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					Physics 2102




Jonathan Dowling


                    Physics 2102
                     Lecture 6
                   Electric Potential II
                                                qi
              Example                    V  k      –Q
                                             i  ri
                                                     +Q
  Positive and negative charges of equal
  magnitude Q are held in a circle of radius R.
  1. What is the electric potential at the center
  of each circle?
                                                      A
  • VA = k3Q  2Q/r  kQ/r
   • VB = k2Q  4Q/r  2kQ/r
   • VC = k 2Q  2Q  / r  0
 2. Draw an arrow representing the
                                                      B
 approximate direction of the electric field at
   the center of each circle.
   3. Which system has the highest electric
   potential energy?                                      C

         UB
   Electric Potential of a Dipole (on axis)
What is V at a point at an axial distance r away from the
midpoint of a dipole (on side of positive charge)?
            Q             Q
  V k            k
             a            a
       (r  )        (r  )                       a
             2            2
             a          a 
        (r  2 )  (r  2 ) 
                                             -Q +Q
   kQ                                                   r
               a        a 
        (r  )(r  ) 
              2        2               Far away, when r >> a:
           Qa                                         p
                                           V 
                a 2
                                                  4 0 r 2
      4 0 (r  )
              2

                4
   Electric Potential on Perpendicular
            Bisector of Dipole
 You bring a charge of Qo = –3C
 from infinity to a point P on the
 perpendicular bisector of a dipole
 as shown. Is the work that you do:    a

  a) Positive?                        -Q +Q d
                                        P
  b) Negative?
  c) Zero?


U= QoV=Qo(–Q/d+Q/d)=0                           -3C
            Continuous Charge
              Distributions
• Divide the charge
  distribution into
  differential elements                   r
• Write down an expression
  for potential from a typical   dq
  element — treat as point
  charge
                                              dq
• Integrate!
                                       V  k
• Simple example: circular                     r
  rod of radius r, total
                                        k        Q
  charge Q; find V at center.
                                        dq k
                                        r        r
          Potential of Continuous Charge
              Distribution: Example
•   Uniformly charged rod
•   Total charge Q               Q/ L       dq   dx
•   Length L
                                              kdx
                                          L
                                   kdq
•   What is V at position P
                              V      
    shown?                          r   0
                                          ( L  a  x)

     x                 P       k  ln(L  a  x)      L
                                                          0
         dx
         L         a
                                         L  a
                               V  k ln      
                                          a 
                 Electric Field & Potential:
                  A Simple Relationship!
Notice the following:           Focus only on a simple case:
                                electric field that points
• Point charge:
                                along +x axis but whose
   – E = kQ/r2
                                magnitude varies with x.
   – V = kQ/r
• Dipole (far away):                         dV
   – E ~ kp/r3
                                      Ex  
   – V ~ kp/r2
                                             dx
• E is given by a DERIVATIVE
  of V!                                      Note:
                       f
                                 • MINUS sign!
                        
• Of course! V   E  ds
                        i
                                 • Units for E --
                                 VOLTS/METER (V/m)
      Electric Field & Potential: Example
• Hollow metal sphere of
  radius R has a charge +q
• Which of the following is                                      +q
  the electric potential V as
  a function of distance r
  from center of sphere?


                             1                       (b)                  1
     V                                          V                    
             (a)             r                                            r



                     r=R                 r                 r=R                r
                                     1
         V     (c)               
                                     r


                     r=R                     r
    Electric Field & Potential: Example
            Outside the sphere:
            • Replace by point charge!
    +q      Inside the sphere:
            • E =0 (Gauss’ Law)
            • E = –dV/dr = 0 IFF V=constant  dV
                                        E
E                 1                           dr
                 2
                 r                          d  Q
                                          k 
                                           dr  r 
                                           Q
                     1
                                        k 2
V                                         r
                     r
      Equipotentials and Conductors
• Conducting surfaces are
  EQUIPOTENTIALs
• At surface of conductor, E is
  normal to surface
• Hence, no work needed to move a
  charge from one point on a
  conductor surface to another
• Equipotentials are normal to E, so
  they follow the shape of the
  conductor near the surface.
       Conductors change the field
            around them!
An uncharged conductor:


A uniform electric field:




An uncharged conductor in the
initially uniform electric field:
                         “Sharp”conductors
• Charge density is higher at
  conductor surfaces that have
  small radius of curvature
• E = s/0 for a conductor, hence
  STRONGER electric fields at
  sharply curved surfaces!
• Used for attracting or getting rid
  of charge:
   – lightning rods
   – Van de Graaf -- metal brush
     transfers charge from rubber belt
   – Mars pathfinder mission --
     tungsten points used to get rid of
     accumulated charge on rover
     (electric breakdown on Mars          (NASA)
     occurs at ~100 V/m)
                          Summary:
• Electric potential: work needed to bring +1C from infinity; units = V
• Electric potential uniquely defined for every point in space --
  independent of path!
• Electric potential is a scalar -- add contributions from individual point
  charges
• We calculated the electric potential produced by a single charge:
  V=kq/r, and by continuous charge distributions : V= kdq/r
• Electric field and electric potential: E= dV/dx
• Electric potential energy: work used to build the system, charge by
  charge. Use W=qV for each charge.
• Conductors: the charges move to make their surface equipotentials.
• Charge density and electric field are higher on sharp points of
  conductors.

				
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posted:10/1/2012
language:English
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