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(B) The resonant frequency of an $LC$ circuit is given by $\omega = \frac{1}{\sqrt{LC}}$.For a coil (inductor),$L = \frac{\mu_0 N^2 A}{\ell}$. If all

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becomes half

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(B) The resonant frequency of an $LC$ circuit is given by $\omega = \frac{1}{\sqrt{LC}}$.For a coil (inductor),$L = \frac{\mu_0 N^2 A}{\ell}$. If all linear dimensions are increased by a factor of $2$,the area $A$ becomes $4A$ and the length $\ell$ becomes $2\ell$. Thus,$L' = \frac{\mu_0 N^2 (4A)}{2\ell} = 2L$.For a parallel plate capacitor,$C = \frac{\epsilon_0 A_c}{d}$. If all linear dimensions are increased by a factor of $2$,the area $A_c$ becomes $4A_c$ and the distance $d$ becomes $2d$. Thus,$C' = \frac{\epsilon_0 (4A_c)}{2d} = 2C$.The new resonant frequency $\omega'$ is $\omega' = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{(2L)(2C)}} = \frac{1}{2\sqrt{LC}} = \frac{\omega}{2}$.Therefore,the resonant frequency becomes half.

An $LC$ circuit consists of a capacitor and a coil with a large number of turns. Suppose all the linear dimensions of all elements of the circuit are increased by a factor of $2$ while keeping the number of turns on the coil constant. How much does the resonant frequency of the circuit change?

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