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(C) The self-inductance $L$ of a solenoid is given by the formula $L = \mu_0 n^2 A l$,where $n$ is the number of turns per unit length,$A$ is the cros

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$27$

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(C) The self-inductance $L$ of a solenoid is given by the formula $L = \mu_0 n^2 A l$,where $n$ is the number of turns per unit length,$A$ is the cross-sectional area,and $l$ is the length of the coil.Given that all linear dimensions are increased by a factor of $k = 3$:$1$. The length $l$ becomes $l' = kl = 3l$.$2$. Since the cross-section is rectangular,if its sides are $a$ and $b$,the area $A = ab$. When dimensions are scaled by $k$,the new area $A' = (ka)(kb) = k^2 A = 3^2 A = 9A$.$3$. The number of turns per unit length $n$ remains constant.Substituting these into the formula for the new inductance $L'$:$L' = \mu_0 n^2 A' l' = \mu_0 n^2 (k^2 A) (kl) = k^3 (\mu_0 n^2 A l) = k^3 L$.Substituting $k = 3$:$L' = 3^3 L = 27L$.Therefore,the self-inductance increases by a factor of $27$.

$A$ coil is wound on a core of rectangular cross-section. If all the linear dimensions of the core are increased by a factor of $3$ and the number of turns per unit length of the coil remains the same,the self-inductance increases by a factor of:

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