A point charge sits at height above the flat face of a semi-infinite linear…
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
A point charge q sits at height d above the flat face of a semi-infinite linear dielectric of permittivity ε=εrε0 filling the region z<0; the region z>0 is vacuum. There is no free surface charge on the interface.
((a)) Guess the image system: for z>0, the real charge q at (0,0,d) plus a single image charge q′ at (0,0,−d), both radiating into vacuum permittivity ε0; for z<0, a single charge q′′ at (0,0,d) radiating into a uniform medium of permittivity ε filling all space. Impose continuity of V and of the normal component of D at z=0 and solve for q′ and q′′ in terms of q and εr.
((b)) Show that as εr→∞, q′→−q and identify the limit.
((c)) The sphere response factor has εr+2 in its denominator. Identify, from the matching algebra of part (a), the single number that plays the analogous role for this flat-interface problem, and explain in one sentence why it is smaller than the sphere's.