Energy Density of a Magnetic Field — Question 3

Question 3

Coaxial magnetic energy is distributed uniformly per logarithmic radius

An ideal coaxial line carries opposite surface currents of magnitude I at radii a and b. The vacuum field between them is $B=\mu_0I/(2\pi r)$ and vanishes elsewhere.

1. Find energy per unit length and external inductance per unit length.

2. If b is doubled while a and I stay fixed, find the additional energy per unit length.

Question 3 - Solution

Coaxial magnetic energy is distributed uniformly per logarithmic radius

(a) Method and calculation

$$\frac U\ell=\int_a^b\frac{1}{2\mu_0}\left(\frac{\mu_0I}{2\pi r}\right)^2 2\pi r\,dr.$$$$\boxed{U/\ell=\frac{\mu_0I^2}{4\pi}\ln(b/a),\quad L/\ell=\frac{\mu_0}{2\pi}\ln(b/a)}.$$

(b) Interpretation and further calculation

$$\Delta(U/\ell)=\frac{\mu_0I^2}{4\pi}[\ln(2b/a)-\ln(b/a)].$$$$\boxed{\Delta(U/\ell)=\frac{\mu_0I^2}{4\pi}\ln2}.$$

Physical check

The added outer annulus stores a finite energy independent of the original outer radius.