What is meant by the term 'mutual inductance' of a pair of coils ? Obtain an expression for the mutual inductance of two long coaxial solenoids, each of length $\displaystyle l$ but having different number of turns $\displaystyle \mathrm{N}_{1}$ and $\displaystyle \mathrm{N}_{2}$ and radii $\displaystyle \mathrm{r}_{1}$ and $\displaystyle \mathrm{r}_{2}\left(\mathrm{r}_{2}>\mathrm{r}_{1}\right)$.

Definition /Meaning of Mutual Inductance $\displaystyle 1$ Expression for Mutual Inductance $\displaystyle 2$ Mutual Inductance-It is the magnetic flux in the secondary coil due to the flow of unit current in the primary coil. Alternatively: It is the emf induced in the secondary coil when rate of change of current in the primary coil is unity. Expression for Mutual Inductance Let \(\displaystyle \mathrm{N}_{1}\) and \(\displaystyle \mathrm{N}_{2}\) be the total number of turns of coils \(\displaystyle \mathrm{S}_{1} \& \mathrm{~S}_{2}\), respectively. When a current \(\displaystyle \mathrm{I}_{2}\) is set up through \(\displaystyle \mathrm{S}_{2}\), it in turn sets up a magnetic flux \[\text { through } \mathrm{S}_{1} \Rightarrow \mathrm{~N}_{1} \phi_{1}=\mathrm{M}_{12} \mathrm{I}_{2} \] \(\displaystyle \mathrm{M}_{12}\) is the mutual inductance of solenoid \(\displaystyle \mathrm{S}_{1}\) w.r.t. \(\displaystyle \mathrm{S}_{2}\). Magnetic field due to \(\displaystyle \mathrm{I}_{2}\) in \(\displaystyle \mathrm{S}_{2}=\mu_{\mathrm{o}}\left(\frac{\mathrm{N}_{2}}{l}\right) \mathrm{I}_{2}\) \[\mathrm{N}_{1} \phi_{1}=\left(\mathrm{N}_{1}\right)\left(\pi \mathrm{r}_{1}^{2}\right)\left(\mu_{\mathrm{o}} \frac{\mathrm{N}_{2}}{l} \mathrm{I}_{2}\right) \]