Video Transcript
An alternating-current circuit has an impedance of 750 ohms. The circuit contains a resistor, an inductor with a 250-ohm inductive reactance, and a capacitor with a 45.0-ohm capacitive reactance. What is the resistance of the resistor? Give your answer to the nearest ohm.
We can begin by sketching this circuit. Here’s our alternating-current supply which we’re told is connected to a resistor, an inductor, and a capacitor all in series. Regarding this circuit, we’re told that it has an impedance of 750 ohms. Impedance is a measure of the overall opposition to the flow of charge in an alternating-current circuit. We say overall because in such a circuit, there’s more than one source of opposition to charge flow. One source is the resistance of the circuit. We call that 𝑅, and that’s what we want to solve for.
But another source is something called reactance. Both the inductor and the capacitor in our circuit have reactance. And we can tell that reactance is like resistance because the units of reactance are ohms, just like those of resistance. To make progress solving for the resistance of the resistor, let’s start by recalling the general equation for impedance 𝑍 of a circuit.
In an alternating-current circuit that has both a resistor, an inductor, and a capacitor, impedance 𝑍 equals the square root of the resistance of the resistor squared plus this term in parentheses here squared. Here, 𝑋 sub L is the reactance of the inductor, while 𝑋 sub C is the reactance of the capacitor. Again, we can think of reactance as similar but not identical to resistance. All three of these quantities then 𝑅, 𝑋 sub L, and 𝑋 sub C have something to do with opposing the flow of charge.
Now in this case, it’s not the impedance 𝑍 that we want to solve for but rather the resistance 𝑅. To do that, we can rearrange this equation so that 𝑅 is the subject. First, we can square both sides. That gives us this equation. And then we subtract the quantity 𝑋 sub L minus 𝑋 sub C squared from both sides. Since we’re adding as well as subtracting that term on the right, the sum of that equals zero. And then as a final step, we take the square root of both sides of the equation. This gives us an equation for the resistance 𝑅 in terms of the impedance, the inductive reactance, and the capacitive reactance.
Our problem statement gives us values for all three of these quantities. Impedance is 750 ohms, inductive reactance is 250 ohms, and capacitive reactant 45.0 ohms. To the nearest whole number, this is 721 ohms. That’s the resistance of the resistor in this circuit.
