Question

(a) Deduce the expression for the torque acting on a dipole of dipole moment $\overrightarrow{p}$ in the presence of a uniform electric field $\overrightarrow{E}$.

(b) Consider two hollows concentric spheres, $S_1$ and $S_2$, enclosing charges 2Q and 4Q respectively as shown in the figure. (i) Find out the ratio of the electric flux through them. (ii) How will the electric flux through the sphere $S_1$ change in a medium of dielectric constant ${}^{\prime }{\epsilon }_r^{\prime }$ if ${}^{\prime }{\epsilon }_r^{\prime }$ is introduced in the space inside $S_1$ in place of air?

Deduce the necessary expression.

Answer 1

(a) Dipole in a Uniform External Field



Consider an electric dipole consisting of charges - q and + q and of length 2a placed in a uniform electric field $\overrightarrow{E}$ making an angle $\theta$ electric field.

Force on charge - q at $A=-q\overrightarrow{E}$ (opposite to $\overrightarrow{E}$)

Force on charge +q at $B=+q\overrightarrow{E}$(along $\overrightarrow{E}$)

Electric dipole is under the action of two equal and unlike parallel force, which give rise to a torque on the dipole.

$\tau$ = Force $\times$ Perpendicular distance between the two forces

$\tau =qE\left(AN\right)=qE\left(2asin\theta \right)$

$\tau =pEsin\theta$

$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$

(b) (i) Charge enclosed by sphere $S_1=2Q$

By Gauss' law, electric flux through sphere $S_1$ is

${\Phi }_1=\frac{2Q}{{\epsilon }_0}$

Charge enclosed by sphere $S_2=2Q+4Q=6Q$

$\therefore {\Phi }_2=\frac{6Q}{{\epsilon }_0}$

The ratio of the electric flux is

$\frac{{\Phi }_1}{{\Phi }_2}=\frac{\frac{2Q}{{\epsilon }_0}}{\frac{6Q}{{\epsilon }_0}}=\frac{2_{\tau }}{6}=\frac{1}{3}$

(ii) For sphere $S_1$, the electric flux is

${\Phi }^{\prime }=\frac{2Q}{{\epsilon }_r}$

$\therefore \frac{{\Phi }^{\prime }}{{\Phi }_1}=\frac{{\epsilon }_0}{{\epsilon }_r}$

$\Rightarrow {\Phi }^{\prime }={\Phi }_1\times \frac{{\epsilon }_0}{{\epsilon }_r}$

$\because {\epsilon }_r>{\epsilon }_0$

$\therefore {\Phi }^{\prime }<{\Phi }_1$

Therefore, the electric flux through the sphere $S_1$ decreases with the introduction of the dielectric medium inside it.