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 hollow concentric sphere, $S_1$ 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 if a medium of dielectric constant '${ϵ}_r$' is introduced in the space inside $S_1$ in place of air?
Deduce the necessary expression.

Answer 1

(a)

Torque due to force acting on +q is ${\tau }_1=-qE\frac{d}{2}sin\theta \hat{k}$

Torque due to force acting on -q is ${\tau }_2=-qE\frac{d}{2}sin\theta \hat{k}$

Total torque acting on dipole is $\tau ={\tau }_1+{\tau }_2$

$\tau =-qEd\sin \theta \hat{k}$

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

(b)

(i)

Flux enclosed in $S_1$, ${\varphi }_1=\frac{2Q}{{\epsilon }_o}$

Flux enclosed in $S_2$, ${\varphi }_2=\frac{6Q}{{\epsilon }_o}$

Ratio of flux, $\frac{{\varphi }_1}{{\varphi }_2}=\frac{1}{3}$

(ii)

For medium ${\epsilon }_r$,

Flux enclosed in $S_1$, ${\varphi }_1=\frac{2Q}{{\epsilon }_o{\epsilon }_r}$

Hence, flux enclosed will reduce.