Download Electric Potential

Chapter 23 - Electric Potential
I.
Introduction
A.
Notice the similarity between the gravitational force and the electrical force:
FG
qq
m1m2
and F  k 1 2 2
2
r
r
We know that the gravitational force is conservative, and because the electrical and gravitational
forces have the same form, the electrical force must also be conservative. Therefore, the electrical
force gives rise to a potential energy, U, which we call the electrical potential energy. This is the
electrical equivalent of the gravitational potential energy
B.
Brief review of potential energy.
Remember that the change in the potential energy equals the negative of the work done by the
conservative force. That is,

B
U  UB  U A  WAB   F  d l
A
or,
WAB  U  U A  UB 


F  dl
B
A
.
In words, the work done by the conservative force moving the object from A to B along an arbitrary
path equals the object’s potential energy at A minus its potential energy at B.
So how do we go about finding the values of the electrical potential energy?
II.
Electric Potential Energy, U
A.
Find an expression that allows us to determine the electric potential energy, U.
Consider a charge q being moved along an arbitrary path from point A to point B in the electric field


shown. The electrical force on the charge is F  qE .
B

E

dl


F  qE
A
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The work done by the electrical force is then written as
W AB  U  U A  U B 
B.
B


 qE  dl  q
A
B
A
 
E  dl ,
Examples using Electric Potential Energy
1.

Suppose a uniform electric field E points directly downward. (How
can you produce a uniform electric field pointing downward?)
a.
B

E
What is the difference in the electric potential energies (UA - UB)
of a charge +q between the positions A and B in the field?
dl
q
yB
yA
A
y=0
b.
Write the electric potential energy function.
c.
At what point, A or B will a positive charge, +q, have a greater electrical potential energy?
At what point, A or B will a negative charge, -q, have a greater electrical potential energy?
d.
An energy equation example: If a charge -q is released from rest at point A, then how fast
will it be traveling when it passes point B -- a distance of (yB – yA)?
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2.
Suppose a point charge +Q creates an electric field in space.
a.
Find the difference in the electric potential energy (UA - UB) of a charge +q between the
positions A and B that are a distances rA and rB from charge the charge +Q.
A
rA
r
rB
B
+Q
b.
Write the equation for the electric potential energy of a charge +q when placed in an electric
field produced by a charge +Q.
c.
At what point, A or B will a positive charge, +q, have a greater electrical potential energy?
At what point, A or B will a negative charge, -q, have a greater electrical potential energy?
d.
Now replace the charge +Q with the charge –Q. Write the equation for the electric
potential energy of a charge +q when placed in an electric field produced by a charge -Q.
e.
At what point, A or B will a positive charge, +q, have a greater electrical potential energy?
At what point, A or B will a negative charge, -q, have a greater electrical potential energy?
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f.
3.
Another energy equation example: Refer to the diagram in part a above. If the charge +q is
released from rest at point A, then how fast is it traveling when it reaches infinity?
How much work must you do in order to move a charge +q from infinity to a distance r from a
charge +Q?
q
Q
r
What if the charge you move is –q?
III. Electric Potential and Electric Potential Difference – very useful concepts in electricity because they
are easily measured.
A.
The electric potential, V.
Definition: the electric potential is the electrical potential energy per unit charge, or
V
Joules
U
, where the units are
 volts.
q
Coulomb
Note: the electric potential is also described as the voltage, and is very closely related to electric
potential energy.
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B.
The electric potential difference between points A and B, VA - VB , is
V  VA  VB 
B
U A  U B W A B 1  


