A long duct has a square cross-section of side

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

The Boundary-Value Problem, Ch The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

a

, its four walls separate plates insulated from one another by narrow gaps at the corners, and its potential depends only on

x

and

y

. The left-hand side (

x=0

) is held at

40\,\mathrm V

; the other three sides (top, bottom, and

x=a

) are earthed at

0\,\mathrm V

. A grid of spacing

a/3

is laid on the square, giving four interior nodes at

(a/3,a/3)

(a/3,2a/3)

(2a/3,a/3)

(2a/3,2a/3)

. Because the boundary data are symmetric under reflection about the horizontal mid-line (

y \to a-y

), the two nodes nearer the live side always share one value, called

A

, and the two nodes nearer the earthed far side always share one value, called

B

. ((a)) Starting from

A=B=0

everywhere inside, carry out three sweeps of the relaxation rule (each interior value replaced by the mean of its four neighbours, using the values from the previous sweep) and give

A

and

B

after the third sweep. ((b)) Find the exact steady-state values of

A

and

B

by writing down and solving the two averaging equations directly.

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