PHY531 Problem Set 1. Due Sept 6, 2001.

**N.B. All homework must be done in Gaussian units.**

First a warm up exercise. Do not turn this in, but be sure to do it. Download my Fortran program surface.f from the class web site. My code calculates the integral of the normal component of the electric field of a unit point charge on the surface of an ellipsoid. Gauss's law tells us that we should get if the charge is inside the ellipsoid and zero if it is outside. Read every line in this code, and then run it for at least 5 cases where the charge is inside and 5 where it is outside the ellipsoid. To get you started, compile the code. On the linux machines you can do this with the command

g77 -O2 -o surface surface.fNext you can run it and expect something like:

prompt%surface x y z of unit charge 1.0 2.0 3.0 semiaxes lengths: a b c 3.0 4.0 5.0 Nu Nv 100 100 integral over ellipsoid/4 pi = 0.99981611 prompt%which gives a result close to one as expected for a charge inside the ellipsoid. An example that puts the charge outside is:

prompt%surface x y z of unit charge 4.0 4.0 4.0 semiaxes lengths: a b c 3.0 4.0 5.0 Nu Nv 100 100 integral over ellipsoid/4 pi = 3.15063198E-06 prompt%

- Jackson Problem 1.7, the result in Gaussian units is

(1) - A filamentary circular current loop of radius is in the plane
centered on the origin. The total current is and is in the counter
clockwise direction when viewed from a position on the positive axis.
- a.
- Write an expression for in cartesian coordinates in terms of delta functions.
- b.
- Write an expression for in cylindrical coordinates in terms of delta functions.
- c.
- Write an expression for in spherical coordinates in terms of delta functions.
- d.
- Calculate explicitly

(2)

- Coulomb's law gives the electrostatic scalar potential to be

(3) - a.
- Since you can pick your coordinate system, to calculate
at any particular you can pick the coordinate system for
with along . In that case,

(4)

(5)

(6) The potential is therefore

(7) - b.
- You can also use
Gauss's law to calculate the electric field , and by integrating
obtain the electrostatic scalar potential.
Show that the Gauss's law expression

(8)

- In your math methods course you should have been (or will be)
introduced to general curvilinear orthogonal coordinate systems.
Let's apply the method to spheroidal coordinates.
We use a metric tensor, but don't be afraid. The metric tensor
here is your friend and you just have to plug into the standard
formula, Eq 11, below. The metric tensor simplifies the
ugly algebra you would otherwise get if you just ground through
the transformation of derivatives using the chain rule.
- a.
- Calculate the metric tensor for the transformation to
oblate spheroidal coordinates
defined by

(9)

and for prolate spheroidal coordinates defined by

(10)

The metric tensor is defined to be

where , , are , , and , and , , and are , , and . Explain why a diagonal metric tensor indicates an orthogonal coordinate system. - b.
- The Laplacian can be derived by using the chain rule.
For an orthogonal coordinate system, the Laplacian simplifies to

(12)

(13)

Show for oblate spheroidal coordinates we get

(14)

(15)

- Find values corresponding to constant , , or along
with the value of and
either prolate or oblate spheroidal coordinates that correspond
to
- a.
- A prolate spheroid, defined by the equation

(16) - b.
- A plane with a circular hole of radius .
- c.
- A oblate spheroid, defined by the equation

(17) - d.
- A disk of radius .

- If you know one solution to a second order linear ordinary differential
equation, you can find the general solution by integration.
To show this, assume that is a nontrivial solution to

(18) As a check, (but substituting this into the differential equation and showing it is the answer gets you zero credit), your result should be

(19) Verify that the method works by applying it to the ordinary Legendre equation

(20)

(21) - Far from
a plane conducting surface at with a circular hole of radius
centered at the origin, the electric field is for large
positive
and zero for large negative . Use oblate spheroidal coordinates
and assuming, correctly, that the potential has the same angular,
, dependence as a constant field in the direction,
so that you can use the known solution for a constant field and the
result of problem 6 to find the form of the general solution.
Calculate the charge density on the top and bottom of the sheet as a
function of the usual cylindrical radial distance.
You can check your
result by comparing with the result in
Jackson's Problem 3.25 after converting it
to Gaussian units.
Note: Make sure your potential is continuous with continuous derivatives as you cross through the hole from positive to negative .