A duct has cross-section ,

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 has cross-section 0<x<a0<x<a, 0<y<b0<y<b, with a=10 cma=10\,\mathrm{cm}. The walls x=0x=0, x=ax=a and y=0y=0 are earthed; the wall y=by=b is held at V0=100 VV_0=100\,\mathrm{V}, with thin insulating gaps where it meets its neighbours. Consider the point PP on the centre line x=a/2x=a/2, a fixed distance a/4=2.5 cma/4=2.5\,\mathrm{cm} below the live wall, so y=b−a/4y=b-a/4. ((a)) Using the series solution with the first three non-zero terms (n=1,3,5n=1,3,5), find the potential at PP for b=ab=a. ((b)) Repeat for b=2ab=2a. ((c)) Repeat for b=4ab=4a, then compare all three with the potential given by the long slot's closed form (b/a→∞b/a\to\infty) at the same distance a/4a/4 from the live wall, and comment on how the finite length bb affects the answer at a point fixed close to the live wall.

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