Energy Density of a Magnetic Field — Question 8

Question 8

Superposed magnetic fields require an energy cross term

Two uniform vacuum fields each have magnitude B and enclose angle theta in the same volume. The physical energy density is computed from their vector sum.

1. Derive the total density, displaying the cross term.

2. Compare parallel, perpendicular, and antiparallel fields with the sum of their separate energy densities.

Question 8 - Solution

Superposed magnetic fields require an energy cross term

(a) Method and calculation

$$u=\frac{|\mathbf B_1+\mathbf B_2|^2}{2\mu_0}=\frac{B_1^2+B_2^2}{2\mu_0}+\frac{\mathbf B_1\cdot\mathbf B_2}{\mu_0}.$$$$\boxed{u=\frac{B^2}{\mu_0}(1+\cos\theta)}.$$

(b) Interpretation and further calculation

$$u_1+u_2=B^2/\mu_0\equiv u_\Sigma.$$$$\boxed{u(0)=2u_\Sigma,\quad u(\pi/2)=u_\Sigma,\quad u(\pi)=0}.$$

Physical check

Field superposition is linear but energy is quadratic. Destructive field superposition cancels local field energy.