Work through the three derivations below in order; each uses the result of the one…
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
Work through the three derivations below in order; each uses the result of the one before it.
((a)) A region contains linear dielectric media, D=εE, and a set of conductors carrying free charge, with all fields vanishing at infinity. Free charge is brought in from infinity in small increments δρf, the surrounding dielectric settling into equilibrium at each stage. Starting from the electrostatic work done moving δρf against the potential V already present, and using ∇⋅δD=δρf, derive δWfree=∫E⋅δDdτ (the integral over all space). Then, holding the dielectric fixed (ε constant) so that only the field changes, integrate to Wfree=21∫D⋅Edτ.
((b)) A model molecule is a charge −qe bound to a fixed positive charge by an isotropic spring of force constant ks. In a uniform applied field E directed along the spring, the charge is pulled to a new equilibrium at displacement x0, giving an induced dipole moment p=αE with α=qe2/ks. Find the work Wspring stored in stretching the spring to x0, and the interaction energy Uint=−pE of the resulting dipole with the applied field. Add the two and show that the molecule's total energy in the field is −21αE2.
((c)) A set of conductors is held at fixed potentials Vi by ideal batteries, with linear dielectrics filling the space between them, and the stored energy is W=21∑iQiVi. One conductor is displaced quasi-statically through a small distance δx, its charge and the charges of the others changing so as to keep every Vi fixed. By computing the work the batteries must supply and balancing it against the change in stored energy and the mechanical work Fxδx delivered by the field, show that Fx=+(∂x∂W)V.