For a duct cross-section 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

For a duct cross-section with earthed walls at

x=0

and

x=a

a=8\,\mathrm{cm}

, and the far wall

y=0

also earthed, someone argues that the choice of which ordinary differential equation carries the minus sign is "a matter of taste, since

X''/X=-Y''/Y

either way". They try

X''=+k^2X

(so

X

is built from real exponentials) with some real

k\neq0

. ((a)) Write

X(x)=Ae^{kx}+Be^{-kx}

and impose

X(0)=0

X(a)=0

. Show that the only solution is

A=B=0

. ((b)) Explain, in the light of (a), why the sign of the separation constant is fixed by the boundary conditions on the earthed walls, and say which pair of walls must carry the oscillatory factor.

Browse the problem library