A series RLC circuit in a radio is in resonance with AM 700 kHz.If the radio station is changed to AM 2100 kHz,by what factor must the capacitance be multiplied to again achieve resonance?
A)9
B)3
C)1
D)1/3
E)1/9

Question

A series RLC circuit in a radio is in resonance with AM 700 kHz.If the radio station is changed to AM 2100 kHz,by what factor must the capacitance be multiplied to again achieve resonance?
A)​9
B)3
C)1
D)1/3
E)​1/9

Quizplus · Accepted Answer

The resonant frequency $f$ of an RLC circuit is given by $f = \frac{1}{2\pi\sqrt{LC}}$, where $L$ is the inductance and $C$ is the capacitance. To maintain resonance when the frequency is tripled (from 700 kHz to 2100 kHz), the product $LC$ must remain constant. Since $L$ is constant, $C$ must be adjusted inversely proportional to the square of the frequency change. Tripling the frequency ($\frac{2100}{700} = 3$) requires the capacitance to be divided by $3^2 = 9$, hence the factor is $1/9$.

A Series RLC Circuit in a Radio Is in Resonance