Derivation of self inductance

The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula

${\displaystyle M_{ij}={\frac {\mu _{0}}{4\pi }}\oint _{C_{i}}\oint _{C_{j}}{\frac {\mathbf {ds} _{i}\cdot \mathbf {ds} _{j}}{|\mathbf {R} _{ij}|}}}$

Derivation

${\displaystyle \Phi _{i}=\int _{S_{i}}\mathbf {B} \cdot \mathbf {da} =\int _{S_{i}}(\nabla \times \mathbf {A} )\cdot \mathbf {da} =\oint _{C_{i}}\mathbf {A} \cdot \mathbf {ds} =\oint _{C_{i}}\left(\sum _{j}{\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathbf {ds} _{j}}{|\mathbf {R} |}}\right)\cdot \mathbf {ds} _{i}}$

where

${\displaystyle \Phi _{i}\ \,}$ is the magnetic flux through the ith surface by the electrical circuit outlined by Cj
Ci is the enclosing curve of Si.
B is the magnetic field vector.
A is the vector potential. [1]

Stokes' theorem has been used.

${\displaystyle M_{ij}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\Phi _{i}}{I_{j}}}={\frac {\mu _{0}}{4\pi }}\oint _{C_{i}}\oint _{C_{j}}{\frac {\mathbf {ds} _{i}\cdot \mathbf {ds} _{j}}{|\mathbf {R} _{ij}|}}}$

so that the mutual inductance is a purely geometrical quantity independent of the current in the circuits.

References

1. ^ Jackson, J. D. (1975). Classical Electrodynamics. Wiley. pp. 176, 263.