Question

-In a series LRC circuit, the frequency at which the circuit is at resonance is f0. If you double the resistance, the inductance, the capacitance, and the voltage amplitude of the ac source, what is the new resonance frequency?

A) 4 f0

B) 2 f0

C) f0

D) f0/2

E) f0/4

Answer 1

f0=1/sqrt(inductance*capacitance)
So doubling both inductance and capacitance makes the new resonance frequency f0/2

Answer 2

When you double capacitance and inductance, the new resonace frequency becomes f0/2.

Resonance frequency

The resonace frequency of RLC series circuit, is the frequency at which the capacitivity reactance is equal to inductive reactance.

Xc = Xl

$\frac{1}{2\pi f_0C} = 2\pi f_0 L\\\\4\pi^2 f_0^2 LC = 1\\\\f_0^2 = \frac{1}{4\pi^2 LC} \\\\f_0 = \sqrt{\frac{1}{4\pi^2 LC} } \\\\f_0 = \frac{1}{2\pi } \frac{1}{\sqrt{LC} }$

where;

• f0 is the resonace frequency
• L is the inductance
• C is the capacitance

When you double capacitance and inductance, the new resonace frequency becomes;

$f_0 = \frac{1}{2\pi } \frac{1}{\sqrt{LC} }\\\\f_1 = \frac{1}{2\pi } \frac{1}{\sqrt{2L (2C)} } \\\\f_1 = \frac{1}{2\pi } \frac{1}{\sqrt{4LC} }\\\\f_1 = \frac{1}{2}( \frac{1}{2\pi } \frac{1}{\sqrt{LC} })\\\\f_1 = \frac{1}{2} (f_0)$

Thus, When you double capacitance and inductance, the new resonace frequency becomes f0/2.

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