Physic Labs

Problem 2

A parallel-plate capacitor has plate area AA, vacuum gap dd, and is initially connected to an ideal source that holds voltage VV. A homogeneous linear dielectric of relative permittivity κ>1\kappa>1 is slowly inserted until it fills the gap; neglect fringing. Let ε0\varepsilon_0 be the vacuum permittivity and C0=ε0A/dC_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.
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Wfield=Wsource−(Uf−Ui)=12(κ−1)C0V2W_{\rm field}=W_{\rm source}-(U_f-U_i)=\frac12(\kappa-1)C_0V^2
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The source's energy becomes increased field energy plus mechanical work. Subtracting the field-energy increase from source work leaves the work delivered to the dielectric.