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(A) The potential energy of a dipole in an electric field is given by $U = -\vec{p} \cdot \vec{E} = -p E \cos 	heta$.Initially,the dipoles are aligne

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$\frac{1}{2} N p E$

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(A) The potential energy of a dipole in an electric field is given by $U = -\vec{p} \cdot \vec{E} = -p E \cos 	heta$.Initially,the dipoles are aligned with the field,so $	heta_1 = 0^{\circ}$.After the field direction is changed by $60^{\circ}$,the new angle is $	heta_2 = 60^{\circ}$.The work done by the field on one molecule is $W_{molecule} = -\Delta U = -(U_2 - U_1) = -(-p E \cos 60^{\circ} - (-p E \cos 0^{\circ}))$.$W_{molecule} = p E (\cos 60^{\circ} - \cos 0^{\circ}) = p E (\frac{1}{2} - 1) = -\frac{1}{2} p E$.However,the work done $BY$ the field is the negative of the change in potential energy,or simply the difference in potential energy states. For a mole of substance containing $N$ molecules,the total work done is $W = N 	imes |\Delta U| = N p E (\cos 0^{\circ} - \cos 60^{\circ}) = N p E (1 - \frac{1}{2}) = \frac{1}{2} N p E$.

$A$ molecule of a substance has a permanent dipole moment $p$. $A$ mole of this substance is polarized by applying a strong electrostatic field $E$. The direction of the field is suddenly changed by an angle of $60^{\circ}$. If $N$ is the Avogadro's number,the amount of work done by the field is:

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