Problem Set 1: Chapter 1 Probs. 1, 2, and 3
NOTES
Problem 1.1. Please be very precise in stating what you assume and exactly what you have proved. You need to assume something about a conductor; let us take as the definition of an ideal conductor that the electric field is zero everywhere in the conductor.
Problem 1.2. In order to prove the general theorem about delta functions located at an arbitrary place (not the origin) you need to extend the definition of the delta function given to
D(alpha; x,y,z; x',y',z') = lim(alpha -> 0) (2*pi)^(-3/2)*alpha^-3*exp(-((x-x')^2+(y-y')^2+(z-z')^2)/ (2*alpha^2)))Problem 1.3. In parts (a)-(c) you may be able to guess the form of the charge density; then you should verify that the integral over all space gives the correct total charge. But in part (d) it is a non- trivial matter to write down a charge density that is in fact uniform over the disk. You need to IMPOSE the condition that the integral of the charge density over any small area of the disk is proportional to the area.
RWB 2-25-96
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Problem Set 2: Ch. 1 Prob. 5 and 6.
Problem 1.5. There are two ways to approach this problem. If you set out to take the Laplacian using the derivatives in terms of r, theta, and phi, this can work, but beware! You are likely to miss something at the origin, because so many terms blow up there.
The other way is to us formal vector calculus without reference to a particular coordinate system. You need to develop some tools, but you have one very important one already:
laplacian(1/r) = -4 pi delta(rvector)Another is
grad(1/r) = - rvector/r^3and so on.
Problem 1-6. Instead of part (d) as given in the text, substitute "(d) What is the capacitance per unit length, in pF/m, of RG58-U coaxial cable? Take the permitivity of the dielectric to be 2.5 times that of air." [If you are not reading this in cyberspace, there should be a piece of RG58-U coax attached; if not, cut a piece out of your cable-TV line and use that.]
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Problem Set 3: Chapter 1 P 10; Chapter 2 P 1
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Problem Set 4: Chapter 2 Problems 3, 13; and,
verify equation 2.5 in Jackson.
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Problem Set 5: Ch 3 P1, and Special Prob. 3.01
Special Problem 3.01. Consider the functions P(xi) discussed in section 3.4 of Jackson.
(a) Substitute the power-series form given, P(xi) = xi^(alpha) Sum(j=0,inf)[aj xi^j] , into equation (3.39) and, by setting equal the coefficients of powers of x, derive the recursion relation for the coefficients, equation (3.40).
(b) Discuss convergence of the series.
(c) How are these functions related to those found on pages 86-87 of Jackson?
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Problem Set 6: Ch. 3 P 2,6,11
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Problem Set 7: Ch. 3 P 12 and Ch. 4 P 1
***** Note: Skip this problem set spring 98, due to computer crash.
Chapter 3 Problem 12. The idea is to try to use the
horrendous Green's function for the double sphere, eq. (3.125) of
Jackson. First get rid of the phi dependence. Remember that
Yl0(theta,phi) = sqrt((2l+1)/(4 pi)) Pl(cos theta)
We are talking about Dirichlet boundary conditions, so you
need to take the partial derivative wrt r of the Green's function - not
too hard.
Then when you do the integral, it should reduce to the same
integral over x that we did in problem 1. I recommend the same trick of
writing the potential on the boundary surface as the sum of a constant
term (V/2 in this case) plus a symmetrical step function like the
example worked through by Jackson.
In case you don't have it, here is what I think is the correct solution to Prob. 1:
Phi = V/2 - V/2 [Sum over l odd] (2l+1) gamma-l Pl(cos theta) *
{(b^(l+1)+a^(l+1))/(b^(2l+1)-a^(2l+1))*r^l -
(b^l+a^l)b^(l+1)a^(l+1)/(b^(2l+1)-a^(2l+1))/r^(l+1)}
where
gamma-l == (integral from 0 to 1) Pl(x)dx
= (-1/2)^((l-1)/2)(l-2)!!/(2((l+1)/2)!)
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Problem Set 8: Ch 4 P 1; Ch. 5 P 2,3
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Problem Set 9: Ch. 5 P 8,11
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Problem Set 10: Ch. 6 P 1,3
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Problem Set 11: Ch. 11 P 1
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Problem Set 12: Ch. 11 2,4,P 16 (note: c=1 units are used), 19
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Problem Set 13: Ch. 11 P 11,13