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MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 1
Chapter 24
Electric Potential
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 2
Electric Potential
1. Potential Difference
2. Potential Due to a System of Point Charges
3. Computing the Electric Field from the Potential
4. V for Continuous Charge Distributions
5. Equipotential Surfaces
6. Electrostatic Potential Energy
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 3
Conservative Force
For conservative forces we can introduce
the concept of a potential energy.
Potential energy is the energy of position
whereas kinetic energy is the energy of
motion.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 4
Gravitational Analogy
The positive charge is used to make the
analogy more transparent.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 5
If you throw a ball straight up in the air its initial kinetic
energy is soon reduced to zero at the top of its motion.
The “loss of kinetic energy” is only apparent. The kinetic
energy is restored to its original value when the ball returns
to its starting position.
The use of the concept of potential energy allows us to
maintain the conservation of total mechanical energy.
Total Mechanical Energy = KE + PE
An Example from Last Semester
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 6
Examples of Forces
Conservative Forces
Gravitation
Electric force
Magnetic force
Spring force
Non-Conservative Forces
Friction
Air resistance
The magnetic force has some restrictions on it since in
general a velocity dependent force is not conservative.
The kinetic energy is transformed into heat
and is then outside the mechanical system.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 7
Work and the Conservative Force
The characteristic of a conservative force is that the work
done on an object moving it, in a force field, from point A
to point B is independent of the path taken between A and
B.
If the path forms a closed loop then the work done is zero.
⋅∫� �
�C
W = F dL = 0
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 8
Potential Energy - Potential Function
There is a potential energy (U) and a potential function
(V) and they are closely related.
� �i
� �
� �i
dU = -F dL
F = qE
dU = -qE dL
≡
∆∫
� �i
� �i
� �i
0
0
b
b a
a0
dU = -q E dL
dUdV = -E dL
q
U∆V = V - V = = - E dL
q
Potential Energy Potential Function
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 9
∆∫� �i
b
b a
a0
U∆V = V - V = = - E dL
q
The Change in Potential a b
The integral is equal to
the difference between
the potential function
evaluated at the end
points.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 10
Potentials for Discrete Charges
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 11
Potential and the Reference Point
The potential
function needs a
common reference
point so that the
calculated potential
differences will
have physical
meaning.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 12
Potential and the Point Charge
ˆ
ˆ
= −
∞
∫ ∫ ∫
∫
�
� � �i i
2
P P P
P ref 2 2
ref ref ref
P
P ref P 2
P refref
ref
kqE = r
r
kq kqV - V = - E dL = - r dL = - dr
r r
dr kq kqV - V V - 0 = -kq =
r r r
kqV = ; as r goes to
rFinal form
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 13
Potential Due to Two Point Charges
Find V at points P1 and P2 ∑ i 1 2
i i 1 2
kq kq kqV = = +
r r r
1
9 -9
1 2P
1 2
kq kq kq 9x10 * 5.0x10V = + = 2 = 2 = 2247 V
r r r .040
2
9 -9 9 -9
1 2P ' '
1 2
kq kq 9x10 * 5.0x10 9x10 * 5.0x10V = + = + = 1200 V
r r .060 .010
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 14
Potential Calculation for 2 Point Charges
∑ i 1 2
i i 1 2
kq kq kqV = = +
r r r
1 2kq kqV = +
x x - a
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 15
Potential for 2 Point Charges
In contrast to the electric field component graphs the potential
graphs, being scalar quantities, depend on the sign of the charge,
there is no direction.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 16
Electric Dipole Geometry for Potential
Calculation
The dipole configuration requires equal and opposite charges.
( )22
kql lV = ; x >
2lx -4
≈ �2 2
kql kpV = ; x l
x x
p = ql is the dipole moment
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 17
3D Image of the Electric Dipole Potential
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 18
Electric Field from the Electric Potential
In general
For one dimension
ˆ ˆ�� � �i i i xdV(x) = -E dl = -E dxi = -(E i)dx = -E dx
x
dV(x)E = -
dx
ˆˆ ˆ ∂ ∂ ∂
∇ ∂ ∂ ∂
� V V VE = - V = - i + j + k = -gradV
x y z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 19
Potentials for Continuous Charge
Distributions
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 20
Potential Due to a Ring Charge
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 21
Potential Due to a Ring Charge
0 0
= = =∫ ∫Q Q
kdq k kQV dq
r r r
2 2 2= +r z a
1 1 12 2 2 22 2 21
2 2
2 21 1
= = = + + +
kQ kQ kQV
z a a az z
z z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 22
Electric Field Due to a Ring Charge
312 2 2 22 2
32 2 2
32 2
3 3 3 22 2 22 23
2 2
2
12
2
ˆ ˆ
ˆ ˆ ˆ ˆ1
1
− −
−
∂ ∂ = − = − + = + ∂ ∂
= = +
= = = = +
+ +
�
�
z
z
z
VE kQ z a kQ z a z
z z
kQzE E k k
z a
kQz kQz kQz aE E k k k k
zzz a az
z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 23
Electric Field Due to a Ring Charge
Far Away from the Ring
32 22
3 32 2
2
3 2 2
3ˆ ˆ ˆ1 12
3 ˆ ˆlim 12
−
→∞
= = + = −
− =
�
z
z
kQz a kQz aE E k k k
z zz z
kQz a kQk k
z zz
This the electric field of a point charge of value Q
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 24
Particle Acceleration Due to a Ring
Charge
The particle and the ring both have the
same sign charge. Therefore the particle
experiences an accelerating force.
