Define the term ‘mutual inductance’ between the two coils. Obtain… - Vidyadip

Question

Define the term ‘mutual inductance’ between the two coils.
Obtain the expression for mutual inductance of a pair of long coaxial solenoids each of length I and radii $r_1$ and $r_2\left(r_2 \gg r_1\right)$.
Total number of turns in the two solenoids are $\mathrm{N}_1$ and $\mathrm{N}_2$ respectively.

✓

Answer

Mutual inductance, between a pair of coils, equals the magnetic flux, linked with one of them, due to a unit current flowing in the other.
Alternate Answer
The mutual inductance, for a pair of coils, equals the emf induced, in one of them, when the current in the other coil is changing at a unit rate.

Let a current $I_2$ flow through the outer coil. The magnetic field due to this current
$ = \mu_{o}\frac{\text{N}_{2}}{l}\times\text{I}_{2}$
The resulting magnetic flux linked with the inner coil
$ = \phi_{12} = \text{N}_{1}.\big(\mu_{o}\frac{\text{N}_{2}}{l}\times\text{I}_{2}\big)\times\pi\text{r}_{1}^{2}$
$ = \bigg(\mu_{o}\frac{\text{N}_{1}\text{N}_{2}}{l}.\pi\text{r}^{2}_{1}\bigg)\text{I}_{2}$
$= M_{12} I_2$
$\therefore\text{M}_{12} = \mu_{o}\frac{\text{N}_{1}\text{N}_{2}}{l}.\pi\text{r}^{2}_{1}.$

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