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

$M_{ij}=\frac{{\mu }_0}{4\pi }{\oint }_{C_i}{\oint }_{C_j}\frac{{\mathbf{d}\mathbf{s}}_i\cdot {\mathbf{d}\mathbf{s}}_j}{\left|{\mathbf{R}}_{ij}\right|}$

Derivation

${\mathrm{\Phi }}_i=\underset{S_i}{\int }\mathbf{B}\cdot \mathbf{d}\mathbf{a}=\underset{S_i}{\int }(\mathrm{\nabla }\times \mathbf{A})\cdot \mathbf{d}\mathbf{a}={\oint }_{C_i}\mathbf{A}\cdot \mathbf{d}\mathbf{s}={\oint }_{C_i}\left(\sum _j\frac{{\mu }_0{I}_j}{4\pi }{\oint }_{C_j}\frac{{\mathbf{d}\mathbf{s}}_j}{\left|\mathbf{R}\right|}\right)\cdot {\mathbf{d}\mathbf{s}}_i$

where

${\mathrm{\Phi }}_i$ 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.

$M_{ij}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{{\mathrm{\Phi }}_i}{I_j}=\frac{{\mu }_0}{4\pi }{\oint }_{C_i}{\oint }_{C_j}\frac{{\mathbf{d}\mathbf{s}}_i\cdot {\mathbf{d}\mathbf{s}}_j}{\left|{\mathbf{R}}_{ij}\right|}$

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

Self inductance

In the self inductance case C_i=C_j. Therefore 1/R diverges and the finite radius of the wire and the distribution of the current in the wire must be taken into account. A generic formula for the self inductance M of a wire loop is available provided that the length l of the wire is much larger than its radius a,

$M=M_{ii}=\frac{{\mu }_0}{4\pi }{\left({\oint }_C{\oint }_{C^{\prime }}\frac{\mathbf{d}\mathbf{s}\cdot {\mathbf{d}\mathbf{s}}^{\prime }}{\left|{\mathbf{R}}_{s{s}^{\prime }}\right|}\right)}_{\left|\mathbf{R}\right|>a/2}+\frac{{\mu }_0}{2\pi }lY+O\left({\mu }_{0}a\right).$

Points with |R| < a/2 now must be excluded from the curve integral. The correction term proportional to l originates from short wire segments which are essentially cylinders. Y=0 if the current flows in the surface of the wire, Y=1/4 if the current is homogeneous in the wire.

The error of the formula is of order μ₀a if the current loop contains sharp corners and of order μ₀a²/R_c for smooth current loops with minimal curvature radius R_c.

In the skin effect case another approximation is hidden in the assumption of constant current density. If wires are close to each other additional currents flow in the surface of the wires (expelling the magnetic field). In this case Maxwell's equation must be solved to determine currents and fields.

References

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