CBSE 2023 · Region 1 · Set 1 · Q28 · 3 marks
A long solenoid of radius $\displaystyle \mathrm{r}$ consists of $\displaystyle n$ turns per unit length. A current $\displaystyle \mathrm{I}=\mathrm{I}_{0} \sin \omega \mathrm{t}$ flows in the solenoid. A coil of N turns is wound tightly around it near its centre. What is :(a)the induced emf in the coil ?(b)the mutual inductance between the solenoid and the coil?
A long solenoid of radius $\displaystyle \mathrm{r}$ consists of $\displaystyle n$ turns per unit length. A current $\displaystyle \mathrm{I}=\mathrm{I}_{0} \sin \omega \mathrm{t}$ flows in the solenoid. A coil of N turns is wound tightly around it near its centre. What is :
(a)
the induced emf in the coil ?
(b)
the mutual inductance between the solenoid and the coil?
Marking-scheme solution
Finding
a) Induced emf
b) Mutual inductance between solenoid and coil
(a)
magnetic field produced in the solenoid near the center
\[\mathrm{B}=\mu_{\mathrm{o}} n \mathrm{I}
\]
Flux linked with the coil wound over solenoid
\[\begin{aligned}
\phi & =\mathrm{NBA}=\mathrm{N} \pi \mathrm{r}^{2} \mathrm{~B} \\
& =\mathrm{N} \pi \mathrm{r}^{2} \mu_{\mathrm{o}} n \mathrm{I}
\end{aligned}
\]
\[\begin{aligned}
\text { Induced emf } \mathrm{e}=\frac{-d \phi}{d \mathrm{t}} & =-\pi \mathrm{r}^{2} \mathrm{Nn} \mu_{\mathrm{o}} \frac{d \mathrm{I}}{d \mathrm{t}} \\
& =-\mu_{\mathrm{o}} \pi \mathrm{r}^{2} \mathrm{nN} \mathrm{I}_{\mathrm{o}} \omega \cos \omega \mathrm{t}
\end{aligned}
\]
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CBSE Class 12 Physics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.