A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5 00 mH and C = 3.50 μF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Video transcript

Welcome back, everybody. We are making observations about a series circuit and we are told that it has an induct er a capacitor, a power source with a variable angular frequency. And we are told that the angular at the angular frequency of omega not the inductive reactant equals the capacitive reactant. So we are tasked with finding a couple of different things here. Part one is that we are going to need to find what that omega not is. And then to, we are going to need to find what the, let's see here. Inductive reactant is and the capacitive reactant is when we have an inductive of 212 milla Henry's. And then when we have a capacitance of 8. micro fare adds. So for this, we are going to need to use two formulas here. We're going to have that the inductive reactant is equal to the induct inst times our particular angular frequency. And for our capacitive reactant, we have that this is equal to one over our capacitance times our angular frequency. Now, remember we are told that at this omega not these two values are equal to one another So let's plug in omega naught to each of these formulas and then set them equal to one another. So let's see, we have L times omega naught for the inductive reactant is equal to one over the capacitance times. Our omega not now, we want to isolate the omega not term. So here's what I'm going to do. I'm going to multiply both sides by omega not over L and you'll see why in just a second on our left side here, these else cancel out on the right side. These cancel out. This leads us to getting the formula that Omega not is equal to the square root of one over our induct inst times our capacitance. So we know these values. Let's go ahead and plug them in. We have that Omega not, is equal to the square root of one on top is just one divided by. And let me make this just a little bit wider here divided by the square root of our induct insp which we need in Henry's. So right now it's in Miller Henry's. So I'm gonna multiply by 10 to the negative third times our capacitance of 8.25 micro fare adds. But similarly, we need it in just fair ads. So this is gonna be multiplied by 10 to the negative six. And when we plug this into our calculator, we get that our omega not is equal to 742.26 radiance per second. Great. So now that we know this value, we can use these two formulas right here to find our inductive react ints and our capacitive reactant. So let's go ahead and do that here starting with the inductive react ints. We have that this is equal to simply our inductive times. Let's see here. Our omega not. So let's go ahead and plug in our values, you know, that are inducted is 220 Mil Henry's or 220 times 10 to the negative third, Henry's times our omega not of 742.26, which when we plug this into our calculator and round, we get 163 oh Maiga's perfect. And now let's move on to the capacitive reactant. This is equal to one over our capacitance times our angular frequency. So this is equal to one over bastions of 8.25 times 10 to the negative six fair ads times are angular frequency of 7.2 point 26 radiance per second, which when we plug into our calculator, we also get 163 omegas. So now we have found our angular frequency and we have found both are inductive and capacitive reactant which corresponds to our final answer. Choice of D. Thank you all so much for watching. Hope this video helped. We will see you well, in the next one.