At what frequency is the reactance of a 20-microfarad capacitor equal to that of a 10-millihenry inductor?
We can call our 20-microfarad value capital 𝐶 and our 10-millihenry inductor capital 𝐿. We want to solve for the frequency at which the reactance of these two elements is equal. We’ll call that frequency 𝑓.
We can begin by recalling the mathematical relationships for the reactance of a capacitor and for an inductor. We recall that reactance is a measure of how much the flow of current is resisted. It’s like resistance and even has units of ohms.
The reactance of a capacitor is defined as one over two 𝜋 times the frequency 𝑓 multiplied by the capacitance 𝐶. And the reactance for an inductor is equal to two 𝜋𝑓 times the inductance 𝐿. Notice that both reactance terms involve the angular frequency 𝜔.
In our exercise, we want to solve for frequency 𝑓 when the reactance of the capacitor is equal to the reactance of the inductor. So when one over two 𝜋𝑓𝐶 is equal to two 𝜋𝑓𝐿, we want to know what is 𝑓.
When we rearrange this equation algebraically to solve for 𝑓, we find it’s equal to one over two 𝜋 times the square root of 𝐿𝐶. Since 𝐿 and 𝐶 are both given in the problem statement, we can plug in those values now to solve for 𝑓.
When we do, making sure to express our inductance in units of henries and our capacitance in units of farads, when we enter these values on our calculator, we find that, to two significant figures, 𝑓 is 360 hertz. That’s the frequency at which the reactance of the capacitor equals the reactance of the inductor.