Question

A circular coil with a cross-sectional area of $4\, cm^{2}$ has $10$ turns. It is placed at the centre of a long solenoid that has $15$ turns/cm and a cross-sectional area of $10\, cm^{2}$, as shown in the figure. The axis of the coil coincides with the axis of the solenoid. What is their mutual inductance?

• A. $7.54\, \mu H$
• B. $8.54\, \mu H$
• C. $9.54\, \mu H$
• D. $10.54\, \mu H$

Answer 1

Answer: (A)

$7.54\, \mu H$

Answer 2

Let us refer to the coil as circuit $1$ and the solenoid as circuit $2$. The field in the central region of the solenoid is uniform, so the flux through the coil is $\phi_{12}=B_{2}A_{1}=\left(\mu_{0}n_{2}I_{2}\right)A_{1}$ where $n_{2}=N_{2} l =1500$ turns/m The mutual inductance is $M=\frac{N_{1}\phi_{12}}{I_{2}}=\mu_{0}n_{2}N_{1}A_{1}$ $=\left(4\pi\times10^{-7}T\,m\,A^{-1}\right)\left(1500\,m^{-1}\right)\left(10\right)\left(4\times10^{-4}\,m^{2}\right)$ $=7.54\times10^{-6}\,H$ $=7.54\,\mu H$