Physics 834: Problem Set #9
Here are some hints, suggestions, and comments on the assignment.
Remember to keep track of the amount of time you spend doing the
(entire) assignment and record this number on your problem solution.
Recent changes to this page:
- 20-Nov-2011 --- Original version.
- Electrostatic potential in hemisphere.
This is a boundary value problem: solve Laplace's equation with
the given boundary conditions. This problem has a lot in common
with the example worked out on pages 142-144 of the Lecture 15
notes (we'll review this example on Monday). That problem had
a full sphere rather than a hemisphere, but the "trick" here is
to solve a full sphere problem that will give you the hemisphere
solution as a byproduct. (Remember that to solve a differential
equation you just need a solution that satisfies the equation
and the boundary conditions.) So how could you choose the
potential on the top and bottom halves of a sphere so that
the top half is V0 and the slice through the
middle is zero? Then look for a solution as in the example.
To solve the necessary integrals, you can use Mathematica but
try also using a recursion relation that let's you do the
integrals directly (because they are total derivatives).
NOTE: It is perfectly ok to solve the hemisphere directly by
imposing the boundary condition on the flat part as opposed to
using the "trick" described above.
- Expansion and application.
- The first part is quite straightforward, once you realize
what r< and r>
let you do. They solve the problem of knowing which of r
and r' is smaller, so you know how to expand the
square root in the generating function. Then it's pretty
much just substitution.
- The application requires you to know that the magnetic
vector potential for a wire with current density J(x)
is μ0/(4π) times the integral of J(x)
times equation (3). So first you have to find
J(x) for a circular loop of radius a that
carries current I. This should involve a couple of
δ functions. Then substitute this and the expansion
for 1/|x-x'|. You can use the delta functions
for two integrals, but you'll have to do the φ integral.
Simplify your final result using results such as Equation (8.53)
- Heating (or cooling) a sphere.
Another boundary value problem. Do a full separation of
variables on this problem, taking into account that we (apparently)
have spherical symmetry. (So what is the dependence on
θ and φ?) When finding solutions to the separated
equations, don't forget the case where the separation constant
is zero (you'll need this to match the boundary conditions!).
Your final answer should be in the form of a sum over
functions in r times functions in t, with all
- Current in a conducting sheet.
- In steady state, how does the charge density change with
time? (Or does it?).
- If we have a circular copper plate, what three
dimensional coordinate system is most appropriate?
You should be able to follow section 8.4 in Lea.
What do you expect for the z dependence of j
- Given the general expansion from part (b), use the
boundary conditions to determine the coefficients.
Try to do the plots in Mathematica using
- Harmonic oscillator.
- Play the game of multiplying by a positive function
p(x) and requiring the new equation have the
standard Sturm-Liouville form, as in the last problem set.
- The weight function should come out as a by-product of
solving part (a). Review the orthonality proof to verify that
the surface terms vanish in this case.
- Revisit our series solution methods. When you get a recurrence
relation for the coefficients of your Taylor expansion, see what
it reduces to for large n (if n is the index of
your expansion). You should be able to sum the series based
on that simplified form, which tells you the asymptotic
form of the series. Does it go to zero for large x?
If not, we need to require the series to terminate, which determines
the possible values of the eigenvalue λ.
- Just plug into your solution and normalize according to
the guideline in the problem.
Physics 834: Assignment #9 hints.
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