Mutual Inductance

Mutual inductance.  When a changing current i_1 in one circuit causes a changing magnetic flux in a second circuit, an emf \mathcal{E}_2 is induced in the second circuit.  \mathcal{E}_2=-M\frac{di_1}{dt} and \mathcal{E}_1=-M\frac{di_2}{dt}, M=\frac{N_2\Phi_{B2}}{i_1}=\frac{N_1\Phi_{B1}}{i_2} – mutual inductance, N_1 – number of turns of coil of the first circuit, \Phi_1 – average magnetic flux through each turn of coil 1.

Self-inductance.  A changing current i in any circuit causes a self-induced emf \mathcal{E}=-L\frac{di}{dt}L=\frac{N\Phi_B}i – depends on the geometry of the circuit and the material surrounding it.

Magnetic field energy.  An inductor with inductance L carrying current I has energy U associated with the inductor’s magnetic field: U=\frac12LI^2.  Magnetic energy density: u=\frac{B^2}{2\mu}.

R-L circuits.  In an R-L circuit the growth and decay of current are exponential with time constant \tau=\frac LR.

L-C circuits.  An L-C circuit undergoes electrical oscillations with an angular frequency \omega=\sqrt{\frac1{LC}}.

L-R-C circuits.  The frequency of damped oscillations \omega'=\sqrt{\frac1{LC}-\frac{R^2}{4L^2}}.

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