Electrodynamics (I): Homework 4 Due: October 30, 2014

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Electrodynamics (I): Homework 4
Due: October 30, 2014
Exercises in Griffiths
3.3 (hand-in 5%), 3.7, 3.8, 3.10(hand in 10%), 3.11(hand-in 5%), 3.16, 3.39(hand-in 10%),
3.41(hand in 10%), 3.42(hand in 10%), 3.47
Ex.1 hand-in
Consider an array
of infinitely Ph501
long wires
that3,lieProblem
on the xz 1
plane.
inceton University
1999
Set
Each wire carries a charg
1
density λ. The direction of wires are all in parallel to the z axis and are equally spaced
in x direction with locations being y = 0 and x = na, n = 0, ±1, ±2, · · ·, as shown in the
1. A grid of infinitely long wires is located in the (x, y) plane at y = 0, x = ±na, n =
0, 1, 2, . . ..following
Each figure.
line carries charge λ per unit length.
Obtain a series expansion for the potential φ(x, y). Show that for large y the field is
just
∞ n
z
that due to a plane of charge density λ/a. By noting that
= − ln(1 − z)
n
n=1
(a) 5 % Using method of separation of variables, find fundamental solutions
of the potentail
for
z complex or real, sum the series to show
V (x, y) for y > 0 and y < 0.
2πx
(b)10 % Consider general2πy
solutions constructed
by fundamental
solutions
−2πy/a
−4πy/a obtained in
φ(x, y) = −λ
+ ln 1 − 2e
cos
+e
a
a
(a). Using appropriate boundary
condition at y = 0, find
V (x, y) as the summation of a
2πy
2πx
mathematical series.
Resum
series to find−the
= −λ
ln the
2 cosh
. of V (x, y).
cosfunction form
a
a
(1)
Show that the equipotentials are circles for small x and y, as if each wire were alone.
1
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