Question #42971

A small magnetic dipole of constant magnitude $m_0$ is fixed at the center of a stationary solid sphere of radius $R$. The sphere is made of a linear paramagnetic material with magnetic susceptibility $\chi$. The dipole is forced by an external agent to rotate with a constant angular velocity $\omega$ in the $xy$-plane, such that its magnetic moment vector is $\vec{m}(t) = m_0(\cos\omega t \hat{i} + \sin\omega t \hat{j})$. A stationary circular wire loop of radius $a$ ($a \gg R$) and electrical resistance $r_0$ is fixed in the $xz$-plane, centered at the origin. The medium outside the sphere is vacuum. Assuming the self-inductance of the loop is negligible and the system is in the quasi-static regime, what is the time-averaged external torque $\langle \tau_{ext} \rangle$ required to maintain the constant angular velocity of the dipole?