Question #43277

A small, thin disk of volume $V_0$ made of a linear dielectric material with relative permittivity $\epsilon_r$ is placed in a non-uniform electric field $\vec{E} = E(x) \hat{i}$. Let $F_{e1}$ be the net electric force on the disk when its normal vector $\hat{n}$ is parallel to the $x$-axis, and $F_{e2}$ be the force when $\hat{n}$ is perpendicular to the $x$-axis.

Similarly, an identical disk made of a linear paramagnetic material with relative permeability $\mu_r$ is placed in a non-uniform magnetic field $\vec{B} = B(x) \hat{i}$. Let $F_{m1}$ be the net magnetic force on the disk when its normal vector $\hat{n}$ is parallel to the $x$-axis, and $F_{m2}$ be the force when $\hat{n}$ is perpendicular to the $x$-axis.

Assume the disk is small enough that the external fields and their gradients are uniform over its volume. Which of the following gives the correct ratios $\frac{F_{e1}}{F_{e2}}$ and $\frac{F_{m1}}{F_{m2}}$?