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 $I$ flows in the short solenoid.

Choose the correct statement.

• A. There is uniform magnetic field ${\mu }_{0}nI$ in the long solenoid.
• B. Mutual inductance of the solenoid 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)

Mutual inductance of the solenoid is $\pi {\mu }_0{r}^{2}nNl$.

The magnetic flux linked with long solenoid of radius $R$ due to short solenoid of radius $r$ having current $I$ is,

$\varphi =MI$. . . . .(1)

The magnetic field due to the short solenoid,

$B={\mu }_{0}nlI$

Magnetic flux linked is,

$\varphi =BA$

$\varphi ={\mu }_{0}nlI\left(\pi r^2\right)$. . . . .(2)

From equation (1) and (2), we get,

$M=\pi {\mu }_0{r}^{2}nl$

For $N$ number of turns,

$M=\pi {\mu }_0{r}^{2}nNl$.

The correct option is B.