A parallel-plate capacitor in vacuum has plates of area

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

Problem

A = 2.0\times10^{-2}\,\mathrm{m^2}

, separated by

d = 1.0\,\mathrm{mm}

, with edge effects ignored so

V

depends only on the coordinate

z

between the plates (

0\le z\le d

). The plate at

z=0

is earthed. The plate at

z=d

is isolated, not connected to any battery, and carries a fixed free charge

Q_{\rm free} = 5.0\,\mathrm{nC}

. Neglect the field outside the plates, so that all of

Q_{\rm free}

sits on the top plate's lower face, the face that looks into the gap. ((a)) Write down the region

\Omega

, the equation holding inside it, and the data on each plate (

z=0

and

z=d

). ((b)) Classify the two pieces of boundary data (Dirichlet, Neumann, or the third, known-charge kind), and say whether the overall problem is Dirichlet, Neumann or mixed.

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