 Solution

ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2012/paper-2/section-a/a4/solution

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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