# Physics 212

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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
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
• 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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