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(A) The torque $	au$ acting on an electric dipole in an external electric field is given by $	au = pE \sin 	heta$,where $	heta$ is the angle betwe

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$pE \sin 	heta, -pE \cos 	heta$

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(A) The torque $	au$ acting on an electric dipole in an external electric field is given by $	au = pE \sin 	heta$,where $	heta$ is the angle between the dipole moment vector and the electric field vector.The potential energy $U$ of the dipole is defined as the work done by an external agent to rotate the dipole from a reference position (where $U = 0$) to the current position $	heta$. Given $U = 0$ at $	heta = 90^o$,the potential energy is:$U = -\int_{90^o}^{	heta} 	au \, d	heta = -\int_{\pi/2}^{	heta} pE \sin 	heta \, d	heta$$U = -pE [-\cos 	heta]_{\pi/2}^{	heta} = pE (\cos 	heta - \cos 90^o)$Since $\cos 90^o = 0$,we get $U = -pE \cos 	heta$.Thus,the torque is $pE \sin 	heta$ and the potential energy is $-pE \cos 	heta$.

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