Question: Q. 7. An electric dipole is held in a uniform electric field.
(i) Show that the net force acting on it is zero.
(ii) The dipole is aligned parallel to the field. Find the work done in rotating it through the angle of $180^{\circ}$.
R [O.D. I, II, III 2012]
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Solution:
Ans. (i)
$$ \begin{aligned} & \vec{F}{\text {net }}=\vec{F}{1}+\vec{F}{2} \ \therefore \quad & \vec{F}{\text {net }}=\overrightarrow{0} \end{aligned} $$
(ii)
$$ \begin{array}{ll} W=-E p\left(\cos \theta_{1}-\cos \theta_{2}\right) & 1 / 2 \ W=2 E p & 1 / 2 \end{array} $$
[CBSE Marking Scheme, 2012]
Detailed Answer :
(i) In an electric dipole with two equal and opposite point charges, $-q$ at point $A$ and $+q$ at point $B$ with distance $2 a$ as shown, so $A B=2 a$ with dipole moment $|\vec{P}|=q(2 a)$
If dipole is kept in an uniform external electric field $\vec{E}$ at angle $\theta$ with direction of $\vec{E}$, then :
(a) force on charge $-q$ at point $A=-q \vec{E}$, opposite to $\vec{E}$ (b) force on charge $+q$ at point $B=+q \vec{E}$, along $\vec{E}$
Hence, net force on dipole will be given as:
$q E-q E=0$
(ii) Work done,
$$ \begin{aligned} W & =E p(1-\cos \theta) \ & =E p\left(1-\cos 180^{\circ}\right) \ & =2 E p \text { joules } \end{aligned} $$
or
Work done on dipole, $W=\Delta U$
$$ \begin{align*} & =p E\left(\cos \theta_{1}-\cos \theta_{2}\right) \ W & =p E\left(\cos 0^{\circ}-\cos 180^{\circ}\right) \ W & =2 p E \tag{1} \end{align*} $$
