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Electric Field
Chapter 29. The Electric Potential
At any time, millions of light
bulbs are transforming
electric energy into light and
thermal energy. Just as
electric fields allowed us to
understand electric forces,
Electric Potential allows us
to understand electric
energy.
Chapter Goal: To calculate
and use the electric
potential and electric
potential energy.
Stop to think 29.1
page 885
Stop to think 29.2
page 889
Stop to think 29.3
page 893
Stop to think 29.4
page 896
Stop to think 29.5
page 898
Electrical Potential Energy
When a test charge is placed in an electric field, it
experiences a force
E
F  qE
q
s
If s is an infinitesimal displacement of test charge, then the work done
by electric force during the motion of the charge is given by
F  s  qE  s
4
Uelec  Uo  Eqs
Uelec  Welec
where s is measured from the
negative plate and U0 is the
potential energy at the negative
plate (s = 0). It will often be
convenient to choose U0 = 0, but the
choice has no physical
consequences because it doesn’t
affect ΔUelec, the change in the
electric potential energy. Only the
change is significant.
Ugrav  Uo  mgy
Electrical Potential Energy
F  qE
α
q
A
ds
s
E
ds
B
B
U  UB  UA  q  E  ds
A
ds is oriented tangent to a path
through space
For all paths:
UB  UA  qsE cos α
The electric force is conservative
7
The potential energy of two point charges
xf
Uf  Ui  Welec    F 1on2 dx
xi
xy
Kq1q 2
1 xf
Kq1q 2 Kq1q 2

dx   Kq1q 2( ) | xi 

2
x
xf
xi
x
xi
The Potential Energy of Point Charges
P886. Consider two point charges, q1 and q2, separated by
a distance r. The electric potential energy is
This is explicitly the energy of the system, not the energy of
just q1 or q2.
Note that the potential energy of two charged particles
approaches zero as r  .
The Potential Energy of a Dipole
P889. The potential energy of an electric dipole p in a
uniform electric field E is
The potential energy is minimum at ø = 0° where the
dipole is aligned with the electric field.
   pE sin 
The Electric Potential
We define the electric potential V (or, for brevity, just the
potential) as
Charge q is used as a probe to determine the electric
potential, but the value of V is independent of q. The
electric potential, like the electric field, is a property of
the source charges.
The unit of electric potential is the joule per coulomb,
which is called the volt V:
Electric Potential in constant electric field
Electric potential is the potential energy per unit charge,
U
V 
q
The potential is independent of the value of q.
The potential has a value at every point in an electric field
Only the difference in potential is the meaningful quantity.
UB UA
sF
VB  VA 


 sE  sE cos α
q
q
q
F  qE
α
q
A
s
B
E
13
The Electric Potential Inside a ParallelPlate Capacitor
The electric potential inside a parallel-plate capacitor is
where s is the distance from the negative electrode.
The electric potential, like the electric field, exists at all
points inside the capacitor.
The electric potential is created by the source charges on
the capacitor plates and exists whether or not charge q is
inside the capacitor.
EXAMPLE 29.7 A proton in a capacitor
QUESTIONS:
EXAMPLE 29.7 A proton in a capacitor
EXAMPLE 29.7 A proton in a capacitor
EXAMPLE 29.7 A proton in a capacitor
Electric Potential for a Point Charge
V  0
Point Charge
V  0
Q
Er  k e 2
r
ds
B
E
r
Q0
B
r
r
r
dr
1
Q
VB  VA   Eds    Er dr  keQ  2  keQ
 ke
r
r
r



