Question

An electric dipole consists of two opposite charges, each of magnitude $1.0\, \mu C$ separated by a distance of $2.0 \, cm$. The dipole is placed in an external field of $10^5 \, NC^{-1}$. The maximum torque on the dipole is

• A. $ \ce{ 0.2 \times 10^{-3} \, N m}$
• B. $ \ce{ 1 \times 10^{-3} \, N m}$
• C. $ \ce{ 2 \times 10^{-3} \, N m}$
• D. $ \ce{ 4 \times 10^{-3} \, N m}$

Answer 1

The Correct Option is D

$1 \times 10^{-6} C$ distance $=2 cm$ Exter field $=10^{5} N / C$ maximum torque on the dipole $=$ ? $\begin{array}{l} q =1 \times 10^{-6} C , \quad 2 a =2 cm \\ \text { or, }=0.02 cm \\ \therefore \quad P & = q \times 2 a \\ & =\left(1 \times 10^{-6}\right) \times 0.02 \\ & =2 \times 10^{-8} cm \end{array}$ Intensity of the external electric field, $E =1.0 \times 10^{5} N / C$
(i) $Z_{\max }=p E=\left(2 \times 10^{-8}\right)\left(10 \times 10^{5}\right)=2 \times 10^{-3} N - m$
(ii) Net work done in turning the dipole from $0^{0}$ to $180^{\circ}$
i.e $W =\int_{0^{\circ}}^{180^{\circ}} \overline{ r } d \theta=\int_{0^{\circ}}^{180^{\circ}} pE \sin \theta d \theta$
$$
\begin{array}{l}
= pE [-\cos \theta]_{0^{0}}^{180^{\circ}} \\
=- pE \left(\cos 180^{\circ}-\cos 0^{0}\right) \\
=2 pE \\
=2 \times\left(2 \times 10^{-8}\right)\left(1 \times 10^{5}\right) J \\
=4 \times 10^{-3} J
\end{array}
$$