Question

(a) Derive an expression for the electric field E due to a dipole of length '2a' at a point at a distance r from the centre of a dipole on the axial line.
(b) Draw a graph of E versus r for r >> a.
(c) If this dipole was kept in a uniform external electric field $E_0$, diagrammatically represent the position of the dipole in stable and unstable equilibrium and write the expressions for the torque acting on the dipole in both the cases.

Answer 1

(a) Consider an electric dipole AB. The charges -q and +q of dipole are situated at A and B respectively, as shown in the figure. The separation between the charges is 2a. Electric dipole moment is given by
p = q. 2a ....(i)
Consider a point P on the axis of dipole at a distance r from mid point O of electric dipole.
The distance of point P from the charge +q at B is, BP = r-a and distance of point P from charge -q at A is AP = r + a
Let $E_1$ and $E_2$ be the electric field strengths at point P due to charges +q and -q respectively.
The resultant electric field due to electric dipole is given by
$E=E_2-E_1$

$=\frac{1}{4\pi {ϵ}_0}.\frac{q}{(r-a{)}^2}-\frac{1}{4\pi {ϵ}_0}.\frac{q}{(r+a{)}^2}E=\frac{1}{4\pi {ϵ}_0}.q[\frac{1}{(r-a{)}^2}-\frac{1}{(r+a{)}^2}]E=\frac{1}{4\pi {ϵ}_0}.q\left(\frac{(r+a{)}^2-(r-a{)}^2}{(r-a{)}^2(r+a{)}^2}\right)E=\frac{1}{4\pi {ϵ}_0}.q\left[\frac{4ra}{(r^2-a^2{)}^2}\right]E=\frac{1}{4\pi {ϵ}_0}.\frac{2pr}{(r^2-a^2{)}^2}\left[\text{from (i)}\right]$
(b) If r>> a,
$E=\frac{1}{4\pi {ϵ}_0}.\frac{2p}{r^3}$

(c) Position of dipole in stable equilibrium :

In stable equilibrium, $\theta =0^{\circ }$
$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}=pEsin\theta =pEsin{0}^{\circ }$
$\overrightarrow{\tau }=0$
Position of dipole in unstable equilibrium :
In unstable equilibrium , $\theta ={180}^{\circ }$
$\overrightarrow{\tau }=pEsin{180}^{\circ }$
$\overrightarrow{\tau }=0$