Question

Derive an expression for electric field due electric dipole at a point on an equatorial line.

Answer 1

Electric field at A due to the charges $q$ and $-q$ is shown in the above figure. We resolve $E$ into horizontal and vertical components. The vertical components $\left(E\sin \theta \right)$ cancel out each other and only the horizontal components survive to give net electric field at A as $2E\cos \theta$.

Electric field at point A, $E_A=2E\cos \theta$

where $E=\frac{q}{4\pi {ϵ}_o(r^2+a^2)}$

We get $E_A=\frac{2q}{4\pi {ϵ}_o(r^2+a^2)}\cos \theta$

From figure, we have $\cos \theta =\frac{a}{\sqrt{r^2+a^2}}$

$\therefore$ $E_A=\frac{2qa}{4\pi {ϵ}_o(r^2+a^2{)}^{3/2}}$

We know dipole moment $P=2qa$

$⟹$ $E_A=\frac{P}{4\pi {ϵ}_o(r^2+a^2{)}^{3/2}}$

For $a<<r$, we can neglect $a^2$ compared to $r^2$.

$⟹$ $E_A=\frac{P}{4\pi {ϵ}_o{r}^3}$