p7310_6
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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-dimensional square of length a, (0 <x<a,0 <y<a) expanding in sine waves with a double sum, like in Eq. 3.167. Now suppose that the potential is specified to be V = 0 on all sides except the side at y = a and Φ(x, a)= V (x). Write down the appropriate formula and look at it – does it not seem to show peculiar behavior as y → a? To be definite, set V (x) = sin(πx/a), do the integral, and you’ll find Φ(x, y) = sin(πx/a)F (y). Plot partial sums of F (y) (summing say the first n terms). What’s going on? Are there alternate version of Green’s functions which will not show this behavior? 3) [25 points] This is a two dimensional problem: Find the Green’s function for a line charge pointing along the z axis, located at (x ′ ,y ′ ) between two infinite grounded conducting plates located at y =0,a in two ways: (a) [10 points] First, use δ(y - y ′ )= 2 a n sin nπy ′ a sin 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 identical equipotentials! (This would make a great clicker question...) (c) [5 points] Close to the wire and far away from the plates, you would expect to see Φ ∼ log r. Take one of the expressons for the Green’s function (this is much easier using the part (a) formula) and see if you can find G ∼ log Δx. Take y = y ′ = a/2 and Δx = x > - x < much less than a. 1
description
electrodynamics
Transcript of p7310_6
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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