The long open slot has earthed side walls

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

The long open slot has earthed side walls x=0x=0, x=ax=a with a=10 cma=10\,\mathrm{cm}, and a live wall at η=0\eta=0 (where η\eta is the perpendicular distance into the slot from the live wall) held at V0=100 VV_0=100\,\mathrm{V}; the slot is open as η→∞\eta\to\infty. Its closed-form potential is V(x,η)=2V0πarctan⁡ ⁣[sin⁡(πx/a)sinh⁡(πη/a)].V(x,\eta)=\frac{2V_0}{\pi}\arctan\!\left[\frac{\sin(\pi x/a)}{\sinh(\pi\eta/a)}\right]. ((a)) On the centre line x=a/2x=a/2, find η\eta for V=90%V=90\%, 75%75\%, 50%50\% and 25%25\% of V0V_0, and describe the shape of the corresponding equipotentials, in particular how they behave near the two corners of the live wall. ((b)) The equivalent closed duct's series needs, for one-volt accuracy on the centre line, 1 term at η=a/2\eta=a/2, 2 terms at η=a/4\eta=a/4, 4 terms at η=a/10\eta=a/10, and 14 terms at η=a/100\eta=a/100. Using these counts, state the two regions where a five-term truncation of that series would still be visibly wrong, and mark them on the sketch.

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