Video: Resonance in Alternating Current Circuits

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.

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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. Hereโ€™s a diagram of our circuit. We have the alternating-voltage source, a resistor of resistance ๐‘…, an inductor of inductance ๐ฟ, and a capacitor of capacitance ๐ถ. Weโ€™ll use the symbol ๐œ” 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 ๐œ” naught ๐ฟ is equal to one divided by ๐œ” naught ๐ถ, where ๐œ” naught is the resonant angular frequency.

If we solve this equality for ๐œ” 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 letโ€™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 wouldnโ€™t change.

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