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Chapter 21
Electric Potential
Reading: Chapter 21
1
Electric Potential Energy
When a test charge is placed in an electric field,
it experiences a force
E
F  qE
sF
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
W  sF F
2
Electric Potential Energy
W  sF F  q sF E
E
F  qE
q
This is the work done by electric field.
s
In this case work is positive.
g
F  mg
s
This is very similar to
gravitational force: the
work done by force is
Because the positive work is done, the potential
energy of charge-field system should decrease.
So the change of potential energy is
U  sF F  q sF E
sF F  mgsF
The change of potential energy is
U  sF F  mgsF
minus sign
3
Electrical Potential Energy
A
F  qE
q
s
E
B
U  UB  UA  Work AB
Work is the same
for all paths
For all paths:
UB  UA  Work AB
The electric force is conservative
4
Electric Potential
Electric potential is the potential energy per unit charge,
V
U
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.
A
UB U A
Work AB
VB  VA 


q
q
q
sF
F  qE
q
s
E
B
5
Electric Potential
To find the potential at every point
 1. we assume that the potential is equal to 0 at some
point, for example at point A, VA  0
 2. we find the potential at any point B from the expression
VB  VA 
Work AB
Work AB

q
q
VB  sF E
A
F  qE
sF
q
s
E
B
6
Electric Potential: Example
Plane: Uniform electric field
σ 0
VA  0
σ
E
2ε
0
B
E
h
s
σh
VB  VA  hE  hE  
2ε
0
A
σ>0
E
s
h
C
σh
VC  VA  hE  hE  
2ε
0
7
Electric Potential: Example
V
σ 0
0
h
V
σ 0
Plane: Uniform electric field
B
E
h
σh
VB  
2ε
0
h
0
A
σ>0
E
h
C
σh
VC  
2ε
0
All points with the same h
have the same potential
8
Electric Potential: Example
Plane: Uniform electric field
The same potential
σh
V 
2ε
0
equipotential lines
σh
V 
2ε
0
9
Electric Potential: Example
Point Charge
V  0
V  0
|Q |
Er  ke 2
r
s
B
E
r
Q0
Q
VB  ke
r
10
Electric Potential: Example
Point Charge
V  0
equipotential lines
Q
Vr  ke
r
11
Units
• Units of potential: 1 V = 1 J/C
– V is a volt
– It takes one joule (J) of work to move a 1-coulomb (C)
charge through a potential difference of 1 volt (V)
 Another unit of energy that is commonly used in atomic and nuclear
physics is the electron-volt
 One electron-volt is defined as the energy a charge-field system
gains or loses when a charge of magnitude e (an electron or a proton) is
moved through a potential difference of 1 volt
1 eV = 1.60 x 10-19 J
12
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
13
Potential Energy: Example
What is the potential energy of point charge q in the field of uniformly
charged plane?
U  QV
Q
σh
V 
2ε
0
h
σ
σQ>0
U
0
σQ
U  qV  
h
2ε
0
h
σQ>0
U
0
repulsion
h
attraction
14
Potential Energy: Example
What is the potential energy of two point charges q and Q?
This can be calculated by two methods:
Uq  qVQ
A
q
Q
r
Q
VQ  ke
r
UQ  QVq
B
q
Q
r
q
Vq  ke
r
The potential energy of
point charge q in the
field of point charge Q
qQ
Uq  qVQ  ke
r
The potential energy of
point charge Q in the
field of point charge q
qQ
UQ  QVq  ke
r
In both cases we have the same expression for the energy. This
expression gives us the energy of two point charges.
qQ
U  ke
r
15
Potential Energy: Example
Potential energy of two point charges:
q
Q
U  ke
r
U
qQ
r
U
σQ>0
repulsion
σQ<0
attraction
0
r
0
r
16
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
17
Potential Energy: Applications: Energy Conservation
For a closed system: Energy Conservation:
The sum of potential energy and kinetic energy is constant
q1
q2
r12,initial
r23,initial
r13
U  U12  U13  U23
- Potential energy
mv 2
T
2
- Kinetic energy
r12,final
q3
r23,final
q2
Example: Particle 2 is released from
the rest. Find the speed of the
particle when it will reach point P.
v2
Initial Energy is the sum of kinetic energy and potential energy (velocity
is zero – kinetic energy is zero)
Einitial  T  U  T  U12  U13  U23  ke
q1q2
r12,initial
q1q3
q2q3
 ke
 ke
18
r13
r23,initial
Potential Energy: Applications: Energy Conservation
For a closed system: Energy Conservation:
The sum of potential energy and kinetic energy is constant
q1
q2
r12,initial
r23,initial
r13
r12,final
q3
r23,final
q2
v2
Final Energy is the sum of kinetic energy and potential energy (velocity
of particle 2 is nonzero – kinetic energy)
Efinal  T  U  T  U12  U13  U23
qq
qq
m2v 22
qq

