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

W_{\rm source}=V(Q_f-Q_i)=(\kappa-1)C_0V^2

problems.proof.analysis

The source voltage is constant, so moving charge $dQ$ requires work $VdQ$ . Integrating over the charge increase gives the result.