Define dipole moment of an electric dipole. Show mathematically that the electric field intensity due to a short dipole at a distance r along its axis is twice the intensity at the same distance along the equatorial axis.

The distance between two negatively-charged dust particles is $1 \mathrm{mm}$ and they repel each other with a force of $4.0\times {10}^{-5} \mathrm{N}$. If the charge on one particle is  $4$ times that of the other, then find out the number of electrons charging them separately. The charge on electron is $\left(5/3\right)\times {10}^{-19 }\mathrm{C}$.