A parallel-plate capacitor of plate area and gap is completely filled with a linear…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter

The Boundary-Value Problem, Ch Energy and force in dielectrics

Problem

A parallel-plate capacitor of plate area

S = 0.020\,\mathrm{m^2}

and gap

d = 2.0\,\mathrm{mm}

is completely filled with a linear dielectric of relative permittivity

\varepsilon_r = 3.0

and carries free charge

Q = 4.0\times10^{-8}\,\mathrm{C}

. Someone computes

W_{\rm free} = \tfrac12\int\mathbf{D}\cdot\mathbf{E}\,d\tau

and announces: "this is the energy stored in the electric field." ((a)) Compute

W_{\rm free}

. ((b)) Compute the energy actually held in the field alone,

\tfrac{\varepsilon_0}{2}\int E^2\,d\tau

. ((c)) Say where the rest of

W_{\rm free}

has gone, and give its value.