Energy Density of a Magnetic Field — Question 6

Question 6

Changing field area reverses the energy trend when flux is constrained

A uniform axial vacuum field occupies volume $A\ell$, with fixed length ell. Neglect fringe fields. Compare doubling A at fixed field B with doubling A at fixed flux $\Phi=BA$.

1. Write the stored energy under each constraint.

2. Find both final-to-initial energy ratios and explain the sign of the change using the formulas.

Question 6 - Solution

Changing field area reverses the energy trend when flux is constrained

(a) Method and calculation

$$B\ \text{fixed}:\quad\boxed{U=\frac{B^2A\ell}{2\mu_0}}.$$$$\Phi\ \text{fixed}:\quad B=\Phi/A\Rightarrow\boxed{U=\frac{\Phi^2\ell}{2\mu_0A}}.$$

(b) Interpretation and further calculation

$$\boxed{U_f/U_i=2\quad(B\ \text{fixed})}.$$$$\boxed{U_f/U_i=1/2\quad(\Phi\ \text{fixed})}.$$

Physical check

The energy change depends on what the external apparatus holds fixed; geometry alone does not specify it.