A duct's cross-section occupies the strip with earthed walls at and

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics

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

Problem

A duct's cross-section occupies the strip

0\le x\le a

with earthed walls at

x=0

and

x=a

, and extends upward without limit in

y

: there is no wall at any finite

y

. The wall at

y=0

is held at some potential

V(x,0)=f(x)

whose detailed shape does not matter here. Because the duct is open at large

y

, the physical requirement replacing a fourth wall is that

V(x,y)\to0

y\to\infty

for every

x

(0,a)

, with

V

bounded throughout. ((a)) Substitute

V(x,y)=X(x)Y(y)

and separate Laplace's equation, writing the two ordinary equations in terms of one constant

\mu

. ((b)) Using only the conditions

X(0)=X(a)=0

on the earthed side walls, show by testing both signs of

\mu

that

\mu

must be negative, and find the allowed values of

k

where

\mu=-k^2

. ((c)) With

\mu=-k^2

, the equation for

Y

Y''=k^2Y

, whose general solution is a sum of two exponentials. Show that only one combination is consistent with

V\to0

y\to\infty

, and write it down.

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