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A series ac circuit contains a 250$\Omega$ resistor, a 15 $\mathrm{mH}$inductor, a 3.5$\mu$ capacitor, and an ac power source of voltageamplitude 45 $\mathrm{V}$ operating at an angular frequency of 360 $\mathrm{rad} / \mathrm{s}$ . (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to theinductor?

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a) 0.302b) $P_{a v}=0.370 \mathrm{W}$c) $P_{a v}=0.370 \mathrm{W}$

Physics 102 Electricity and Magnetism

Chapter 22

Alternating Current

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Electromagnetic Induction

Cornell University

Rutgers, The State University of New Jersey

Hope College

McMaster University

Lectures

04:44

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug electrical appliances into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

11:31

In electrical engineering, a direct current (DC) circuit is an electrical circuit operating with a constant voltage (or current), as opposed to alternating current (AC) circuits. Direct current may flow in a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams.

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31.30. A series ac circuit…

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A series ac circuit contai…

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An $R L C$ series circuit …

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A circuit consists of a re…

15:48

06:06

An $L-R-C$ series circuit …

11:48

06:44

before we do anything, let's get the voltage, our mess from the voltage amplitude. So the voltage our mass is equal to the voltage amplitude over the square to two and playing in the value here, which is 45 bolts. We get a voltage our mess of 31 point ables, and we'll set that aside for now. So the power factor that they're referring to impart A is equal to co sign a fly or fives the phase angle and the phasing Ole is given my well the tangent, the phase anguish and, say is given by the reactions of the in Dr Nice The reactant. So the faster over the resistance. And so let's figure out what these qualities are. The reactions of the induct ER is equal to Omega Times L and playing in the values that were given 360 radiance per second. And for l. We have 15 times 10 to the minus third Henry's, and so this is equal to 5.4 homes. And then for X c, we know it to be won over Omega Psi, and so this is equal to one over 3 60 ratings for second times the capacitance, which is 3.5 times 10 to the minus six carats. And so this gives 794 owns. And so now we can plug these two things along with the resistance that they give us to figure out what this bias. We have to take an inverse tangent at the end. But if you do that, you get that fine is negative 72.4 degrees. And so now we can take the CO side of this to figure out what the power factor is. Since this here is the power factor and we get that the power factor. What? Take it go son is equal to 0.302 in part B. We're going to need to calculate the impedance in order to calculate the power going through the circuit here. And so it's going to calculate that I get Z is equal to the square of r Squared plus Excel mine sexy. This is how you calculate and Pete and summon r l C circuit plugging in. Everything we already know are from the problem. And we calculate these two things in party. So this gives 827 homes. Now, in order defying the arm s current, we need to take the voltage. Our mess, My pens go Chin out. OK, Australian the voltage are miss and we need to divide by the impedance which we just found. We found the voltage. Armah says the first thing we did it's up here. So employing those two things in gives enormous current of 0.0 35 appears And now I can finally use the formula for the power. It's equal to PR mess times. I are miss times the power factor and so plugging in everything that we know we have 0.370 wants in Parsi. We need to figure out the average power deliver to all three circuit elements. Now the average power delivered to the resistor is equal to the Army's current. Oh, it's good to know our most current squared times the resistance and then play in our values. Here we 0.370 wants, which you'll notice is the exact same of this. And the takeaway here is that there's no power delivered to the other elements. So the arson say there's no average power. The average power delivered to the capacitor. And to the in DR Zero. It's all delivered to the resistance, and that completes the problem.

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