qE  d l ,
q
q
q

A
or,
B
 
V  VA  VB  E  d l

.
A
Note also that
WAB  qV  qVA  VB  .
This is the work done by the electric field moving a charge q from point A to point B. In many cases,
this expression is much simpler to use when calculating work done.
C.
The electric potential for a point charge. Find the electric potential difference (or the voltage)
between two points, A and B, in the electric field produced by a point charge +Q.
A
rA
r
+Q
rB
B
1.
At what position (A or B) is the electric potential (voltage) higher?
2.
If the charge +Q is replaced by the charge –Q, what position (A or B) is at a higher potential?
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D.
3.
Usually a position is chosen where the potential (voltage) is defined to be zero (V =0). (This is
very much like choosing the position where the potential energy is zero.) In some cases, this
position is taken to be an infinite (  ) distance away from the charge. If this is the case, then
what is the electric potential at a distance r from the charge Q?
4.
Suppose you chose the electric potential to be zero at a distance R from the charge. What is the
electric potential at a distance r from the charge?
5.
What is the electric potential of a negative point charge (-Q) at a distance r from it, and where
the electric potential is chosen to be zero at infinity?
The electric potential for multiple point charges.
For charges q1, q2, … at distances r1, r2, … from point P, the total electric potential is
V  V1  V2  ....  k
q1
q
 k 2  ....
r1
r2
,
or is just the scalar sum of the electric potentials due to each point charge.
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1.
An electric dipole lies on the y-axis. Find the electric potential at point P that has coordinates (x,
y).
P
y
(x, y)
+q
l/2
x
l/2
-q
2.
Two charges, q1 = +2 C and q2 = -5 C, are placed 4 cm apart on the x-axis. Find the position(s)
on the x-axis where the electric potential is zero. (Are there points off the x-axis where the
electric potential is zero?)
q2
q1
x
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3.
How much work must you do in order to move three charges from infinity to the corners of an
equilateral triangle with sides of length a?
q1
a
a
q2
E.
q3
a
Electric potential calculations for continuous charge distributions
V
P
ri
qi
dV =
dq
1 dq
k
4 o r
r
, where dq = dl = dA = d(Vol)
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1.
A charge, Q, is uniformly distributed on a rod of length L. Find the electric potential at a
distance x on the perpendicular bisector of the rod.
y
L/2
x
L/2
2.
Charge Q is uniformly distributed around a ring of radius R. Find the electric potential a
distance x from the center of the ring on the line passing through the center of the ring and
perpendicular to the plane of the ring.
y
x
R
z
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3.
Find the potential on a line passing through the center of a disk perpendicular to the disk where
the disk has a uniform charge density  The radius of the disk is R.
y
x
z
IV. Knowing what the electric field looks like for various charge distributions, find the potential
difference between two points in the electric field.
B
 
VA  VB  E  d l

A
1.
For two large, oppositely charged parallel plates separated by a distance d, what is the potential
difference between the plates? What do the electric field lines of force look like? What should
the equipotentials look like? Remember that the electric field is uniform between the plates, i.e.,

E is constant.
B

E
d
A
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2.
Find the electric potential both outside and inside a solid conducting sphere of radius R with a
charge Q. Choose V = 0 at infinity.
R
3.
Find the electric potential both outside and inside a solid sphere of radius R with a charge
density  = constant. Choose V = 0 at infinity.
R

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4.
A long cylindrical, metal rod of radius R has a uniform linear charge density . If the potential
is chosen to be zero on the surface of the rod, find the potential both inside and outside the rod.
R
V.
If you know the electric potential can you find the electric field?
A.
Remember that the potential difference between two points is related to the electric field as:
B
 
VA  VB  E  d l

A
.
When rearranged:
B
B
 
VB  VA   E  d l   Es ds

A

A
.
Take the differential of this expression and rearrange.
Es  
B.
dV
ds .
Examples of the gradient calculations:
1.
Point charge,
V
kq
r
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V
2.
Ring of charge,
kQ
x2  R2
VI. Charge sharing
Suppose conductors are brought into contact or are connected together by a wire. In the electrostatic
situation, the E field in the conducting material will be zero, and therefore, all points in and on the
conductors will be at the same potential (Do you remember why?). This idea allows us to find how
the charge density, , varies as a function of the curvature of the conducting surface.
Example: Two solid conducting spheres of radii R1 = 10 cm and R2 = 20 cm have charges of Q1 = +2
C and Q2 = –5 C, respectively. The spheres are brought into contact, and then separated again.
How much charge is on each sphere, what is the surface charge density on each sphere, and what is
the potential of each sphere?
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