 ke 1 2  ke 1 3  ke 2 3
2
r12,final
r13
r23,19final
Potential Energy: Applications: Energy Conservation
For a closed system: Energy Conservation:
q1
The sum of potential energy and kinetic
energy is constant
q2
r12,initial
r23,initial
r13
r12,final
q2
Final Energy = Initial Energy
q3
ke
q1q2
r12,initial
r23,final
v2
q1q3
q2q3
q1q3
q 2 q3
m2v 22
q1q2
 ke
 ke

 ke
 ke
 ke
r13
r23,initial
2
r12,final
r13
r23,final
q2q3
q2q3
m2v 22
q1q2
q1q2
 ke
 ke
 ke
 ke
2
r12,initial
r12,final
r23,initial
r23,final
v2 
2
m2

 1
1
k
q
q

 e 1 2 

 r12,initial r12,final

 1
1

k
q
q

 e 2 3 

 r23,initial r23,final

 
 
20
Electric Potential of Multiple Point Charge
Q
Vr  ke
r
P
q4
q1
q2
q3
q3
q1
q2
q4
Vr  V1  V2  V3  V4  ke  ke
 ke
 ke
r1
r2
r3
r4
The potential is a scalar sum.
The electric field is a vector sum.
21
Spherically Symmetric Charge Distribution
Uniformly distributed charge Q
Q
V ?
22
Spherically Symmetric Charge Distribution
V  0
V  0
r a
ds
Q0
B
E
Q
Er  ke 2
r
r
Q
VB  ke
r
23
Important Example: Capacitor
    
     
    
     
E 0
σ
E
ε0
E 0
σ>0
E
σ>0
σ<0
V ?
σ<0
24
Important Example
V 0
E 0
x
σ
E
E
ε0
σ 0
x
x
σ 0
E 0
σ
V  x
ε
0
σ
V  Eh   h
ε
0
V
h
x
σ
 h
ε0
25
Electric Potential: Charged Conductor
 The potential difference between A and B is
zero!!!!
 Therefore V is constant everywhere on the
surface of a charged conductor in equilibrium
– ΔV = 0 between any two points on the
surface
 The surface of any charged conductor is an
equipotential surface
 Because the electric field is zero inside the
conductor, the electric potential is constant
everywhere inside the conductor and
equal to the value at the surface
26
Electric Potential: Conducting Sphere: Example
Vr  ke
Q
r
Vsphere  ke
for r > R
Q
R
for r < R
The potential of
conducting sphere!!
27
Conducting Sphere: Example
What is the potential of conducting
sphere with radius 0.1 m and
charge 1μC ?
Vsphere
6
Q
10
C
 ke  9  109
 90000V
R
0.1m
28
Chapter 21
Capacitance
29
Capacitors
 Capacitors are devices that store electric charge
• A capacitor consists of two conductors
– These conductors are called plates
– When the conductor is charged, the plates carry
charges of equal magnitude and opposite directions
• A potential difference exists between the plates due to
the charge
Q
A
A
Q
B
B
Q - the charge of capacitor
V  VA  VB - a potential difference of capacitor
30
Capacitors
• A capacitor consists of two conductors
conductors (plates)
A
Plate A has the SAME potential at all points
because this is a conductor .
B
Plate B has the SAME potential at all points.
So we can define the potential difference
between the plates:
V  VA  VB
31
Capacitance of Capacitor
A
Q
C
V
 The SI unit of capacitance is the farad (F) = C/V.
B
 Capacitance is always a positive quantity
 The capacitance of a given capacitor is constant
and determined only by geometry of capacitor
The farad is a large unit, typically you will see
microfarads (μF ) and picofarads (pF)
32
Capacitor: Parallel Plates
E 0
E
σ
ε0
E 0
1
σ 0
E
2
σ 0
V 0
V 
σ
x
ε
0
V 
σ
h
ε
0
Q0
Q
σ
S
E
Q  0
The potential difference:
V 
σ
Q
h=
h
ε
εS
0
0
The capacitance:
ε
Q
0S
C

