Set 6–due 7 October

The midterm will be Friday, October 16, 7-830 PM in G-2B47.

“You can observe a lot just by watching.” – Yogi Berra

1) Jackson 2.23. [20 points] (a)–10, (b)–5, (c)–5.

2 [20 points] Write down the Dirichlet Green’s function for a two-dimensionalsquare of length a, (0 < x < a, 0 < y < a) expanding in sine waves with adouble sum, like in Eq. 3.167. Now suppose that the potential is specified to beV = 0 on all sides except the side at y = a and Φ(x, a) = V (x). Write down theappropriate formula and look at it – does it not seem to show peculiar behavioras y → a? To be definite, set V (x) = sin(πx/a), do the integral, and you’llfind Φ(x, y) = sin(πx/a)F (y). Plot partial sums of F (y) (summing say the firstn terms). What’s going on? Are there alternate version of Green’s functionswhich will not show this behavior?

3) [25 points] This is a two dimensional problem: Find the Green’s function fora line charge pointing along the z axis, located at (x′, y′) between two infinitegrounded conducting plates located at y = 0, a in two ways: (a) [10 points]First, use

δ(y − y′) =2

a

∑n

sinnπy′

asin

nπy

a(1)

and (b) [10 points] begin by writing

δ(x − x′) =1

2π

∫∞

−∞

dk exp(ik(x − x′)) (2)

It’s hard to believe that the answers to (a) and (b) would produce identicalequipotentials! (This would make a great clicker question...)(c) [5 points] Close to the wire and far away from the plates, you would expectto see Φ ∼ log r. Take one of the expressons for the Green’s function (this ismuch easier using the part (a) formula) and see if you can find G ∼ log ∆x.Take y = y′ = a/2 and ∆x = x> − x< much less than a.

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