Electric Dipoles using Dirac's Delta function

[TheWire247]

Homework Statement

In the lectures, we considered a dipole, made of two charges ±q at a separation d. Using
Dirac's δ function, write the charge density for this dipole.

Evaluate the charge (monopole moment), dipole moment, and quadrupole moments Q, p,
and Qij in the multipole expansion for this case and show that p agrees with the dipole moment.

Now consider four charges, all in the xy plane, arranged in a square, centred at the origin
and edges parallel to the coordinate axes, all of magnitude q. Two charges, at opposite ends, are positive, the other two negative.

Find the quadrupole moment for this arrangement. Explain briefly, without calculation, why
the monopole and dipole moments vanish.

Homework Equations

τ=PxE <=electric dipole equation

The Attempt at a Solution

Many of us have been staring at this problem for hours with no success. Any help or pointers in the right direction would be very much appreciated

 

[Tarti]

The charge distribution of a pure (mathematical) dipole in terms of the $\delta$-distribution is given by

$\rho_D(\mathbf r )=-\mathbf{p}\cdot\nabla\delta(\mathbf r - \mathbf r_D )$ ,

which is not hard to show if you solve Gauss's law for the electrostatic potential - can you do that calculation? Hint: confirm your result here :)

Can you now find the higher order multipole moments if you express the given charge distributions in terms of sums over $\delta$-distributions?

If you do not know how to do this, first think of a way to express a single charge using a $\delta$-distribution.