Question #42754

A composite point dipole consisting of an electric dipole moment $\vec{p}$ and a magnetic dipole moment $\vec{m}$ is located at position $\vec{r} = d \hat{i}$. A point charge $+Q$ is fixed at the origin. A non-uniform external magnetic field $\vec{B} = \alpha (x \hat{i} - y \hat{j})$ exists in the region, where $\alpha$ is a positive constant. The dipole moments are parallel to each other ($\vec{p} \parallel \vec{m}$) and lie in the $xy$-plane, making an angle $\theta$ with the positive $x$-axis. An external agent rotates the composite dipole quasi-statically about its center of mass (which remains fixed at $\vec{r} = d \hat{i}$) from $\theta = 0$ to $\theta = \pi/2$. Let $W$ be the total work done by the external agent during this rotation, and let $|\vec{\tau}_O|$ be the magnitude of the net torque exerted by the electric and magnetic fields on the composite dipole about the origin at an arbitrary angle $\theta$. Which of the following gives the correct expressions for $W$ and $|\vec{\tau}_O|$?