Physics 212 Lecture 4, Slide 1

1
Physics 212 Lecture 6
Today's Concept Electric Potential Defined in
terms of Path Integral of Electric Field
Extra 1
Preflight 1
Preflight 3
Extra 2
Preflight 7
Extra 3
Homework
Preflight 5
Music
Preflight 9
Preflight5_11
2
Things you identified as difficulties

• Zero of Potential?????
• Totally Arbitrary
• But, cant choose at infinity if E doesnt fall
  off fast enough
• e.g., Cylinders DVab ln(ra/rb) and Planes
  DVab (ra-rb)

2) Conductors and Insulators What do we need for
now?? Insulators Charge cannot move, charge
distribution must be given to you Conductors
Charge free to move, E 0 in conductor at
equilibrium, excess charge on surface,
surface is equipotential
3
ACT
Case A
d
Case B
2d

• In case A two charges which are equal in
  magnitude but opposite in charge are separated by
  a distance d. In case B the same charges are
  separated by a distance 2d. Which configuration
  has the highest potential energy?
• Case A
• Case B
• The potential energy is the same since the total
  charge is zero in both cases.

4
ACT Discussion

• As usual, choose U 0 to be at infinity

Case A
d
UB gt UA
5
Preflight 5_11

• The charge will move in the direction that
  requires least work so the absolute value of the
  potential energy will be smaller
• All systems, whether economic, chemical,
  physical, or otherwise seek to minimize potential
  energy, this system is no different.
• The path of constant potential energy would
  create a state of equilibrium where nothing is
  changing, which is what nature desires.

6
Preflight 5_11

• THE ANSWER I DIDNT SHOW YOU
• Potential energy is relative to an arbitrary 0,
  so the absolute value has nothing to do with it,
  since the charge has no idea which arbitrary
  point we chose for 0. It does know which
  direction will make its potential energy more
  negative, since that is the direction of the net
  force upon it, so it will move that direction
  seeking equilibrium.

Simulation
7
Preflight 7
A B C
D
E
8
Preflight 9
If the charge is going with the flow of the
electric field, this is as if it were rolling
down the hill, decreasing the potential
energy. The work done by the field is positive,
hence potential energy decreases.
D
E
9
Problem Suppose you have a solid conducting
sphere of radius a carrying a net charge Q. Find
E everywhere.
Important Facts
Q
1) Charges in a conductor will move if there is
an E field.
a
2) The E field in a conductor is always ZERO.
3) Gauss law says that if E 0 everywhere inside
the sphere then all points in the sphere are
neutral.
4) All of the extra charge must be uniformly
spread on the outside surface of the sphere.
Simulation
10
Preflight 1
Consider a solid conducting sphere of radius a.
Which of the following graphs best describes the
magnitude of the electric field as a function of
distance r from the center of the sphere?
A
C
B
(Notice that they are all identical for r gt a)
D
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Preflight 1
Consider a solid conducting sphere of radius a.
Which of the following graphs best describes the
magnitude of the electric field as a function of
distance r from the center of the sphere?
Common Misconception
The E field inside depends on how much charge is
enclosed by r, so it increases linearly outside
the sphere we can treat it as a point charge
which is proportional to 1/r2
Point a. can not be more than one value of E.
D
12
Gauss Law
For r gt a
Q
a
r
D
13
Problem Suppose you have a conducting spherical
shell of radius a carrying a net charge Q. Find
E everywhere.
Important Facts
Q
1) Charges in a conductor will move if there is
an E field.
2) Just like in the case of the the solid sphere,
all of the extra charge must be uniformly spread
on the outside surface of the sphere.
3) The E field is identical to that of the solid
sphere everywhere.
Simulation
14
Preflight 5
If the solid conducting sphere is replaced by a
hollow conducting spherical shell having the same
radius and carrying the same charge, which of the
following would change
A B C D E
conducting shell
solid conductor
15
Preflight 5
Many people picked 1 3
Your reasons were mostly like this
electric field inside would be zero also,
electric potential would be constant because e
field is zero
The right argument for the wrong answer !!
16
Problem Suppose you have a solid conducting
sphere of radius a carrying a net charge Q. Find
V everywhere.
Important Facts
Q
1) V can be determined once we know E
a
We know E everywhere, so we can find V everywhere
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Q
Lets find the change in V going from r 0 to r
a
a
0
E 0
DV 0 between any two points in a conductor
V does not change inside a conductor
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Preflight 3
Consider a solid conducting sphere of radius a.
Which of the following graphs best describes the
electric potential as a function of distance r
from the center of the sphere?
A
C
B
(Notice that they are all identical for r gt a)
19
Preflight 3
Common Misconception
The potential increases the closer it gets to the
surface, which is where the charge is located.
The electric potential will be greatest in the
center of the sphere since potential is a scaler
and the sum of all potentials from around the
sphere is greatest in the center.
B
A
C
D
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Now lets find the change in V going from r a to
r gt a
21
If V(a) 0 this looks like
22
Now lets find the change in V going from r 8 to
r gt a
0
23
If V(8) 0 this looks like
24
When V is chosen to be 0 at r 8
When V is chosen to be 0 at r 0
Same exact curve just shifted
25
Homework Problem
cross-section
a4
a3
Q

• Point charge q at center of concentric conducting
  spherical shells of radii a1, a2, a3, and a4.
  The inner shell is uncharged, but the outer shell
  carries charge Q.
• What is V as a function of r?

a2
a1
q
metal
metal

• Conceptual Analysis
• Charges q and Q will create an E field throughout
  space

• Strategic Analysis
• Spherical symmetry Use Gauss Law to calculate E
  everywhere
• Integrate E to get V

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Homework Problem Quantitative Analysis
cross-section
a4
a3
Q

• r gt a4 What is E(r)?
• (A) 0 (B) (C)

a2
a1
q
(D) (E)
metal
metal
27
Homework Problem Quantitative Analysis
cross-section
a4
a3
Q

• a3 lt r lt a4 What is E(r)?
• (A) 0 (B) (C)

a2
a1
q
(D) (E)
metal
metal
How is this possible??? -q must be
induced at ra3 surface
28
Homework Problem Quantitative Analysis
cross-section
a4
a3
Q

• Continue on in.
• a2 lt r lt a3

a2
a1
q
metal
metal

• To find V
• Choose r0 such that V(r0) 0 (usual r0
  infinity)
• Integrate !!

29
Homework Problem Quantitative Analysis
cross-section
a4
a3
r gt a4
Q
a2
a1
a3 lt r lt a4
q
metal
metal
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Music

• Who is the Artist?
• Eric Clapton
• Jefferson Airplane
• Santana
• Gregg Allman
• Jeff Beck

Fine Live Album Many Favorites
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Extra 1
32
Extra 2
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Extra 3