Inductance - Derivation From Faraday's Law of Inductance

Derivation From Faraday's Law of Inductance

The inductance equations above are a consequence of Maxwell's equations. There is a straightforward derivation in the important case of electrical circuits consisting of thin wires.

Consider a system of K wire loops, each with one or several wire turns. The flux linkage of loop m is given by

Here N_m denotes the number of turns in loop m, Φ_m the magnetic flux through this loop, and L_m,n are some constants. This equation follows from Ampere's law - magnetic fields and fluxes are linear functions of the currents. By Faraday's law of induction we have

where v_m denotes the voltage induced in circuit m. This agrees with the definition of inductance above if the coefficients L_m,n are identified with the coefficients of inductance. Because the total currents N_ni_n contribute to Φ_m it also follows that L_m,n is proportional to the product of turns N_mN_n.