A parallel-plate capacitor in vacuum has plates at and

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 The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

A parallel-plate capacitor in vacuum has plates at z=0z=0 and z=d=5.0 cmz=d=5.0\,\mathrm{cm}, edge effects ignored so V=V(z)V=V(z) only. The plate at z=0z=0 is earthed, V=0V=0. The plate at z=dz=d is held by a battery at V=100 VV=100\,\mathrm{V}. A probe near the earthed plate also reports dVdz∣z=0=−40 V m−1\left.\dfrac{dV}{dz}\right|_{z=0}=-40\,\mathrm{V\,m^{-1}}. Working submitted for marking reads: "The general harmonic function of zz alone is V(z)=c1z+c2V(z)=c_1z+c_2. From V(0)=0V(0)=0, c2=0c_2=0. From V(d)=100 VV(d)=100\,\mathrm{V}, c1=100/0.05=2000 V m−1c_1=100/0.05=2000\,\mathrm{V\,m^{-1}}. But the probe gives dV/dz∣0=−40 V m−1dV/dz|_0=-40\,\mathrm{V\,m^{-1}}, i.e. a second value c1=−40 V m−1c_1=-40\,\mathrm{V\,m^{-1}}. Since both come from valid boundary data, I will use their average, c1=12(2000−40)=980 V m−1c_1=\tfrac12(2000-40)=980\,\mathrm{V\,m^{-1}}, as the best estimate, so V(z)=980zV(z)=980z." Find the error in this working and state what should have been concluded instead.

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