Classical Electrodynamics · Problem 5.28

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · mutual-inductance-coaxial-loops

Problem

Show that the mutual inductance of two circular coaxial loops in a homogeneous medium of permeability

\mu

M_{12} = \mu\sqrt{ab}\left[\left(\frac{2}{k} - k\right)K(k) - \frac{2}{k}E(k)\right]

where

k^2 = \frac{4ab}{(a+b)^2 + d^2}

and

a

b

are the radii of the loops,

d

is the distance between their centers, and

K

and

E

are the complete elliptic integrals. Find the limiting value when

d \ll a, b

and

a \approx b