Do calculation
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 25
Potential Due to a Disk Charge
For simplicity we only consider potentials along the z-axis
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 26
Potential Due to a Disk Charge
2
2σ σ π
σπ
=
= =
=
kdqdV
r
dq dA ada
QR
( )
( )
( )
12 2 2
12 2 2
0 0
2
2
2
2
( ) 2 1 1
πσ
σπ
π σ
= =+
= =+
= + −
∫ ∫V R
k adakdqdV
r z a
adadV V k
z a
RV z k z
z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 27
Potential Close to a Disk Charge
2
2( ) 2 1 1π σ
= + −
RV z k z
z
Very close to the disk z<<R. The disk
appears to effectively be an infinite plane.
2
2
0
( ) 2 1 1 2 1
( ) 2 2 2
π σ π σ
π σ π σ π σ
= + − ≈ −
= − = −
R RV z k z k z
z z
V z k R k z V k z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 28
The Absolute Value Operation
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 29
Potential Due to an Infinite Plane of
Charge
The potential function is required to be continuous but its
derivatives need not be continuous.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 30
The Electric Field Due to an Infinite
Plane of Charge (z>0)
( ) ( )
( )
0
0 0
( ) 2
2 2
12
4 2
π σ
π σ π σ
σπ σ
πε ε
= −
∂ ∂= − = =
∂ ∂
= =
n
n
V z V k z
VE z k z k
z z
E z
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 31
Potential Due to a Thin Spherical Shell
No electric field inside, therefore
V is constant inside the sphere.
Do calculation
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 32
Potential Due to a Thin Spherical Shell
2
2
ˆ For r > R,
ˆ
= − =
= −
�� �i
�
i
kQdV E dl E r
r
kQdV r dl
r
2
2
−
+∞ +∞
= − = − =∫ ∫p pr r
p
p
kQ kQV dr kQ r dr
r r
For r < R
( )2
20 −
+∞ +∞
= − = − − =∫ ∫ ∫p pr rR
p
R
kQ kQV dr kQ r dr dr
r R
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 33
Potential Due to a Uniformly Charged
Solid Sphere
3
2
,
,
< =
> =
r
r
kQFor r R E r
R
kQFor r R E
R
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 34
Potential For an Infinite Line Charge
Here we are going
to use our
knowledge of the
field to determine
the potential.
Due to the geometry
here we are unable
to place the
reference point at
infinity.
Since it is the “difference in potential” that has a physical meaning, whatever
reference point we pick will drop out upon taking the difference of potential.
⋅∫� �P
P Ref
Ref
∆V = V - V = - E dL
ref
P ref P
RV - V = V - 0 = 2kλln
R
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 35
Equipotential Surfaces
The equipotentials (gray
lines) are represented by
contours along which the
value of the potential
function is constant.
Potential lines and
electric field lines are
everywhere
perpendicular.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 36
Equipotential Surfaces
Close to the surface of
the conductor the
equipotential surfaces
follow the shape of the
conductor.
The surface of a
conductor is an
equipotential surface.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 37
Equipotential Surfaces-Point Charges
Field Lines
Potential Lines
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 38
Potentials Due to Spherical Shells
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 39
Potentials Due to Spherical Shells
Do calculation
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 40
Primitive Method of Storing a Large
Charge or Creating a High Potential
Spherical conductor on the end of an
insulating rod
Spherical conductor is charged positively by
removing a quantity of electrons.
Touching the charged sphere to the inner
surface of the large hollow conducting sphere
causes electrons to transfer to the small
spherical conductor - the large sphere is left
with a net positive charge.
Repeat the process to build up the charge.
Eventually the process saturates.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 41
The Van de Graff Generator
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 42
Robert J. Van de Graaff
1901-1967
The energies produced by Van de Graaff
atomic particle accelerators are limited to
about 30 MeV, even with tandem generators
accelerating doubly charged (for example
alpha) particles.
More modern particle accelerators using
different technology produce much higher
energies, thus Van de Graaff particle
accelerators have become largely obsolete.
They are still used to some extent for
graduate student research at colleges and
universities and as ion sources for high
energy bursts.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 43
Ernest O. Lawrence
1901- 1958
Ernest Orlando Lawrence was an American physicist and
Nobel Laureate, known for his invention, utilization, and
improvement of the cyclotron atom-smasher beginning in
1929, based on his studies of the works of Rolf Widerøe, and
his later work in uranium-isotope separation for the
Manhattan Project.
