Problem 2

A parallel-plate capacitor has plate area

A

, vacuum gap

d

, and is initially connected to an ideal source that holds voltage

V

. A homogeneous linear dielectric of relative permittivity

\kappa>1

is slowly inserted until it fills the gap; neglect fringing. Let

\varepsilon_0

be the vacuum permittivity and

C_0=\varepsilon_0A/d

. (a) Find the initial and final charge and field energy. (b) Find the work done by the source during insertion. (c) Find the mechanical work done by the field on the dielectric using energy conservation.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{Q_f=\kappa C_0V,\;U_i=\tfrac12C_0V^2,\;U_f=\tfrac12\kappa C_0V^2,\;W_{\rm source}=(\kappa-1)C_0V^2,\;W_{\rm field}=\tfrac12(\kappa-1)C_0V^2}

problems.proof.analysis

The field-energy increase is half the source work; the other half is mechanical work pulling the dielectric inward. The expressions have the required charge and energy dimensions.

problems.proof.pitfall. Keep the sign convention consistent and retain only physically meaningful roots or directions.