Question 6A100

The resonant frequency ($f$) of an LC circuit is inversely proportional to the square root of the product of its inductance (L) and capacitance (C). This fundamental relationship means that to double the frequency, the term $\sqrt{LC}$ must be halved. For $\sqrt{LC}$ to be halved, the product $LC$ must be reduced to one-fourth (1/4) of its original value. Let's examine the options: * **A) Make C one third of its original value:** The new product $LC$ would be $L \times (C/3) = LC/3$. This would increase the frequency by a factor of $\sqrt{3}$, not 2. * **B) Make L and C both half their original values:** The new product $LC$ would be $(L/2) \times (C/2) = LC/4$. This quarters the $LC$ product. Taking the square root, $\sqrt{LC/4} = \sqrt{LC}/2$, which means the $\sqrt{LC}$ term is halved. Halving $\sqrt{LC}$ doubles the resonant frequency. * **C) Decreasing the value of both L and C in any proportion so that their product will be one-half of the original values:** If the product $LC$ becomes $LC/2$, the frequency would increase by a factor of $\sqrt{2}$, not 2. Therefore, reducing both L and C by half is the correct method to double the resonant frequency.