8.02
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Spring 2013
Hand in your problem set in your section slot in the boxes outside the door of 32-082 or 26-152 depending on which is your classroom. Make sure you clearly write your name, section, and table number on your problem set.
Text: Dourmashkin, Belcher, and Liao; Introduction to E & M MIT 8.02 Course Notes
Revised.
Week Three:
Electric Potential
W03D1 T Feb 19
W03D2 W/R Feb 20/21
Problem Set 2 Due Tuesday Feb 19 at 9 pm
Monday Schedule: Faraday Law Exploration
Reading
W03D3 F Feb 22
Reading
Week Four: Equipotentials and Energy; Exam 1
W04D01 M/T Feb 26/27
Electric Potential, Discrete and Continuous Charges;
Configuration Energy
Course Notes: Sections Sections 4.1-4.3
PS03: Electric Potential
Course Notes: Sections Sections 4.7-4.10
Reading
W04D2 W/R Feb 27/28
W04D3 F Mar 1
Potential and Gauss’s Law; Equipotential Lines and
Electric Fields
Course Notes: Sections 3.3-3.4, 4.4-4.6, 4.10.5
Exam 1 Review
No Class

Problem 1: Electric Potential and Electric Potential Energy
If you place a negatively charged particle in an electric field, the charge will move
1.
from higher to lower electric potential and from lower to higher potential energy.
2.
from higher to lower electric potential and from higher to lower potential energy.
3.
from lower to higher electric potential and from lower to higher potential energy.
4.
from lower to higher electric potential and from higher to lower potential energy.
Explain your reasoning.
Problem 2 Electric Potential Difference
Four point-like objects with charges, equal to located on the vertices of a square whose sides have length origin. The
x
-axs is horizontal and the
y
-
2 Q
, -
2 Q
-axis is vertical.
, +
Q
, and
a
-
Q
. The point
, respectively, are
P
is located at the
(a) What are the x - and the y -components of the electric field
E
at point
P
?
(b) What is the electric potential
V
at point
P
, assuming that
V
=
0 at infinity?
(c) A fifth point-like object with charge +
9 Q
is slowly moved in from infinity to point P .
An external agent in moving the charged object in this process must do how much work?

Problem 3 Energy Stored in a Uniformly Charged Spherical Shell
Consider a spherical shell of radius
R
We want to know the potential energy
that has charge
U of this sphere
Q
distributed uniformly on the surface.
of charge. You can make use of the fact that outside a spherical distribution of charge, the electric field and electric potential is the same as if all the charge were at the center. We shall calculate the stored potential energy by determining the work done in assembling the charge distribution.
(a) If during this charging process, the shell has charge agent do in moving an amount of charge
dq
q
, how much work does an external
from infinity and adding it to the shell?
(b) Determine the potential energy stored in the shell when the charging the charge on the shell is
Q
. (You will need to set up an calculate a simple integral.)
(c) An interstellar dust grain, roughly spherical with a radius of negative charge such that the potential difference
V
( R )
-
V (
¥
)
R
=
3
´
10
0.15 V
-
7 m
, has acquired a
. (i) How many extra electrons has it picked up? (ii) What is the magnitude of the electric field on its surface?
Problem 4: Energy Minimization
Consider two metal shells of radius R
1
and R
2 these radii. (a) Given a total amount of charge
, quite far apart from one another compared with
Q
, which we have to divide between the shells, how should it be divided to make the potential energy of the resulting charge distribution as small as possible? Hint: Use your results from problem 3. First calculate the potential energy when charge
Q
q is placed on one shell and charge minimize the energy has a function of
q
q
is placed on the other shell. Then
. You may assume that any charge put on these shells distributes itself uniformly. (b) When you have found the optimal distribution of charge, what is the potential difference between the shells?
Problem 5: Fusion and Fission One way of thinking about fusion and fission is simply thinking about electrical effects and not considering nuclear effects.
(a) If you model a proton as a uniformly charged sphere of radius about a fermi
(
1 fm
=
1
´
10
-
15 m ), how fast would two protons have approach each other in order to fuse
(come into contact)?
(b) If you had 1 kilogram of 236
92
U
(approximate radius
7 fm
) and it underwent fission, in which each parent nucleus split into two adjacent nuclei of half the charge and half the volume, how much energy would be released? Convert this to units of kilotons of TNT
(
1 kiloton TNT
10
7
=
4.18
´
10
12
J
). It is estimated that the recent meteor (mass approximately kg
) that exploded over Siberia released energy released from 1 kilogram of 236
92
U
?
500 kilotons TNT.
. How does this compare to the

Problem 6: Energy Stored in a Spherical Distribution of Charge
(a) A spherical volume of radius
R
is filled with charge of uniform density r
.
Determine the potential energy
U of this sphere of charge by building up the sphere layer by layer, making use of the fact that outside a spherical distribution of charge, the electric field and electric potential is the same as if all the charge were at the center. In order ti understand the differential contribution to the stored energy, consider a uniformly charged sphere of radius of uniform density
r and charge density charge r
. How much charge is contained in the sphere? Now add a uniformly charged spherical shell of radius r and thickness
dr
. How much charge is contained in that shell? How much work does an external agent do in moving that amount of charge infinity and adding it to the sphere? Is this the same as the increase in potential energy
dq
dU
from
of the charge distribution? Now integrate until you have built up the entire sphere to find the potential energy
U of this sphere
of charge.
(b) At the beginning of the twentieth century the idea that the rest mass of the electron might have a purely electrical origin was very attractive, especially when the equivalence of energy and mass was revealed by special relativity. One defect of this model is that nothing holds the charge together.
Imagine the electron as a ball of charge of constant volume density r out to some maximum radius m e r
0
. Set the potential energy of this system equal to
=
9.11
´
10
-
31
kg is the mass of the electron, and
c
=
2.998
´
10
8 m
× s
-
1 m e c
2
, where
is the speed of light in vacuum. (i) Determine r
0 but the radius is doubled?
. (ii) What happens to the mass if the charge density is keep constant
Problem 7 Charged Rod
A thin rod extends along the x -axis from
x
= d to
x uniformly distributed along its length.
= d . The rod carries a positive charge
Q a) Find the electric potential at a point
x
> d along the x have chosen your zero reference point for your potential.
-axis. Indicate clearly where you b) A positively charged particle of mass point x
=
4 d
m and charge
q
is released from rest from the
. What is the speed of the particle when it reaches the point x
=
7 d ?

Problem 8: Charged Disk
A thin disk, radius 3cm surface charge density of
, has a circular hole of radius
on the disk.
1cm
in the middle. There is a uniform
(a) What is the electric potential at the center of the hole? (Assume zero potential at infinite distance form the center.)
(b)
An electron, starting from rest at the center of the hole, moves out along the axis, experiencing no forces except repulsion by the charges on the disk. What is the speed of the electron that it ultimately gains? The mass of the electron is
9.11
´
10
-
31
kg .
(c) A non-conducting disk of radius R lies in the xy -plane and carries a non-uniform charge density .
Determine the electric potential on the axis of the disk a distance z from its center. Check the limit where z R
to see that you recover the point-charge limit. Hint: the following integral may be helpful:
 dr r
2  z
2
 ln

 r
 r
2  z
2

