Inductance/derivation of self inductance

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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}|}

[edit] 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 Cj
Ci is the enclosing curve of Si.
B is the magnetic field vector.
A is the vector potential.

Stokes' theorem has been used.

M_{ij} \ \stackrel{\mathrm{def}}{=}\ \frac{\Phi_{ij}}{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 inductance is a purely geometrical quantity independent of the current in the circuits.

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