A duct of 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 of cross-section 0<x<a0<x<a, 0<y<b0<y<b has walls x=0x=0, x=ax=a, y=0y=0 earthed and the wall y=by=b held at V0=100 VV_0=100\,\mathrm{V}, with thin insulating gaps at the corners. Consider the point (a2, b−a4)\left(\dfrac{a}{2},\, b-\dfrac{a}{4}\right), a distance a/4a/4 from the live wall on the centre line. ((a)) Evaluate the potential at this point for b=ab=a (a square duct) and for b=3ab=3a, each to within 1 V1\,\mathrm{V}, stating how many non-zero series terms are needed in each case. ((b)) The corresponding point of the long open slot (the same live wall, with the opposite earthed wall removed to infinity) has potential 54.5 V54.5\,\mathrm{V} from the closed-form sum. Comment on how the two duct values found in (a) compare with this slot value as b/ab/a increases.

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