A duct has cross-section

PHYSICS · Subjective · 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=0.10 ma=0.10\,\mathrm{m}, b=0.30 mb=0.30\,\mathrm{m}. Walls x=0x=0, x=ax=a, y=0y=0 are earthed. The wall y=by=b is held at V(x,b)=V0sin⁡(πx/a),V0=80 V.V(x,b)=V_0\sin(\pi x/a),\qquad V_0=80\,\mathrm{V}. Find the potential at (a/2, b/2)(a/2,\,b/2).

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