What Is the Dipole Moment of a Paired Charged Ring System?

[rock_pepper_scissors]

Homework Statement

Consider the charge distribution of a uniformly charged ring of radius $R$ and charge $Q$ at a distance $d$ above the origin and a uniformly charged ring of radius $R$ and charge $-Q$ at a distance $d$ below the origin.

(a) Calculate the dipole moment of this distribution.

(b) Find the electric potential due to this distribution along an axis passing through the centres of the rings. Does the dipole moment you obtained from (a) make sense given your expression for the electric potential?

Homework Equations

The Attempt at a Solution

(a) My intuition is that for each infinitesimal volume element of charge in the bottom ring, I need to find its dipole moments for all the volume elements of charge in the top ring.

The dipole moment of the distribution is the sum of all these dipole moments.

Is my intuition correct?

 

[kuruman]

The elegant way would be to write the volume charge distribution using Dirac delta functions and use the definition of the dipole moment. If you are not familiar with Dirac delta functions, you might consider element dipoles, dp consisting of +dq and -dq one above the other at the same θ.