A free point charge is embedded in a thick block of a uniform linear dielectric of…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter

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

Problem

A free point charge q=5.0 nCq = 5.0\,\mathrm{nC} is embedded in a thick block of a uniform linear dielectric of relative permittivity εr=4.0\varepsilon_r = 4.0, at a depth h=2.0 mmh = 2.0\,\mathrm{mm} below the block's flat face; above the face there is vacuum. A student draws a sphere of radius r=1.0 mmr = 1.0\,\mathrm{mm} centred on the charge, so that the sphere lies entirely inside the dielectric, and writes D=q4πr2 at every point of that sphere,D = \frac{q}{4\pi r^2}\ \text{at every point of that sphere}, arguing that Gauss's law for D\mathbf{D} involves free charge only and the only free charge is the point charge. Which one of the following is the correct assessment? (a) The flux of D\mathbf{D} out of the sphere is qq, but DD is not q/4πr2q/4\pi r^2 at each point of it, because the arrangement is not spherically symmetric: the bound charge on the block's face is not distributed symmetrically about the charge. (b) This is right: D=q/4πr2D = q/4\pi r^2 on the sphere, because the sources of D\mathbf{D} are the free charges only and the free charge here is a single point charge. (c) The flux of D\mathbf{D} out of the sphere is q/εrq/\varepsilon_r, because the sphere lies wholly inside a medium of relative permittivity εr\varepsilon_r. (d) The flux of D\mathbf{D} out of the sphere is zero, because the bound charge induced around the point charge cancels it.

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