Question 831: Electrostats & Capacitance

Question

An electric dipole of moment $\overrightarrow{\mathrm{p}}$ is lying along a uniform electric field $\overrightarrow{\mathrm{E}}$. The work done in rotating the dipole through $\frac{\pi^{\mathrm{c}}}{3}$ is $\left[\sin 30^{\circ}=\cos 60^{\circ}=0 \cdot 5, \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2\right]$

Accepted Answer

To find the work done in rotating an electric dipole in a uniform electric field, use the formula:
$$ W = -\Delta U = - (U_f - U_i), $$
where $U$ denotes the potential energy of the dipole in the electric field, which is given by:
$$ U = -\overrightarrow{p} \cdot \overrightarrow{E} = -pE \cos 	heta, $$
with $	heta$ being the angle between the dipole moment $\overrightarrow{p}$ and the electric field $\overrightarrow{E}$.
Initially, the dipole is aligned with the field, so $	heta_i = 0$, and
$$ U_i = -pE \cos 0 = -pE. $$
After rotating through an angle of $\frac{\pi}{3}$, $	heta_f = \frac{\pi}{3}$, and
$$ U_f = -pE \cos\left(\frac{\pi}{3}ight) = -pE \cdot \frac{1}{2}. $$
The change in potential energy, $\Delta U$, is then:
$$ \Delta U = U_f - U_i = \left(-pE \cdot \frac{1}{2}ight) - (-pE) = -pE \cdot \frac{1}{2} + pE = \frac{pE}{2}. $$
Therefore, the work done is:
$$ W = -\Delta U = -\left(\frac{pE}{2}ight) = \frac{pE}{2}. $$
The correct answer is Option D:
$$ \frac{pE}{2}. $$

Answer

3 pE

Answer

$\sqrt{2} \mathrm{pE}$

Answer

pE

Answer

$\frac{\mathrm{pE}}{2}$

Electrostats & Capacitance

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