19
The Electric Potential of a Point Charge
Let q be the source charge, and let a second charge q', a
distance r away, probe the electric potential of q. The
potential energy of the two point charges is
By definition, the electric potential of charge q is
The potential extends through all of space, showing the
influence of charge q, but it weakens with distance as 1/r.
This expression for V assumes that we have chosen V = 0 to
be at r = .
Electric Potential: Example
Point Charge
V  0
equipotential lines
Q
Vr  ke
r
21
The Electric Potential of a Charged Sphere
In practice, you are more likely to work with a charged
sphere, of radius R and total charge Q, than with a point
charge. Outside a uniformly charged sphere, the electric
potential is identical to that of a point charge Q at the
center. That is,
Or, in a more useful form, the potential outside a sphere
that is charged to potential V0 is
The Electric Potential of Many Charges
The electric potential V at a point in space is the sum of the
potentials due to each charge:
where ri is the distance from charge qi to the point in space
where the potential is being calculated.
In other words, the electric potential, like the electric
field, obeys the principle of superposition.
EXAMPLE 29.10 The potential of two charges
EXAMPLE 29.10 The potential of two
charges
Potential and Potential Energy
• If we know potential then the potential energy of point charge
q is
U  qV
(this is similar to the relation between electric force and electric field)
F  qE
26
Potential Energy: Example
Potential energy of two point charges:
q
Q
U  ke
r
U
qQ
r
U
σQ  0
σQ  0
attraction
repulsion
0
r
0
r
27
Potential Energy: Example
Find potential energy of three point charges:
q2
q1
r12
r13
r23
U12  ke
U  U12  U13  U23
q1q2
r12
U13  ke
q1q3
r13
U23  ke
q2q3
r23
q3
U  U12  U13  U23  ke
qq
qq
q1q2
 ke 1 3  ke 2 3
r12
r13
r23
28
Electric Potential of Continuous Charge Distribution
• Consider a small charge element dq
– Treat it as a point charge
• The potential at some point due to this
charge element is
dq
dV  ke
r
• To find the total potential, you need to
integrate to include the contributions
from all the elements
V  0
dq
V  ke 
r
The potential is a scalar sum.
The electric field is a vector sum.29
The potential of a ring of charge
ri  R  z
2
2
1 dq
1
dq
1
V   Vi  



4 0 ri 4 0 R2  z 2 4 0
i 1
N
Q
R2  z 2
Spherically Symmetric Charge Distribution
V  0
V  0
r a
Er  ke
ds
Q
r
a3
r
Q
Er  k e 2
r
E
C
B
Q0
B

a


r
r
a
VB  VA   Eds   Er dr   Er dr   Er dr 
a

2


Q
dr
Q
Q
Q
r
2
2
 ke 3  rdr keQ  2  ke 3 (a  r )  ke  ke
3  2 
a r
r
2a
a
2a 
a 31
r
Spherically Symmetric Charge Distribution
Vr  ke
Q
r
Q
r2 
VB  ke
3  2 
2a 
a 
r a
r a
32
quiz1
The positive charge is
the end view of a
positively charged glass
rod. A negatively charged
particle moves in a
circular arc around the
glass rod. Is the work
done on the charged
particle by the rod’s
electric field positive,
negative or zero?
A. Positive
B. Negative
C. Zero
C
Quiz 2
A proton is
released from
rest at point B,
where the
potential is 0
V. Afterward,
the proton
A. moves toward A with a steady speed.
B. moves toward A with an increasing speed.
C. moves toward C with a steady speed.
D. moves toward C with an increasing speed.
E. remains at rest at B.
B
Quiz 3
Rank in order,
from largest to
smallest, the
potentials Va to Ve
at the points a to
e.
A.
B.
C.
D.
E.
Vd = Ve > Vc > Va = Vb
Vb = Vc = Ve > Va = Vd
Va = Vb = Vc = Vd = Ve
Va = Vb > Vc > Vd = Ve
Va = Vb = Vd = Ve > Vc
D
Quiz 4
Rank in order, from
largest to smallest,
the potential
differences ∆V12, ∆V13,
and ∆V23 between
points 1 and 2, points
1 and 3, and points 2
and 3.
A.
B.
C.
D.
E.
∆V13 > ∆V12 > ∆V23
∆V13 = ∆V23 > ∆V12
∆V13 > ∆V23 > ∆V12
∆V12 > ∆V13 = ∆V23
∆V23 > ∆V12 > ∆V13
B

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