Dielectrics and Gauss' Law — Question 10

Question 10

A potential profile can expose an impossible charge-free dielectric model

A plate gap has prescribed potential $V(z)=V_0[1-(z/d)^2]$ for $0<z<d$, with V0>0. First suppose there is no free bulk charge and the lower electrode supplies a nonzero positive constant displacement D0.

1. Infer the required relative permittivity and determine whether it stays finite at z=0.

2. Instead impose uniform finite kappa. Find the free volume density and both electrode free densities, assuming no exterior field.

Question 10 - Solution

A potential profile can expose an impossible charge-free dielectric model

(a) Method and calculation

$$E_z=-dV/dz=2V_0z/d^2,\quad D_z=D_0.$$$$\kappa(z)=\frac{D_0d^2}{2\epsilon_0V_0z}\to\infty\quad(z\to0^+).$$

(b) Interpretation and further calculation

$$D_z=2\epsilon_0\kappa V_0z/d^2\Rightarrow\rho_f=2\epsilon_0\kappa V_0/d^2.$$$$\sigma_{f,0}=D_z(0)=0,\quad\sigma_{f,d}=-D_z(d)=-2\epsilon_0\kappa V_0/d,$$which cancels the integrated free bulk charge per area.

Physical check

Potential, material law and source distribution cannot be specified independently; Gauss law supplies the consistency constraint.