A long rectangular duct has cross-section

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Separation of variables in Cartesian geometry

Problem

0<x<a

0<y<b

with

a=10\,\mathrm{cm}

and

b=30\,\mathrm{cm}

(

b/a=3

). The walls

x=0

x=a

and

y=0

are earthed; the wall

y=b

is held at

V_0=150\,\mathrm{V}

, with thin insulating gaps at the corners. On the centre line

x=a/2

, consider three points at perpendicular distances

\eta=5\,\mathrm{cm}

2\,\mathrm{cm}

and

0.5\,\mathrm{cm}

below the live wall (so

y=b-\eta

). Because

b\gg\eta

for all three points,

\sinh(n\pi y/a)/\sinh(n\pi b/a)\approx e^{-n\pi\eta/a}

. Using this approximation, find how many non-zero terms of the series are needed at each point so that the first omitted term is smaller than

1\,\mathrm{V}

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