A collection of point charges

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential

The Boundary-Value Problem, Ch The multipole expansion of the potential

Problem

A collection of point charges, of unspecified individual sizes but summing to a total charge

Q=6.0\,\mathrm{nC}

, is fixed at various positions around a circular ring of radius

b=5.0\,\mathrm{cm}

that lies in the

xy

-plane, centred on the origin. Find the potential at a point on the

z

-axis a distance

z=1.5\,\mathrm{cm}

from the centre, using the interior multipole expansion kept to and including its

\ell=2

term. Compare with the exact potential of a total charge

Q

smeared uniformly around such a ring,

V=Q/4\pi\varepsilon_0\sqrt{z^2+b^2}

, and comment on the size of the discrepancy.

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