Two results above both come out with a permittivity ratio in a denominator: the…
PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Electric fields in matter
The Boundary-Value Problem, Ch Linear dielectrics and the boundary-value problem with matter in it
Problem
Two results above both come out with a permittivity ratio in a denominator: the sphere response factor (εr−1)/(εr+2) and the half-space image charge, whose ratio is (εr−1)/(εr+1). This problem derives both and asks why the two denominators differ.
((a)) A sphere of uniform linear dielectric, relative permittivity εr, sits in a uniform applied field E0. Its own uniform polarisation produces a uniform depolarising field −P/3ε0 inside the sphere, and P=ε0(εr−1)Ein. Use self-consistency, Ein=E0−P/3ε0, to derive Ein=εr+23E0 and identify the sphere response factor (εr−1)/(εr+2).
((b)) A point charge q stands at height d above a flat dielectric half-space of permittivity ε=ε0εr filling z<0, with vacuum in z>0 and no free charge on the interface. Take the potential in z>0 to be that of q at (0,0,d) plus an image q′ at (0,0,−d), and the potential in z<0 to be that of a single charge q′′ at (0,0,d) in a medium of permittivity ε filling all space. Impose continuity of V and of the normal component of D at z=0 for every point on the plane, and derive q′=−qεr+1εr−1.
((c)) Both denominators are of the form εr−1+(constant): 3 for the sphere, 2 for the image charge. In two or three sentences, explain why the constants differ: name the physical feature the sphere problem has that the flat-interface problem does not.
((d)) Evaluate, for εr=3: the ratio Ein/E0, the sphere response factor, and q′/q.