Homework Assignments Physics 442
Assignment 1
7.1, 7.2, 7.5, 7.7, 7.11, 7.12
7.100 Read through the brief history handout and find all of the references to Michael Faraday. Use them to write a brief summary of his scientific contributions.
Assignment 2
Physics 122 problems: 1-23;
7.13, 7.21, 7.20, 7.24
Assignment 3
7.25, 7.27, 7.29
7.101 Read through the brief history handout and find as many references as you can to the discovery of electric current and resistance. Names you might want to look for are Galvani, Volta, Ohm, Davy, Cavendish, Kirchoff, Seebeck, and Peltier. This list of names may be exhaustive, or it may not. See if you can find some more.
Assignment 4
Physics 122 problems: 24-28;
7.34
7.102 Tell what Joseph Henry, Emil Lenz, and James Joule contributed to our knowledge of electromagnetism.
Assignment 5
1. The quasistatic approximation to Maxwell's equations is obtained simply by leaving out the displacement current term in Ampere's law. This is a valid approximation at low frequencies, where low frequency means that the frequency is low enough that electromagnetic waves at that frequency would have wavelengths much longer than the system being studied.
(a)
Use Maxwell's equations in this approximation, together with
Ohm's law in the form
to obtain a diffusion equation for 
:
(b) Without using any complicated Physics 318 mathematics, estimate how long it would take for current to diffuse into a long copper wire of radius 1 mm after a driving electric field has been turned on.
2. Use the diffusion equation you got in Problem 1 to study the
problem of the diffusion of current into a long cylindrical conductor.
At time t=0 an electric field 
is applied to a long conductor of
radius a and conductivity 
in the z-direction.
This electric field
is constant in time.
What happens is that initially the current in the wire flows in a very
thin layer on the outer edge of the wire, but as time goes on the current
diffuses into the bulk of the conductor.
To find out exactly how this current diffusion occurs, we solve the diffusion
equation in Problem 2 by separation of variables, as you learned in Physics 318.
The boundary conditions in this problem are a little tricky, however, so we
will do it in sort of a weird way.
(a) Find the steady state distribution of 
by setting the
time derivative in the diffusion equation to zero:
(Note: this doesn't look quite like you think 
ought to
look. The reason is that it is of the 
-component
of a vector, and this changes the operator. The above equation is
correct for the diffusion of .)
Hint: try powers of r.
When you find the steady solution for , take its curl
to find the steady solution for 
.
Finally, use Ohm's law to write this steady state solution for
the current in terms of the applied electric field .
(b) Now assume that the separable solutions for
are of the form
plug this form into the diffusion equation, and show that
where 
is a certain combination of 
,
, and 
.
Also show that the separable form for
involves the Bessel function 
.
(c) The idea now is to build the general solution for
as a superposition of many solutions from
part (b), all with different values of , plus
the final steady state solution:
What we want to have happen is to have no current
inside the metal at time t=0, so the sum part
above must cancel with the steady part at t=0.
We can get a complete set of functions to cancel
the final steady state current by choosing
the 
's so that 
.
(This part is tricky, and there may be more than
one way to choose the 
's. This one
works, so just trust me.)
To determine the 
's take the curl of
the above form for to get a similar
form for .
At t=0 we have 
, so we can multiply
the form for by 
and integrate from
0 to a.
The orthogonality of the Bessel functions makes this integral
just pick off the n=m term in the sum, and the integral
of 
against the steady state term can be gotten from
the integral
The orthogonality relation for the Bessel functions
is
Your formula for the 's will involve 
evaluated at the zeroes of .
That's OK-just let the zeroes of be
denoted by the symbol 
and get a symbolic
final solution.
Here is a partial list of the zeroes of 
in the
format 
.
Higher-order zeroes (as well as some of these) are well
approximated by the formula
(d) Now use Maple or Mathematica to make plots of current and magnetic field as functions of radius at various times after the electric field is turned on so that you can see how the current diffuses into the conductor.
