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 Incoming links: Transverse electrostatic field inside a dielectric cylinder

Past exam of the physics B course of the University of Cambridge / 2012 / Paper 2 / Section A / A4 / Solution 2026-10-20

Neglect ends and take the exterior as vacuum. With polar angle measured from the applied field, regularity inside and the far-field condition allow

ϕ_{in} = - B E_{0} s cos θ, ϕ_{out} = - E_{0} s cos θ + A E_{0} a^{2} \frac{c o s θ}{s} .

Continuity of potential gives

B = 1 - A

; continuity of normal electric displacement, with no free surface charge, gives

ϵ_{r} B = 1 + A

. Hence the transverse electrostatic field inside a dielectric cylinder is uniform, parallel to the applied field, with magnitude

E_{in} = \frac{2 E _{0}}{1 + ϵ _{r}} .

This cylindrical result differs from the spherical depolarization factor. It returns

E_{0}

for

ϵ_{r} = 1

and tends to zero for large positive

ϵ_{r}

.

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