Energy Density of a Magnetic Field — Question 9

Question 9

A uniform field minimizes energy at a prescribed mean flux density

In a region of volume V, an axial vacuum field has spatially varying scalar magnitude $B(\mathbf r)=\bar B+\delta B(\mathbf r)$ with $\int\delta B\,d\mathcal V=0$. The mean field is prescribed.

1. Express total magnetic energy in terms of mean field and variance.

2. Find the minimum and the excess energy when the rms variation equals half the mean field magnitude.

Question 9 - Solution

A uniform field minimizes energy at a prescribed mean flux density

(a) Method and calculation

$$U=\frac1{2\mu_0}\int(\bar B^2+2\bar B\delta B+\delta B^2)d\mathcal V.$$$$\boxed{U=\frac{V\bar B^2}{2\mu_0}+\frac{V\langle\delta B^2\rangle}{2\mu_0}\ge U_{\min}=\frac{V\bar B^2}{2\mu_0}}.$$

(b) Interpretation and further calculation

$$\sqrt{\langle\delta B^2\rangle}=|\bar B|/2\Rightarrow\Delta U=V\bar B^2/(8\mu_0).$$$$\boxed{\Delta U/U_{\min}=1/4,\quad U=1.25U_{\min}}.$$

Physical check

The cross term integrates to zero by the definition of the mean; equality requires a uniform field almost everywhere.