Question #88717

A modified tangent galvanometer contains a magnetic needle of mass $m$ and magnetic dipole moment $\vec{\mu}$. The needle is pivoted at the origin of a coordinate system $(x,y,z)$, but its center of mass is displaced from the pivot by a distance $r_0$ along the axis of the needle such that its position vector is $\vec{r}_{cm} = \frac{r_0}{\mu} \vec{\mu}$. The galvanometer is placed in a non-inertial laboratory frame which has a spatially varying acceleration $\vec{a} = \alpha x \hat{i}$, where $\alpha$ is a positive constant. The true external magnetic field $\vec{B}$ in the laboratory is uniform.

An observer in this non-inertial frame attempts to explain the equilibrium orientations of the needle purely through magnetic interactions. To do so, they define an 'effective magnetic field' $\vec{B}_{eff}$ such that the total torque on the needle is given by $\vec{\tau} = \vec{\mu} \times \vec{B}_{eff}$.

If this observer applies Gauss's Law for Magnetism to their effective field ($\nabla \cdot \vec{B}_{eff} = \mu_0 \rho_m$), what is the magnitude of the fictitious volumetric magnetic charge density $|\rho_m|$ they would deduce?