Depolarization field

The electric field produced by the bound charge of a polarized body is its depolarization field. For a uniformly polarized ellipsoid in vacuum, the internal depolarization field is uniform and opposes the relevant component of the electric polarization: $E_{\mathrm{dep},i}=-N_i{P}_i/{ϵ}_0$. Its dimensionless geometric factors obey $N_x+N_y+N_z=1$.

For a homogeneous linear isotropic dielectric in vacuum, $P_i={ϵ}_0({ϵ}_r-1)E_{\mathrm{in},i}$. Combining this with the depolarization field and the imposed uniform electric field gives $E_{\mathrm{in},i}=E_{0,i}/[1+N_i({ϵ}_r-1)]$. A sphere has $N_i=1/3$. An infinite slab has $N=1$ normal to its faces and $N=0$ in its plane. A very long circular rod has $N=0$ along its axis and $N=1/2$ transverse to it. Slab and rod statements are limiting geometries, with end effects neglected; a finite cylinder does not generally have an exactly uniform internal electric field.