The reactance of the capacitor needed in an alternating current circuit is 120 ohms. The capacitor to be used has a 75-microfarad capacitance. What frequency must the alternating current have? Give your answer to the nearest hertz.
We’re talking here about a capacitor that makes up some part of an alternating current circuit. The capacitance 𝐶 of the capacitor is given as 75 microfarads or seventy-five one millionth of a farad. We’re also told of something called the capacitor’s reactance. Symbolized 𝑋 sub c, this is a value which, notice, has units of ohms. This is consistent with the fact that reactance has to do with opposing the flow of charge. It’s similar though not identical to resistance.
We want to solve for the frequency, we’ll call it 𝑓, of our alternating current source such that this circuit with a capacitance of 75 microfarads indeed does have a capacitive reactance of 120 ohms. These three variables — frequency, capacitive reactance, and capacitance — are related by a mathematical expression. In general, capacitive reactance equals one over two 𝜋 times frequency times capacitance. In our case, it’s not the capacitive reactance but rather the frequency we want to solve for. If we multiply both sides of this equation by 𝑓 over 𝑋 sub c, then on the left-hand side, the capacitive reactance cancels out, and on the right the frequency cancels. That leaves us with this expression. And recall that we’re given the capacitance and capacitive reactance of our circuit.
We now substitute those given values into this expression: 75 times 10 to the negative sixth farads for 𝐶 and 120 ohms for 𝑋 sub c. This fraction equals a result in units of hertz. Rounded to the nearest whole number, it’s 18 hertz. The frequency the alternating current in the circuit must have is 18 hertz.