An electric dipole (dipole moment = p) is placed in a uniform electric field in stable equilibrium position at rest. Now it is rotated by a small angle and released. The time after which it comes to t

Question

An electric dipole (dipole moment = p) is placed in a uniform electric field in stable equilibrium position at rest. Now it is rotated by a small angle and released. The time after which it comes to the equilibrium position again (for first time) is t. If the moment of inertia of the dipole about the axis of rotation is x π 2 t 2 pE ​ , then find the value of x.

TestHub · Accepted Answer

For small oscillations of a dipole in a uniform field, $τ = - pEθ$ , so $T = 2 π \frac{I}{pE}$ .

The first return to equilibrium takes $t = \frac{T}{4} = \frac{π}{2} \frac{I}{pE}$ , giving $I = \frac{4 t ^{2} pE}{π ^{2}}$ .

Comparing with $I = x \frac{t ^{2} pE}{π ^{2}}$ , $x = 4$ .

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