Question #43176

This question tests the fundamental definitions of electromagnetic energy density in linear media and the rigorous application of boundary conditions.

Students often rely on the heuristic that electric and magnetic energy densities are equal ($u_E = u_B$), which holds for freely propagating EM waves in vacuum but not necessarily for static fields in a medium.

A lethal trap is incorrectly assuming the magnetic energy density formula is $u_B = \frac{1}{2}\mu B^2$ (analogous to $u_E = \frac{1}{2}\epsilon E^2$), which places $\mu_r$ in the numerator and yields a ratio of $4/5$.

Another trap is misapplying boundary conditions by assuming $\vec{H}$ is continuous across a boundary perpendicular to $\vec{B}$, which also incorrectly yields $4/5$.

By forcing the use of a diamagnetic medium ($\mu_r < 1$) with static crossed fields, the student must recognize that $B_\perp$ is continuous and correctly apply $u_B = \frac{B^2}{2\mu}$, leading to an increased magnetic energy density relative to the vacuum state.