Transverse electrostatic field inside a dielectric cylinder ($E_{\mathrm{in}}=2E_0/(1+{ϵ}_r)$)

For an infinitely long circular dielectric cylinder in a uniform transverse field in vacuum, the inside electrostatic potential is linear in the transverse coordinate. Matching the tangential electric field and normal electric displacement field to the outside dipolar solution gives the displayed uniform internal field. The result assumes no free interface charge and neglects end effects; an exterior dielectric changes the permittivity ratio.