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29 Dec 2011 18:58:39 IST

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explanation and derivation of mutal induction.

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Viraat Gupta

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9 Mar 2012 23:11:06 IST

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inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit. This property also is called self-inductance to discriminate it from mutual inductance, describing the voltage induced in one electrical circuit by the rate of change of the electric current in another circuit.

Viraat Gupta

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9 Mar 2012 23:12:05 IST

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

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

The symbol μ₀ denotes the magnetic constant (4π×10⁻⁷ H/m), C_i and C_j are the curves spanned by the wires, R_ij is the distance between two points.

Viraat Gupta

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9 Mar 2012 23:12:50 IST

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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.

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

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

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