Electromagnetic Induction and Alternating Current 3 Question 3
3. The total number of turns and cross-section area in a solenoid is fixed. However, its length $L$ is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to
(a) $1 / L$
(b) $L^{2}$
(c) $L$
(d) $1 / L^{2}$
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Answer:
Correct Answer: 3. (a)
Solution:
- (a) Self inductance $L _{sol}$ of a solenoid is given by
$$ L _{sol}=\mu _0 n^{2} \pi r^{2} L $$
(Here, $n=N / L$ and $L=$ length of solenoid)
$$ \begin{array}{ll} \text { or } & L _{sol}=\frac{\mu _0 N^{2} \pi r^{2}}{L} \\ \text { Clearly, } & L _{sol} \propto \frac{1}{L} \end{array} $$
( $\because$ All other parameters are fixed)
NOTE We can determine expression of $L$ as follows
$$ \varphi=N B A=L _{\text {sol }} l $$
But for a solenoid, $B=\mu _0 n l, A=\pi^{2}$
$$ \begin{array}{ll} \therefore & L _{\text {sol }} l=\mu _0 n / \pi^{2} N \\ \text { or } & L _{\text {sol }}=\mu _0 n^{2} \pi^{2} L=\mu _0 \frac{N^{2}}{L} \pi r^{2} \end{array} $$
