Coefficient of Mutual Induction or Mutual Inductance :
It is known that the magnetic flux linked with the secondary coil is directly proportional to the current flowing through the primary coil.
i.e.,
$\phi_s \propto I_p$
or, $\phi_s = MI_p \quad \dots (1) $
where $M$ is a constant of proportionality called \textbf{co-efficient of mutual induction or Mutual inductance}.
If $I_p = 1$, then $M = \phi_s$.
Thus, magnetic inductance of two coils or circuits is defined as the magnetic flux linked with the secondary coil due to the flow of unit current in the primary coil.
According to Faraday's law of electromagnetic induction,
$ \varepsilon_s = - \frac{d\phi_s}{dt} $
Using equation (1), we get
$ \varepsilon_s = - \frac{dMI_p}{dt}$
$ \varepsilon_s= -M \frac{dI_p}{dt} \quad \dots (3) $
or
$ M = - \frac{\varepsilon_s}{\frac{dI_p}{dt}} \quad \dots (4) $
If $-\frac{dI_p}{dt} = 1$, then $M =\varepsilon_s$.
Thus, mutual inductance of two coils can be defined as the induced e.m.f. produced in the secondary coil due to unit rate of decrease of current in the primary coil.
S.I. unit of mutual inductance is henry (H).
$ 1H = 1Wb A^{-1} = 1V A^{-1}s$
Thus, mutual inductance of a pair of coils is said to be 1 henry (H) if 1 volt e.m.f. is induced in a coil due to $1 \text{ A s}^{-1}$ decrease of current in the neighbouring coil.