nm A = area dΦ E 215. Am pere -Maxwell Law: B ds μ o I I D μ o I μ o ε o dt Tm 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 Nm 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 dΦ B N dΦ B N d B A cos θ dt dt dt E = electromotive force (emf), N = number of turns or coils, B = magnetic flux, B = magnetic field strength, A = area, = angle between B and A dΦ B Note: negative sign indicates that the EMF an d oppose each other. dt 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 dΦ B dcos w t b. EMF: E N N B A N B A ω sin ω t dt dt 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 dA ε 4 π k Q Q enclosed a. Gauss’ Law: enclosed o dA = 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: Bd A 0 B = magnetic field strength, d( A ) = area differential c. Fara day’s Law: ): E dΦ B N dΦ B N d B A cos θ dt dt dt E = electromotive force (emf), N = number of turns or coils, B = magnetic flux, B = magnetic field strength, A = area, = angle between B and A dΦ B Note: negative sign indicates that the EMF an d oppose each other. dt dΦ E d. Am pere -Maxwell Law: B ds μ o I I D μ o I μ o ε o dt Tm 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 Nm differential dΦ B dI 222. Self-Induced EMF: E L N L dt dt 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 dΦ B Note: negative sign indicates that the EMF an d oppose each other. dt 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 Tm 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 dI 2 228. Induced EMF (Mutual In ductance): E1 M2 on1 dt 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 2C 2 2C 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 dQ d. Current vs. Time for an LC Circuit: I = -Qmax sin[(t) + ] dt 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 rmsVrmscos = 0.5VmaxI max cos Pavg = average power, I rms = rms current, Vrms = rms voltage, I max = maximum current, Vmax = maximum voltage, = phase angle 2π g. Angular Velocity: ω 2π f T = angular velocity, f = frequency, T = period ω h. Frequency of Alternating Current: f 2π = 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 dt max cos ω t L ωL i L = instantaneous current, Vrms = rms voltage, = angular velocity, L = inductance, t = time Vmax V l. Maximum Current in Inductor: max I max ωL 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: = tan1 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 ωC 2πf 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: dq iC ω C Vmax cos ω t 2 π f C Vmax cos ω t dt 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 maxsin [(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: = tan1 L R = phase angle, XL = inductive reactance, XC = capacitive reactance, R = resistance X XC VR g. Power Factor: = tan1 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 sinω 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 sinω 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 2π 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 = VIcos 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 hc 241. Planck’s Equation (Quanta of Energy): E hf λ E = energy, h = 6.626 x 10-34 Js, 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 Js, c = 3 x 108 m/s, = wavelength 243. Kinetic Energy Transfe rred to an Electron (Com pton Effect): K = 0.5mv2 = (hfi) – (hfs) K = kinetic energy, m = 9.11 x 10 -31 kg, v = velocity of electron, hfi = energy of incident photon, hfs = energy of scattered photon h 244. de Broglie Wavelength: λ mv = wavelength, h = 6.626 x 10-34 Js, m = mass of particle, v = velocity of particle h 245. Heisenberg Uncertainty Principle: Δx Δp 4π x = uncertainty in measurement of particle’s position, h = 6.626 x 10 -34 Js 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 Js, 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 Js, n = principal quantum number 1 1 251. Energy of Photon Emitted by Hydrogen Atom: h f 13.6 eV 2 2 n m hf = energy, h = 6.626 x 10-34 Js, 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 12 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 12 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 λt 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 λN A = activity, = disintegration constant, N = number of nuclei 260. Activity as a Function of Time: A A o e λt 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