V
h
33
Capacitor: Parallel Plates: Assumptions
• Main assumption - the electric field is uniform
• This is valid in the central region, but not at the ends of the plates
• If the separation between the plates is small compared to the length
of the plates, the effect of the non-uniform field can be ignored
34
Capacitors with Dielectric (Insulator)
Dielectric (insulator) inside capacitor. Capacitance = ?
Q
Q  CV
dielectric
C ?
h
Q
Dielectric (insulator)
• The molecules that make up the
dielectric are modeled as dipoles
• An electric dipole consists of two
charges of equal magnitude and
opposite signs
• The molecules are randomly
oriented in the absence of an
electric field
36
Dielectric in Electric Field
 Dielectric in External Electric Field
 The molecules partially align with
the electric field
 The degree of alignment of the
molecules with the field depends on
temperature and the magnitude of
the field
Polarization
37
Capacitors with Dielectric
Dielectric inside capacitor. Capacitance = ?
Q
Q  CV
dielectric
C ?
h
Q
The electric
E0 field
 Eind inside
dielectric
38
Capacitors with Dielectric
Dielectric inside capacitor. Capacitance = ?
Q
Q  CV
dielectric
C ?
h
Q
If we have the same charge as without dielectric
then the potential difference is increased, since
Q
V  h(E0  Eind )
Q
without dielectric it was
V0  hE0
V  V0
Q
Q
C


C

0
then
V
V0
Capacitance is increased.
39
Capacitors with Dielectric
Dielectric inside capacitor. Capacitance = ?
Q
Q  CV
dielectric
C ?
h
Q
 Capacitance is increased
Q
Q
 To characterize this increase the
coefficient (dielectric constant of material)
is introduced, so
C  κC0
(this is true only if dielectric completely fills
the region between the plates)
40
41
Type of Capacitors: Tubular
• Metallic foil may be interlaced
with thin sheets of paper
• The layers are rolled into a
cylinder to form a small
package for the capacitor
42
Type of Capacitors: Oil Filled
• Common for high- voltage
capacitors
• A number of interwoven
metallic plates are immersed
in silicon oil
43
Type of Capacitors: Electrolytes
• Used to store large amounts
of charge at relatively low
voltages
• The electrolyte is a solution
that conducts electricity by
virtue of motion of ions
contained in the solution
44
Type of Capacitors: Variable
• Variable capacitors consist of
two interwoven sets of metallic
plates
• One plate is fixed and the
other is movable
• These capacitors generally
vary between 10 and 500 pF
• Used in radio tuning circuits
45
Capacitor: Charging
 Each plate is connected to a terminal of the
battery
 The battery establishes an electric field in the
connecting wires
 This field applies a force on electrons in the
wire just outside of the plates
 The force causes the electrons to move onto
the negative plate
 This continues until equilibrium is achieved
 The plate, the wire and the terminal are all
at the same potential
 At this point, there is no field present in the
wire and there is no motion of electrons
Battery- produces the fixed voltage –
the fixed potential difference
46
Chapter 21
Capacitance and Electrical Circuit
47
Electrical Circuit

A circuit diagram is a simplified
representation of an actual circuit
 Circuit symbols are used to
represent the various elements
 Lines are used to represent wires
 The battery’s positive terminal is
indicated by the longer line
48
Electrical Circuit








Conducting wires.
In equilibrium all the points of the
wires have the same potential
49
Electrical Circuit
V
 


V
The battery is characterized by the voltage –
the potential difference between the contacts of
the battery
In equilibrium this potential difference is equal to
the potential difference between the plates of the
capacitor.
Then the charge of the capacitor is
Q  CV
V
 
If we disconnect the capacitor from the battery the
capacitor will still have the charge Q and potential
difference V
50
Electrical Circuit
V
 
Q  CV
V
 

V

If we connect the wires the charge will disappear
and there will be no potential difference
V  0
51
Energy Stored in a Capacitor: Application
Q2 1
1
U
 QV  C(V )2
2C 2
2
One of the main application of capacitor:
 capacitors act as energy reservoirs that can be
slowly charged and then discharged quickly to
provide large amounts of energy in a short pulse
Q
Q  CV
Q
52
Electric Potential and Electric Field
 Can we find electric field if we know electric potential?
VB  VA  x E
A
E
x
B
VB  VA
V
E

x
x
x
V
E
x
53
Electric Potential and Electric Field
V
E
s
 Equipotential lines are everywhere perpendicular to the electric field.
Equipotential lines
54