PES 1120 Spring 2014, Spendier
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Lecture today:
1) Conductors
Announcement: Prof. Karen Livesey will fill in for me Monday 17th. Lecture will be
held in the usual room at usual time. HW 3 will be due Feb. 19th. So please look at
your HW 3 in advance and ask questions this Friday. I will not hold office hours
next week M/Tue/W due to travel. You can always e-mail with a specific question as
well.
Last time we calculated the electric field due to a sphere of radius R with UNIFROM
charge distribution. We used Gauss' Law to calculate E for r < R and r > R.


 
q enc
E
r

dA




0
We got the following for E(r):
This is all assuming that the charge is distributed uniformly through the volume. But hit
is not very realistic, especially for conductor in which charges can move.
Conductors
In our first lecture this semester we talked about conductors and insulators:
An insulator such as glass or paper is a material in which electrons are attached to some
particular atoms and cannot move freely. On the other hand, inside a conductor, electrons
are free to move around. When a conductor acquires an excess charge, the excess charge
moves about and distributes itself about the conductor in such a manner as to reduce the
total amount of repulsive forces within the conductor. When the sum of all electrostatic
forces on the electrons is zero

F  0
then


F
E 0
q
PES 1120 Spring 2014, Spendier
Lecture 10/Page 2
then the electric field inside the material is zero and the electrons won't move around.
This happens in electrostatic equilibrium. Electrostatic equilibrium is the condition
established by charged conductors in which the excess charge has optimally distanced
itself so as to reduce the total amount of repulsive forces. Once a charged conductor has
reached the state of electrostatic equilibrium, there is no further motion of charge about
the surface.
(1) The electric field is zero inside a conductor.
If we place a solid spherical conductor in a constant external field the positive and
negative charges will move toward the polar regions of the sphere, thereby inducing an
electric field inside the conductor that points in the opposite direction of the outside
electric field. Since charges are mobile, they will continue to move until the inside field
completely cancels outside field. At electrostatic equilibrium,
must vanish inside a conductor.
(2) Any net charge must reside on the surface.
If there were a net charge inside the conductor, then by Gauss’s law, the electric field
inside the conductor would no longer be zero there. Therefore, all the net excess charge
must flow to the surface of the conductor.
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Lecture 10/Page 3

(3) E is normal to the surface just outside the conductor.
Magnifying any surface makes it look like an infinite sheet!
Example 1) Calculate the magnitude of the electric field on the surface of a
conductor
To compute the field strength just outside the conductor, consider the Gaussian pillbox:
Using Gauss’s law, we obtain
This is the magnitude of the electric field for a conducting surface.
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(4) If a neutral metal shell is placed around a charge, the positive and negative

charges rearrange so that inside the metal E  0.
Take the Gaussian surface to be in the middle of the metal shell.

Since E  0    0  qenc  0
Therefore the charges in the metal must have rearranged so that there is a surplus of
positive charges on the inside edge and a surplus of negative charges on the outside edge.
If the negative point charge in the center is -q, then +q accumulates on the inside of the
metal shell and -q on the outside.
The negative charges on the outside of the shell arrange so they are maximum distance

apart. Why? Because they do not see the inside charge in equilibrium, E  0 inside the
shell. This means the electric field outside the spherical shell is equal in magnitude for a
given radius r.
The electric field lines produced look like those for a point charge.
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Example 2) A spherical conducting shell has inner radius a and outer radius b. At the
center is a point charge +Q = 1 μC. The shell is given a total charge of +2 μC, how much
of that charge is on the inner surface and how much is on the outer surface?
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Example 3:
Concentric Spherical Shells A small conducting spherical shell with inner radius a
and outer radius b is concentric with a larger conducting spherical shell of inner radius c
and outer radius d. The inner shell has total charge +q, and the outer shell has charge +5q.
Using Gauss’s law find the total charge on
(i) the inner surface of the small shell;
(ii) the outer surface of the small shell;
(iii) the inner surface of the large shell;
(iv) the outer surface of the large shell?
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