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Electric Potential
Read: Chapter 17
Review of Potential
Potential Energy is an energy associated with the interaction of a pair of particles within a system.
An isolated single particle system has only rest and kinetic energies, E = mc2 + K.
When analyzing energy, you must define a system — the system consists of all those objects that are not considered part
of the surroundings.
~
The work done on (or by) a particle is the scalar product of force and displacement: W = F~ · d.
Work can be external or internal. External work is work done on the system by the surroundings; internal work is the work
done by forces excusively internal to the system.
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Electric Potential
Review of Potential
The energy of a system can change only if work is done on the system by the surroundings (+Wext ) or by the system on the
surroundings (−Wext ).
Given constant rest energy, the change in system energy ∆Esys consists of the change in kinetic energy minus any internal
work: ∆Esys = ∆K − Wint . Energy conservation requires
∆Esys = Wext = ∆K − Wint .
Potential energy is defined by the relation
∆U ≡ −Wint .
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Electric Potential
Isolated Systems
An isolated system is a system where the influence of the surroundings is negligible or nonexistent, therefore
∆Ksys + ∆U = 0 .
Question: In this isolated system, does K of the proton increase or decrease?
What about U ?
−
+
~
E
~e
F
B
A
Exercise: Calculate ∆U of the proton.
∆x
Now replace the proton with an electron. Answer the same questions and
calculate ∆U of the electron.
−
+
~
E
~e
F
B
A
∆x
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Electric Potential
Potential Difference in a Uniform Field
~ is related to F~e through the relation E
~ = F~e /q, the potential difference ∆V is related to a change in potential
Much like E
energy ∆U through the relation ∆V = ∆U/q. Potential difference describes the amount of energy per unit charge needed to
move a charged object from one point to another.
~ can be expressed as V /m.
The unit of potential difference is the volt: V = J/C. Using volts, the units of E
The potential difference in a uniform electric field is given by the scalar product
~ · ∆~l .
∆V = −E
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Electric Potential
Potential Difference in a Uniform Field
Example: For the path from A to B, calculate:
∆~l
A
B
(a) the electric potential difference.
~
E
(b) ∆U for a proton on this path.
(c) ∆U for an electron on this path.
~ of 100 N/C, what is ∆V if you move 2 meters
Example: Given a uniform E
◦
at an angle of 30 to the electric field?
2m
d~l
30◦
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Electric Potential
Potential Difference in a Uniform Field
V
A
B
x
~ can be obtained from V by taking its negative gradient
Mathemaically, E
∂
∂
∂
~
E = −∇V = −
x̂ +
ŷ +
ẑ V (x, y, z) .
∂x
∂y
∂z
~ x.
At point A, the small positive gradient results in small negative E
~ at B?
Question: What is the sign and relative magnitude of E
~ = −∇V , what conclusion can we draw about V inside a conductor?
Question: Since E
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Electric Potential
Sign of ∆V
~ is doing work on the system, then ∆V < 0. If work is being done by the system against the electric field, then ∆V > 0.
If E
Consider a proton in an electric field. The above statements correspond to:
~ ∆V decreasing, K increasing.
1. Proton moving in direction of E:
~ ∆V increasing, K decreasing.
2. Proton moving opposite E:
~ ∆V = ∆K = 0.
3. Proton moving ⊥ E:
Question: How is this similar to gravitational potential?
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Electric Potential
~
∆V in nonuniform E
~ will not be uniform. If so, follow a procedure similar to that of dealing with charge distributions:
Generally speaking, E
~ and sum.
break up the space into regions of uniform E
~3
E
~2
E
~1
E
A
B
C
D
x
∆VA→D = −
3
X
~ i · ~xi
E
i=1
= (−E1x (B − A)) + (−E2x (C − B)) + (−E3x (D − C))
~ varies continuously as a function of position. In this situation, the “pieces” of uniform E
~ become
Most realistically, E
infinitessimally small and the sum becomes an integral.
~
E(x)
A
D
x
Z
D
~
E(x)
· d~x
∆V = −
A
The most general expression for ∆V is
Z
∆V (x, y, z) = −
f
~
E(x,
y, z) · d~l(x, y, z) .
i
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Electric Potential
Example: Disk and Shell
+Q2
RD
1
2
RS
d
−Q1
A plastic spherical shell of radius RS carries a charge −Q1 , while a thin circular glass disk of radius RD RS carries charge
+Q2 . The disk and shell are separated by a distance d RD . Find V2 − V1 .
Question: What simplifying assumptions do the conditions of the problem allow you to make?
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Electric Potential
Path Independence
Electric potential difference is path independent — the potential difference between two points is independent of the path
chosen between them.
Path independence is a characteristic of conservative forces; in processes involving only conservative forces, energy is conserved. When energy is conserved, the work done along any closed loop is zero.
~ = −∇V , E
~ is a
Any force F~ that can be written as the gradient of some potential Φ, F~ = −∇Φ, is conservative. Since E
conservative force and ∆V is independent of path.
Question: Air resistance can be modeled as a velocity dependent force F~air = −b~v 2 where b is some constant. Prove that
F~air is not conservative.
10
Electric Potential
Potential at a Single Location
Infinitely far away from all charged particles, V = 0. We commonly define the potential at a specific location V (r) as the
potential difference between r and the point at infinity: V (r) = V (r) − V (∞) = V (r) − 0.
We obtain V (r) through the integral relation
Z
r
V (r) = −
~
E(r)
· d~r .
∞
Since V is path independent, choose the most convenient path for integration. Note that ∞ is the lower limit, as it must be
the initial point.
Since it is only potential difference that is physically meaningful, we are free to redefine the zero of potential.
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Electric Potential
∆V in an Insulator
~ plates
E
The net field inside the plastic (commonly referred to as a dielectric) is the
~ is reduced in magnitude inside the
sum of the external and induced fields. E
dielectric.
~ net
E
~ dipoles
E
The dielectric constant K is defined by the relation
~ dielectric ≡ 1 E
~ ext ,
E
K
and is always ≥ 1. Since K is independent of position, the potential difference ∆V obeys a similar relation:
∆Vdielectric =
1
∆Vvacuum .
K
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Electric Potential
Energy Density
The energy density u describes an amout of energy per unit volume. The concept
of energy density allows us to think of energy being “stored” in the electric field. To
see, consider the work done moving one plate of a capcitor.
−
−
−
−
−
−
−
−
−
−
+
+
+
+
+
F~ext
+
F~plate
+
+
+
+
∆s
Energy density is a fundamental quantity, and is given by the relation
u=
1 ~2
0 E .
2
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