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| The mutual inductance by circuit ''i'' on circuit ''j'' is given by the double integral ''[[Franz Ernst Neumann|Neumann]] formula''
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| :<math> M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|} </math> | |
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| ==Derivation ==
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| :<math> \Phi_{i} = \int_{S_i} \mathbf{B}\cdot\mathbf{da} = \int_{S_i} (\nabla\times\mathbf{A})\cdot\mathbf{da}
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| = \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 </math>
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| where
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| :<math>\Phi_i\ \,</math> is the [[magnetic flux]] through the ''i''th surface by the [[electrical circuit]] outlined by ''C''<sub>''j''</sub>
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| :''C<sub>i</sub>'' is the enclosing curve of S<sub>''i''</sub>.
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| :''B'' is the [[magnetic field]] vector.
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| :''A'' is the [[vector potential]]. <ref>{{cite book |last=Jackson |first=J. D. |title=Classical Electrodynamics |url= |date=1975 |accessdate= |edition= |publisher=Wiley |pages=176, 263 |isbn= }}</ref>
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| [[Stokes' theorem]] has been used.
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| :<math> 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}|} </math>
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| so that the mutual inductance is a purely geometrical quantity independent of the current in the circuits.
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| ==Self inductance ==
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| In the self inductance case ''C<sub>i</sub>=C<sub>j</sub>''. 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'',<ref name="rden12">{{cite arXiv |last=Dengler |first=R. |eprint=1204.1486 |title=Self inductance of a wire loop as a curve integral|year=2012 }}</ref>
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| :<math> M = M_{ii} = \frac{\mu_0}{4\pi} \left ( \oint_{C}\oint_{C'} \frac{\mathbf{ds}\cdot\mathbf{ds}'}{|\mathbf{R}_{ss^{\prime }}|}\right )_{|\mathbf{R}| > a/2} | |
| + \frac{\mu_0}{2\pi}lY + O\left( \mu_0 a \right ).</math>
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| 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
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| wire, ''Y=1/4'' if the current is homogeneous in the wire.
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| The error of the formula is of order ''μ<sub>0</sub>a'' if the current loop contains sharp corners and of
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| order ''μ<sub>0</sub>a<sup>2</sup>/R<sub>c</sub>'' for smooth current loops with minimal curvature radius ''R<sub>c</sub>''.<ref name="rden12"/>
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| In the skin effect case another approximation is hidden in the assumption of constant current density.
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| If wires are close to each other additional currents flow in the surface of the wires (expelling the magnetic field).
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| In this case Maxwell's equation must be solved to determine currents and fields.
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| ==References== | |
| {{reflist}}
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| [[Category:Electrodynamics]]
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