Inductors and Inductance — Question 3

Question 3

A coaxial circuit has external inductance per length set by a radius ratio

An ideal coaxial circuit carries +I on the surface r=a and return current -I on r=b. Between them $B=\mu_0I/(2\pi r)$; elsewhere B=0. Consider length ell and ignore end effects. Each radial flux strip links the circuit once.

1. Find its inductance per unit length.

2. Find how that quantity changes when both radii double, and when only b doubles.

Question 3 - Solution

A coaxial circuit has external inductance per length set by a radius ratio

(a) Method and calculation

$$\Lambda=\ell\int_a^bB(r)dr=\frac{\mu_0I\ell}{2\pi}\ln(b/a).$$$$\boxed{L/\ell=\frac{\mu_0}{2\pi}\ln(b/a)}.$$

(b) Interpretation and further calculation

$$(a,b)\to(2a,2b):\quad\boxed{(L/\ell)\prime=L/\ell}.$$$$b\to2b:\quad\boxed{\Delta(L/\ell)=\mu_0\ln2/(2\pi)}.$$

Physical check

The result excludes internal magnetic flux because the currents were explicitly confined to surfaces.