Four charges of magnitude sit at and

PHYSICS · Multi-part · 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

Four charges of magnitude

+q

sit at

(\pm a,0,0)

and

(0,\pm a,0)

, and a charge

-4q

sits at the origin, all in the

xy

-plane. Someone argues: "the total charge is

4q-4q=0

, and by the symmetry of the square the dipole moment is also zero, so the potential of this arrangement is zero everywhere outside it." ((a)) Find the error in this argument. ((b)) Compute the quadrupole moment tensor

Q_{ij}

of this arrangement (it is diagonal, with

Q_{xx}=Q_{yy}

) and give the leading term of the potential, showing how it depends on the polar angle

\theta

measured from the

z

axis and on

r

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