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Q.

A dipole consisting of two charges +q and -q joined by a massless rod of length $ℓ$ , is seen oscillating with a small amplitude in a uniform electric field of magnitude E. The period of oscillation

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a

$is proportional to \frac{1}{\sqrt{E}} but \neq π \sqrt{\frac{m ℓ}{3 qE}}$

b

$is proportional to 1 / E$

c

$is π \sqrt{\frac{m ℓ}{3 qE}}$

d

$is proportional to E$

answer is C.

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Detailed Solution

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$\begin{array}{l} I = \frac{m ℓ^{2}}{12} \\ Restoring torque = qE ℓ \sin θ \approx qE ℓ θ = \frac{d^{2} θ}{{dt}^{2}} \frac{m ℓ^{2}}{12} \\ ∴ ω = \sqrt{\frac{qE ℓ}{I}} = \sqrt{\frac{qE ℓ \times 12}{m ℓ^{2}}} = \frac{2 π}{T} \\ \Rightarrow T = π \sqrt{\frac{m ℓ}{3 qE}} \end{array}$

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