Question

Derive an expression for the electric potential at any point along the axial line of an electric dipole.

Answer 1

The electric field due to a dipole on the axial line.
- you can see that, if the point $P$ is far away i.e., $r>>l$, then
we can reduce the expression to $(2.K.q.l)/r^3$
where $q.l$ is defined as the electric dipole moment of the dipole of length $l$
If $P$ is outside the dipole,
$\overrightarrow{E}$ at $P$ due to $+q=\frac{kq}{(r-\frac{l}{2}{)}^2}\left(\hat{x}\right)$
$\overrightarrow{E}$ at $P$ due to $-q=\frac{+kq}{(r+\frac{l}{2}{)}^2}(-\hat{x})$
Total electric field at $P,E=kq[\frac{1}{(r-\frac{l}{2}{)}^2}-\frac{l}{(r+\frac{l}{2}{)}^2}]$
$E=\frac{2kqlr}{(r^2-\frac{l^2}{4}{)}^2}\left(\hat{x}\right)$