The mutual inductance M of two coupled inductances L₁ and L₂ is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance:
M = E_2m / (di₁/dt)
M = E_1m / (di₂/dt)

If the self induced voltages of the inductances L₁ and L₂ are respectively E_1s and E_2s for the same rates of change of the current that produced the mutually induced voltages E_1m and E_2m, then:
M = (E_2m / E_1s)L₁
M = (E_1m / E_2s)L₂
Combining these two equations:
M = (E_1mE_2m / E_1sE_2s)^½ (L₁L₂)^½ = k_M(L₁L₂)^½
where k_M is the mutual coupling coefficient of the two inductances L₁ and L₂.

If the coupling between the two inductances L₁ and L₂ is perfect, then the mutual inductance M is:
M = (L₁L₂)^½