Electric Potential
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Flashback to PHY113…
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Work done by a conservative force is
independent of the path of the object.
Gravity and elastic forces are examples.
This leads to the concept of potential energy and
helps us avoid tackling problems using only
forces.
xf
W = ∫ Fs dx = − ΔU
xi
„
Electrostatic forces are also conservative.
Electric Potential Energy
B
v
E
v
v
F = q0 E
v v
ΔU = − ∫ F.d s = −q0
path
v v
∫ E.d s
path
A
q0
…but F is a conservative force
…so the path we take
does not matter
v w
ΔU = −q0 ∫ E.d s
B
A
Electric Potential
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Work, ΔU, is dependent on the magnitude of the
test charge, q0.
We’d like to have a quantity independent of the
test charge and only an attribute of the electric
field.
U
V=
q0
v v
ΔU
ΔV =
= − ∫ E.d s
q0
A
Electric Potential
(J/C=V)
Potential Difference
between A and B
B
P
VP = − ∫ E.ds
∞
Electric Potential at P
Unit Pit Stop
Potential
V =J C
Energy
J = N ⋅m
Electric Field
N ⎛ V .C ⎞⎛ J ⎞
N C= ⎜
⎟ =V m
⎟⎜
C ⎝ J ⎠⎝ N .m ⎠
Electron-Volt
1eV = e(1V ) = 1.6 ×10 −19 C (1 J C ) = 1.6 ×10 −19 J
(
)
Electric Potential in a Uniform Field
B
B
B
A
A
A
ΔV = − ∫ E.ds = − ∫ E cos 0°ds = − E ∫ ds
ΔV = − Ed
ΔU = −q0 Ed
Electric field lines point to decreasing
potential.
A positive charge will lose potential energy
and gain kinetic energy when moving in the
direction of the field.
Equipotential Surfaces
D
B
s
E
B
B
A
A
ΔV = − ∫ E.ds = −E.∫ ds = −E.s
θ
C
A
ΔV = −( E.s) AC − ( E.s)CB
ΔU = −q0 (E.s) AC − q0 (E.s) CB
ΔV = − Es AC cos 0° − EsCB cos 90°
ΔU = −q0 Es AC cos 0°
ΔV = − Es cos θ
ΔU = −q0 Es cos θ
D
D
A
A
ΔVAD = − ∫ E.ds = −E.∫ ds = −E.s = −( E.s) AC − ( E.s)CD = − Es cos θ
No work is done moving a charged particle perpendicular to
a field (along equipotential surfaces)
Equipotential Surfaces
Uniform Field
Point Charge
Electric Dipole
Electric Potential of a Point Charge
B
VB − VA = − ∫ E.ds
A
E.ds = k e
q
q
q
ˆ
r
.
d
s
=
k
ds
cos
θ
=
k
e 2
e 2 dr
2
r
r
r
VB − VA = − ∫ Er dr
⎡1 1⎤
dr
k
q
=
− ⎥
e ⎢
2
r
r
⎣ B rA ⎦
rA
rB
VB − VA = −ke q ∫
V A= ∞ = 0
q
V = ke
r
Two Point Charges
⎛ q1 q2 ⎞
VP = ke ⎜⎜ + ⎟⎟
⎝ r1 r2 ⎠
A System of Point Charges
q1q2
U = ke
r12
⎛ q1q2 q2 q3 q3q1 ⎞
⎟⎟
U = ke ⎜⎜
+
+
r23
r31 ⎠
⎝ r12
Concept Question
Two test charges are brought separately into the vicinity of
a charge +Q. First, test charge +q is brought to a point a
distance r from +Q. Then this charge is removed and test
charge –q is brought to the same point. The electrostatic
potential energy of which test charge is greater:
1. +q
2. –q
3. It is the same for both.
Getting From V to E
E
B
V = − ∫ E.ds
A
E
dV
Er = −
dr
dV = −E.ds
∂V
In general: E x = −
∂x
dV
Ex = −
dx
∂V
Ey = −
∂y
∂V
Ez = −
∂z
Electric Potential of a Dipole
2ke qa
VP = 2
2
x −a
2ke qa
V≈
x2
(x >> a)
dV 4ke qa
Ex = −
=
dx
x3
Between the Charges
2ke qx
VP = 2
a − x2
⎛ a2 + x2 ⎞
dV
⎟
Ex = −
= −2ke q⎜ 2
2 ⎟
2
⎜
dx
⎝ (a − x ) ⎠
Electric Potential Due to
Continuous Charge Distributions
Start with an infinitesimal
charge, dq.
dq
dV = ke
r
Then integrate over the
whole distribution
dq
V = ke ∫
r
Electric Potential Due to
a Uniformly Charged Ring
keQ
V=
E=
x2 + a2
(x
keQx
2
+a
)
2 3/ 2
Electric Potential Due to
a Finite Line of Charge
keQ ⎛⎜ l + l 2 + a 2
ln
V=
⎜
l
a
⎝
⎞
⎟
⎟
⎠
Electric Potential Due to
a Uniformly Charged Sphere
Er = k e
Q
r2
r
r>R
kQ
Er = e 3 r
R
r<R
r
VD − VC = − ∫ Er dr =
R
keQ ⎛
r2 ⎞
⎜⎜ 3 − 2 ⎟⎟
VD =
2R ⎝
R ⎠
(
keQ 2 2
R −r
3
2R
)
r
dr
r2
∞
VB = − ∫ Er dr = − keQ ∫
∞
VB = ke
Q
r
VC = k e
Q
R
Potential Due to a Charged
Conductor
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Charges always reside at
the outer surface of the
conductor.
The field lines are always
perpendicular to surface.
Then E.ds=0 on the
surface at any point.
Which means, VB-VA=0
along the surface.
The surface is an
equipotential surface.
Finally, since E=0 inside
the conductor, the
potential V is constant and
equal to the surface value.
Connected Charged Conducting
Spheres
q1 r1
=
q2 r2
E1 r2
=
E2 r1
Cavity Within a Conductor
B
VB − VA = − ∫ E.ds = 0
A
We can always find a
path where E.ds is
non-zero.
But, since ΔV=0 for all
paths, E must be zero
everywhere in the cavity.
A cavity without any charges enclosed by a
conducting wall is field free.
Summary
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Charges can have different electric
potential energy at different points in an
electric field.
Electric potential is the electric potential
energy per unit charge.
All points inside a conductor are at the
same potential.
For Next Class
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Reading Assignment
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Chapter 26 – Capacitance and Dielectrics
WebAssign: Assignment 3