Video Transcript
A circuit consists of a resistor, a
capacitor, and an inductor all of which are in series. An alternating-voltage source is
connected to the circuit, and an alternating current is generated. How does the resonant frequency of
the circuit change if the inductance of the inductor is increased? (a) The resonant frequency
decreases. (b) The resonant frequency
increases. (c) The resonant frequency does not
change.
The question is asking us about the
resonant frequency of an alternating-current circuit. Specifically, for a series circuit
with a resistor, capacitor, and inductor, the question is asking us what will happen
if the inductance of the inductor is increased. Here’s a diagram of our
circuit. We have the alternating-voltage
source, a resistor of resistance 𝑅, an inductor of inductance 𝐿, and a capacitor
of capacitance 𝐶. We’ll use the symbol 𝜔 for the
angular frequency of the voltage source.
Recall that resonance will occur in
this circuit when the difference between inductive and capacitive reactances, that
is, the total reactance, is zero. In other words, resonance is when
the inductive and capacitive reactances are equal. We also have the formulas that
relate angular frequency to reactance as inductive reactance is angular frequency
times inductance and capacitive reactance is one divided by angular frequency times
capacitance. If we equate these expressions, as
will be true at the resonant frequency, we get that 𝜔 naught 𝐿 is equal to one
divided by 𝜔 naught 𝐶, where 𝜔 naught is the resonant angular frequency.
If we solve this equality for 𝜔
naught, we find that the resonant angular frequency is equal to one divided by the
square root of the inductance of the inductor times the capacitance of the
capacitor. This formula relates resonant
frequency to inductance, so let’s use it to answer our question. As inductance is increased, the
square root of inductance times capacitance is increased. So the denominator of our fraction
is getting larger, which means the value of the overall fraction is getting
smaller. But the value of this fraction is
just the resonant frequency. So as the inductance of the
inductor increases, the resonant frequency decreases. Interestingly, we can see from our
formula that the resonant frequency would also decrease if we increase the
capacitance of the capacitor. But if we change the resistance of
the resistor, the resonant frequency wouldn’t change.
