A circuit containing a capacitor and an inductor in series has a resonant frequency of 575 kilohertz. The inductor in the circuit has an inductance of 1.25 henries. What is the capacitance of the capacitor? Give your answer in scientific notation to two decimal places.
Let’s say that this is the circuit we’re working with. It has a capacitor and inductor. And since this is an alternating current circuit, it has a variable voltage supply. We’re told that the resonant frequency of this circuit, we’ll call it 𝑓 sub 𝑅, is 575 kilohertz. That’s 575,000 hertz. We’re also told that the inductance of our inductor is 1.25 henries. The henry is the standard SI unit of inductance. Knowing all this, we want to solve for the capacitance of the capacitor.
We’re going to do this assuming that our circuit is indeed at its resonant frequency. This is the frequency at which the overall opposition to charge flow in the circuit is minimized. A circuit’s resonant frequency is achieved when what is called the reactance of the capacitor and the reactance of the inductor are equal. All that to say, the resonant frequency in a circuit depends on its capacitance and its inductance. These three quantities are related through this equation. The resonant frequency of a circuit equals one over two 𝜋 times the square root of its inductance times its capacitance. Note that in our scenario, we know the resonant frequency of our circuit and its inductance. It’s the capacitance we want to solve for.
So let’s rearrange this resonant frequency equation to solve for 𝐶, the capacitance. If we multiply both sides of this equation by the square root of 𝐶 over 𝑓 sub 𝑅, then on the left-hand side, that resonant frequency cancels out. And on the right, the square root of capacitance cancels. We therefore have that the square root of 𝐶 equals one over two 𝜋 𝑓 sub 𝑅 times the square root of 𝐿. Then, if we square both sides, we get this equation for the capacitance: one over four 𝜋 squared times 𝑓 sub 𝑅 squared times 𝐿.
Let’s now recall that we know the resonant frequency of our circuit as well as its inductance. We then substitute those values into this expression: 575 times 10 to the third hertz for the resonant frequency and 1.25 henries for the inductance. When we enter this expression on our calculator, we find a result in scientific notation to two decimal places of 6.13 times 10 to the negative 14th farads. This is the capacitance of the capacitor in our circuit.