An earthed conducting cylinder of radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Cylindrical geometry, edges, and what a field does to a dipole

Problem

a = 3.0\,\mathrm{cm}

, infinitely long, is placed with its axis along

z

in a region that would otherwise carry a uniform field

E_0 = 2.0\times10^5\,\mathrm{V\,m^{-1}}

pointing along

x

(transverse to the cylinder's axis, so the unperturbed potential is

V_{\rm ext} = -E_0 s\cos\phi

). ((a)) Starting from the cylindrical Laplacian, separate Laplace's equation with no dependence on

z

and no free charge in the region

s \ge a

, and write down the radial equation that results. ((b)) Write the general two-dimensional exterior solution and argue, from the required behaviour as

s\to\infty

and the symmetry of the problem, which single angular term (

m

) survives and what its far-field coefficient must be. ((c)) Apply the boundary condition on the cylinder's surface to fix the remaining coefficient, and hence write

V(s,\phi)

for

s \ge a

. ((d)) From

V(s,\phi)

find the radial field

E_s(s,\phi)

, and give its largest value on the cylinder's surface.

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