Question

$A$ short solenoid (length $l$ and radius $r$ with n turns per unit length) lies well inside and on the axis of a very long, coaxial solenoid (length $L$, radius $R$ and $N$ turns per unit length, with $R> r).$ Current I follows in the short solenoid. Choose the correct statement.

• A. There is uniform magnetic field $\mu _0nI $ in he long solenoid
• B. Mutual inductance of the solenoids is $πμ_0r^{2} nNl$
• C. Flux through outer solenoid due to current $I$ in the inner solenoid is proportional to the ratio $R / r.$
• D. Mutual inductance of the solenoids is $πμ_{0}rRnNlL / (rR)^{1/2} .$

Answer 1

Answer: (B)

We use reciprocatory theorem to calculate mutual inductance of the system of solenoids

Let outer solenoid carries a current $I_{1}$.
Flux linked with small solenoid $(2)$ is proportional to the current in large solenoid $(1)$.
$\Rightarrow \phi_{21} \propto I_{1}$
$ \Rightarrow (n l)\left(\mu_{0} N I_{1}\right)\left(\pi r^{2}\right) \propto I_{1}$
Proportionality constant is the coefficient of mutual inductance $M$ of system.
Note that $M_{12}=M_{21}$ in all cases.
So, $n / \mu_{0} N I_{1} \pi r^{2}=M I_{1}$
or $M=\pi r_{0} r^{2} n N I$