Derivation of self inductance

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

$M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|}$

Derivation

$\Phi_{i} = \int_{S_i} \mathbf{B}\cdot\mathbf{da} = \int_{S_i} (\nabla\times\mathbf{A})\cdot\mathbf{da} = \oint_{C_i} \mathbf{A}\cdot\mathbf{ds} = \oint_{C_i} \left(\sum_{j}\frac{\mu_0 I_j}{4\pi} \oint_{C_j} \frac{\mathbf{ds}_j}{|\mathbf{R}|}\right) \cdot \mathbf{ds}_i$

where

$\Phi_i\ \,$ is the magnetic flux through the ith surface by the electrical circuit outlined by C_j

C_i is the enclosing curve of S_i.

B is the magnetic field vector.

A is the vector potential.

Stokes' theorem has been used.

$M_{ij} \ \stackrel{\mathrm{def}}{=}\ \frac{\Phi_{i}}{I_j} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|}$

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