Lawrence had a long career at the University of California,
where he became a Professor of Physics. In 1939, Lawrence
was awarded the Nobel Prize in Physics for his work in
inventing the cyclotron and developing its applications.
Chemical element number 103 is named "lawrencium" in
Lawrence's honor.
Source: http://en.wikipedia.org/wiki/E._O._Lawrence
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 44
Source: http://en.wikipedia.org/wiki/E._O._Lawrence
Lawrence read a journal article about producing very high-energy particles
required for atomic disintegration by means of a succession of very small
"pushes."
In an attempt to make the accelerator more compact, Lawrence decided to
use a circular accelerating chamber situated between the poles of a large
electromagnet. The magnetic field would hold the charged particles in a
planar spiral path as they were accelerated between the two semicircular
electrodes. These electrodes would be sandwiched between the magnetic
poles and connected to an alternating potential. After about a hundred
circular orbits, the particles would impact the target as a beam of high-
energy particles. Lawrence had discovered a method for obtaining particles
of very high energy without the use of any high voltage.
The Beginning of the Cyclotron
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 45
Giant calutron plants developed at Lawrence's laboratory were used at Site X during
World War II to purify uranium for use in the first atomic bomb. Alpha Track Calutron at
the Y-12 Plant (Site X) at Oak Ridge, Tennessee from the Manhattan Project, used for
uranium enrichment.
Alpha Track Calutron at the Y-12 Plant (Site X) at
Oak Ridge, Tennessee
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 46
Ernest O. Lawrence
1901- 1958
Robert J. Van de Graaff
1901-1967
Linear
Acceleration
Circular
Acceleration
Van de Graaff was jealous of
his more successful rival -
Lawrence. In later years at
HVE his engineers could not
even discuss a design that
involved particles traveling
in a circular motion.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 47
Dielectric Breakdown
High E-Field
Low E-Field
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 48
“Air by nature is an Insulator. For it to become a Conductor, it must experience Dielectric
Breakdown. This occurs if a voltage of 30,000V per centimeter is applied to the air. The air is
then Ionized. In a Jacobs Ladder, Dielectric breakdown occurs when a voltage of 30,000 V is
supplied to the bottom of the Rods.”
“An Electric Arc is formed between the Rods and rises along with the hot air. As the arc length
increases, its power consumption increases and there is insufficient Voltage to maintain the
Electric Arc. The arc extinguishes and the cycle repeats with a new arc forming at the bottom
of the rods.”
Source: http://macao.communications.museum/eng/Exhibition/secondfloor/moreinfo/2_1_1_JacobsLadder.html
Jacob’s Ladder serves no
scientific purpose but
every mad scientist in the
movies had to have at least
one in his lab.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 49
Dielectric Strengths
Source: http://en.wikipedia.org/wiki/Dielectric_strength
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 50
Two Charged Spherical Conductors
The thin conducting wire connecting the two spheres serves
the same purpose as allowing the two spheres to touch.
This contact allows charge to transfer (remember only the
electrons move) between the spheres and to bring the
conductor surfaces to the same potential.
Do calculation
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 51
U - Electrostatic Potential Energy
• The electrostatic energy of a
system of charges is equal to the
Work required to assemble the
charges from infinity.
• The Work is the difference in
potential energy between the
charge at infinity and at its final
position.
• Since U at infinity is zero, the
work is just U evaluated at the
final charge positions.
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 52
U - Electrostatic Potential Energy
∑n
i i
i=1
1U = q V
2
∑n
i i
i=1
1U = QV
2
1U = QV
2
For point charges
A conductor with a charge Q
at potential V
For a system of conductors
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 53
Extra Slides
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 54
Conservative Forces and Potentials
from Vector Analysis
Work around a closed loop = 0
Stokes Theorem
Therefore a potential function V exists for a conservative force.
( )
( )
( )
⋅
⋅ = ∇ ⋅
∇ ⋅ ⇒ ∇
∇ ∇ ∇
∫
∫ ∫
∫
��
�� � � �
� � � ��
� � � �
�
�
C
C S
S
W = F dl = 0
F dl × F da
× F da = 0 × F = 0
F = - V since × V = 0
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 55
Vector Analysis
ψ��
φ and are scalar functions
F and G are vector functions
∇
∇ ⋅
∇
�
� � � �
� � � �
φ= grad φ= gradient of φ
F = div F = divergence of F
× F = curl F = curl of F
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 56
Vector Analysis
Gradient
Divergence
Curl
ˆˆ ˆφ φ φ∂ ∂ ∂
∇∂ ∂ ∂
�φ= i + j + k
x y z
∂∂ ∂∇ ⋅
∂ ∂ ∂
� �yx z
FF FF = + +
x y z
ˆˆ ˆ
∂ ∂ ∂∇ ×
∂ ∂ ∂
� �
x y z
i j k
F =x y z
F F F
MFMcGraw-PHY 2426 Ch24d-Electric Potential-Revised 8/23/2012 57