An electric dipole of mass m , charge q , and length $l$ is placed in a uniform electric field $\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \hat{\mathrm{i}}$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

(D) $2 \pi \sqrt{\frac{\mathrm{~m} l}{2 \mathrm{qE}_{0}}}$

(A) $\frac{1}{2 \pi} \sqrt{\frac{2 \mathrm{~m} l}{\mathrm{qE}_{0}}}$

(B) $2 \pi \sqrt{\frac{\mathrm{~m} l}{\mathrm{qE}_{0}}}$

(C) $\frac{1}{2 \pi} \sqrt{\frac{\mathrm{~m} l}{2 \mathrm{qE}_{0}}}$

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Elasticity & SHM

An electric dipole of mass m , charge q , and length

l

is placed in a uniform electric field

E = E_{0} \hat{i}

. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

Answer: (D)

A dipole consists of two charges

+ q

and

- q

, each of mass

m

, fixed at the ends of a rigid massless rod of length

l

, placed in a uniform field

E = E_{0} \hat{i}

. At equilibrium the dipole moment

p = q l

is aligned along

E

.

Displacing the dipole by a small angle

θ

, the restoring torque about the centre is

τ = - p E_{0} sin θ \approx - q l E_{0} θ

The moment of inertia about the centre, with a charge of mass

m

at each end, a distance

l /2

from the centre, is

I = 2 m (\frac{l}{2})^{2} = \frac{m l ^{2}}{2}

Newton's second law for rotation gives

I \ddot{θ} = - q l E_{0} θ ⟹ \ddot{θ} = - \frac{2 q E _{0}}{m l} θ

which is SHM with

ω^{2} = \frac{2 q E _{0}}{m l}

The time period is

T = \frac{2 π}{ω} = 2 π \frac{m l}{2 q E _{0}}

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Single correct option