An electric dipole of length 4 cm, when placed with its axis making an angle of 60° with a uniform electric field, experiences a torque of $4\sqrt{3} N m$. If the dipole has charges of $±8 nC$, then the potential energy of the dipole in the electric field is:

$-4 J$

The correct answer is Option (3) → $-4 J$

Given:

Dipole length $l = 4 \, \text{cm} = 0.04 \, \text{m}$

Torque $\tau = 4\sqrt{3} \, \text{Nm}$

Charge $q = 8 \, \text{nC} = 8 \times 10^{-9} \, \text{C}$

Angle with field $\theta = 60^\circ$

Dipole moment: $p = q \cdot l = 8 \times 10^{-9} \cdot 0.04 = 3.2 \times 10^{-10} \, \text{C·m}$

Torque relation: $\tau = p E \sin \theta \;\Rightarrow\; E = \frac{\tau}{p \sin \theta}$

$\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$

$E = \frac{4\sqrt{3}}{3.2 \times 10^{-10} \cdot (\sqrt{3}/2)} = \frac{4}{1.6 \times 10^{-10}} \approx 2.5 \times 10^{10} \, \text{N/C}$

Potential energy: $U = -p E \cos \theta$

$\cos 60^\circ = 0.5$

$U = -(3.2 \times 10^{-10}) \cdot (2.5 \times 10^{10}) \cdot 0.5 = -4 \, \text{J}$

Answer: $U = -4 \, \text{J}$