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 ${\mu }_{0}nI$ in he long solenoid
• B. Mutual inductance of the solenoids is $\pi {\mu }_0{r}^{2}nNl$
• 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 $\pi {\mu }_{0}rRnNlL/(rR{)}^{1/2}.$

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 $(1)$.
$\Rightarrow {\varphi }_{21}\propto I_1$
$\Rightarrow (nl)\left({\mu }_{0}N{I}_1\right)\left(\pi r^2\right)\propto I_1$
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 $M=\pi r_0{r}^{2}nNI$