Question #42966

A rectangular container of length $L$ is placed along the $x$-axis from $x = 0$ to $x = L$. It is filled with an incompressible diamagnetic fluid of density $\rho$ and magnetic susceptibility $\chi$ (where $\chi < 0$). The system is permeated by a non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$, where $B_0$ is a positive constant and $\hat{k}$ is the unit vector in the vertically upward $z$-direction. The container is given a constant horizontal acceleration $\vec{a} = -a_0 \hat{i}$ ($a_0 > 0$). In the steady state, the free surface of the fluid is observed to be perfectly horizontal at the edge $x = L$. Assuming the fluid does not spill, what is the difference in height of the fluid surface between the edges $x = L$ and $x = 0$, given by $\Delta h = h(L) - h(0)$? (Here $g$ is the acceleration due to gravity and $\mu_0$ is the vacuum permeability).