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 capacitance of the capacitor is increased? (A) The resonant frequency increases. (B) The resonant frequency decreases. (C) The resonant frequency does not change.
Here, we’re told that our circuit consists of a resistor, a capacitor, and an inductor. And there’s an alternating voltage supply. This supply will vary the voltage with some frequency. And it may be at what is called the resonant frequency of the circuit. If we symbolize the resonant frequency as 𝑓 sub 𝑟, we can recall that mathematically 𝑓 sub 𝑟 is one over two 𝜋 times the square root of 𝑙 times 𝑐. Here, 𝑙 is the value of our inductor and 𝑐 is the capacitance of our capacitor.
In this example, we’re to imagine that the capacitance of the capacitor increases. We want to know what effect will that increase have on the resonant frequency of the circuit. Looking at our mathematical expression for resonant frequency, we see that if a value in the denominator, the capacitance, gets larger, then if nothing else changes, this fraction overall will decrease. That is, if capacitance 𝑐 increases, then 𝑓 sub 𝑟, the resonant frequency, decreases. This is indicated by answer option (B). In this circuit, if the capacitance of the capacitor is increased, the resonant frequency decreases.