Assignment 6
7.35, 7.36, 7.37, 7.41, 7.42
Assignment 7
7.45, 7.46;
Start building a DC motor with the strongest magnets you
can make, powered only by a 9-volt battery.
No permanent magnets allowed.
I will supply you with wire and some big nails, if you
want.
Assignment 8
Physics 122 problems: 29-52;
Time Dependent Circuit problems: 1-4
Assignment 9
Physics 122 problems: 53-64
Time Dependent Circuit problems: 5-8
Assignment 10
8.1, 8.3, 8.4, 8.7, 8.9, 8.10
Assignment 11
8.12, 8.18, 8.19
DC motor due.
Assignment 12
8.20, 8.21, 8.22, 8.23, 8.24
Assignment 13
8.25, 8.29, 8.30
Find formulas for the phase and group velocities of
electromagnetic plasma waves having the following
dispersion relation:
Express these velocities in terms of 
, 
and
c only; k may not appear.
(To eliminate k, use the dispersion relation.)
In particular, tell whether these two wave speeds are above
or below the speed of light.
Also compare them as approaches cutoff, i.e.,
as 
from above.
Assignment 14
8.31, 8.32, 8.33, 8.35, 8.42
Waveguide Problem:
Using the wave equations for 
and V,
the Lorentz gauge condition, and the boundary
conditions, derive the properties of TM modes
in a rectangular waveguide.
Just follow the procedure we followed in class
for the TE modes.
(You may take as given that
and 
.)
In particular, show the following:
(a)
and
(b)
(c) Show that
TM
and TM
modes cannot exist.
(d) Make some kind of a rough 3-d sketch of the electric
and magnetic fields of a TM
mode in a rectangular
waveguide.
I think the best way to do this is to make sketches of
both 
and in the yz-plane
for the y and z components of the fields, and
also a contour plot of 
in the yz-plane.
Then make a side-view sketch of the -lines
in the xyplane.
Maple and Matlab can help here.
Assignment 15
9.2, 9.3
Assignment 16
9.4, 9.5, 9.6, 9.11, 9.12
One of the ways of doing delicate
experiments in electrodynamics is to trap a small
number of charged particles in electromagnetic
traps called Penning traps.
A strong magnetic field keeps the particles from
escaping radially and electrostatic fields confine
the particles axially.
It is often desirable to cool the particles, and one
way of doing this is just to let the particles
radiate their kinetic energy away as they are accelerated
in a circle by the strong magnetic field.
Estimate the time it takes for the following charged particles
to cool appreciably (estimate this time as E/(dE/dt) where
E is the kinetic energy of the particles.)
(a) Electrons in a 1 kG magnetic field.
(b) Singly ionized Beryllium ions in a 6 T magnetic field.
Assignment 17
9.15, 9.16, 9.17
Assignment 18
9.19, 9.20, 9.22, 9.23, 9.25
Assignment 19
10.2, 10.3, 10.4, 10.9, 10.11
Assignment 20
10.13, 10.16, 10.18, 10.21, 10.24, 10.28
Assignment 21
10.29, 10.31, 10.32, 10.33
Consider two neutrons, one at rest and the other moving toward
the stationary one at 
.
Both particles are on the x-axis.
They collide, and after the collision one of them
(call it particle 1)
has positive
y-velocity and makes angle 
with the x-axis,
while the other (call it particle 2)
has negative y-velocity and makes angle
with respect to the x-axis.
(Treat both and as positive angles in this
problem.)
(a) Assume that is known and write down a set of five
equations that determine 
, 
, 
, 
and
in the final state.
(b) Change variables using 
and
(this makes the equations lots easier
to write since m and c now disappear).
Beat on your equations until you get a single
equation for 
.
Solve this single equation for 
for
between 0 and 90 degrees and graph it.
Maple should do this very nicely.
(c) Also solve for 
and 
and make graphs of them as well.
A particularly nice thing to do is to display
and 
on the same graph.
(d) Finally, make plots of 
and 
vs.
, where 
and 
are the speeds of
the two particles in the final state.
Assignment 22
10.36, 10.37, 10.39, 10.40, 10.47, 10.50, 10.53