An electric dipole of mass m,charge q,and len | vedclass

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(D) The torque acting on the dipole in an electric field is $	au = p E_0 \sin 	heta$. For small oscillations,$\sin 	heta \approx 	heta$,so $	au =

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$2 \pi \sqrt{\frac{m l}{2 q E_0}}$

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(D) The torque acting on the dipole in an electric field is $	au = p E_0 \sin 	heta$. For small oscillations,$\sin 	heta \approx 	heta$,so $	au = p E_0 	heta$. Since $p = q l$,we have $	au = q l E_0 	heta$. The restoring torque is $	au = -I \alpha$,where $I$ is the moment of inertia about the center of mass. For a dipole of two point masses $m/2$ at distance $l/2$ from the center,$I = 2 	imes (m/2) 	imes (l/2)^2 = \frac{m l^2}{4}$. Equating the magnitudes,$I \frac{d^2 	heta}{dt^2} = -q l E_0 	heta$. Comparing with the $SHM$ equation $\frac{d^2 	heta}{dt^2} + \omega^2 	heta = 0$,we get $\omega^2 = \frac{q l E_0}{I} = \frac{q l E_0}{m l^2 / 4} = \frac{4 q E_0}{m l}$. Wait,re-evaluating the moment of inertia: If the dipole consists of two masses $m$ at ends,$I = 2 m (l/2)^2 = \frac{m l^2}{2}$. Then $\omega^2 = \frac{q l E_0}{m l^2 / 2} = \frac{2 q E_0}{m l}$. The time period $T = \frac{2 \pi}{\omega} = 2 \pi \sqrt{\frac{m l}{2 q E_0}}$.

An electric dipole of mass $m$,charge $q$,and length $l$ is placed in a uniform electric field $\vec{E} = E_0 \hat{i}$. When the dipole is rotated slightly from its equilibrium position and released,the time period of its oscillations will be:

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