A long conducting duct has 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 Separation of variables in Cartesian geometry

Problem

a=12\,\mathrm{cm}

. Two adjacent walls,

x=0

and

y=0

, are earthed. The other two,

x=a

and

y=a

, which meet at the corner

(a,a)

, are both held at the same potential

V_0=90\,\mathrm{V}

; thin insulating strips separate each live wall from the earthed wall it touches (there is no potential jump at the shared corner

(a,a)

, since both walls there are at

V_0

). ((a)) Decompose the problem into two one-live-wall duct problems, each of the type already solved: problem A has

x=a

V_0

with

x=0

y=0

and

y=a

earthed; problem B has

y=a

V_0

with

x=0

x=a

and

y=0

earthed. Write down the series solution for each. ((b)) Without summing any series, use the four-fold superposition of the single-live-wall problem (four rotated copies of one live wall adding to

V_0

everywhere, each contributing exactly

V_0/4

at the centre) to state the exact potential at the centre

(a/2,a/2)

of this duct with two adjacent live walls. ((c)) Using the first two non-zero terms of each series, evaluate

V

(3a/4,\,a/4)

, one quarter of the way in from the

y=0

wall and three-quarters of the way across towards the

x=a

wall.

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