A large parallel-plate capacitor has its gap completely filled by two layers with…

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

Electricity, Magnetism and Electromagnetism, Ch Dielectrics: polarisation, bound charge and the displacement field

Problem

A large parallel-plate capacitor has its gap completely filled by two layers with faces parallel to the plates. Layer 1, next to the left plate, is an electret of thickness

t_1 = 1.0\ \mathrm{mm}

carrying a fixed uniform polarisation of magnitude

P_0 = 3.0\times10^{-6}\ \mathrm{C\,m^{-2}}

pointing from the left plate towards the right plate. Layer 2 is a linear dielectric of relative permittivity

\varepsilon_r = 4.0

and thickness

t_2 = 2.0\ \mathrm{mm}

. The left plate carries the uniform free surface charge density

+\sigma_f

and the right plate

-\sigma_f

, with

\sigma_f = 1.0\times10^{-6}\ \mathrm{C\,m^{-2}}

. Take the

x

axis from the left plate to the right plate; edge effects are ignored, so all fields are along

x

and uniform within each layer. ((a)) Using Gauss's law for

\mathbf{D}

, show that

D_x = \sigma_f

in both layers. ((b)) Obtain expressions for

E_x

in layer 1 and in layer 2 in terms of

\sigma_f

P_0

\varepsilon_0

and

\varepsilon_r

. ((c)) Evaluate

E_x

in each layer with the numbers given. ((d)) Find

V(\text{left}) - V(\text{right})

. ((e)) What free surface charge density on the left plate would make the field inside the electret zero? Explain in one sentence what this says about the bound charge.

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