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An $R L C$ circuit with $R=20.0 \Omega, L=295 \mathrm{mH},$ and $C=49.7 \mu \mathrm{F}$ is connected to an ac generator with an rms voltage of 21.5 $\mathrm{V}$ . Determine the average power delivered to this circuit when the frequency of the generator is (a) equal to the resonance frequency, (b) twice the resonance frequency, and (c) half the resonance frequency.

a) 23.1 $\mathrm{W}$b) 3.94 $\mathrm{W}$c) 3.94 $\mathrm{W}$

Physics 102 Electricity and Magnetism

Chapter 24

Alternating-Current Circuits

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Electromagnetic Induction

Alternating Current

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the average power that is delivered to the circuit is Eco's true, the R E master voltage kinds, the rmx current times. Their co sign off the fees with No, Remember that, Ari Master voltage, Is it close to the area? Maskell rent times the imperatives then they are a mask or rent is the question the voltage decided by the impudence. Therefore, the average power can be reading mass. Um, I remember squared, divided by the impedance times they call Sign off the face, but the course hand off the phase. Is it close to the resistance divided by the pendants and the average power? Is it close to the R A mask voted squared, divided by the square times the resistance In the first item the circuit is driven out residence. Then remember that at residence both react. Ince's are equal and therefore the people's easy goes to the resistance. Then, in this case, the average power is it close to? They are a master voltages squared, divided by the resistance. And these is itcause True, 21.5 squared, divided by 20 which is approximately 23.1. What on the next item, free things off the circuit is it goes to two times in the resonant frequency as the resonant frequency is because to one divided by the square, root off L C. Then to this day frequents often a circuit. The videos is equal to the square root r squared. Plus, the difference between both reactors is squared. Before planning anything in this equation, let us take a look at the reactors is the capacity reactors is equals to one divided by the frequency times the capacitance These equals true one divided by well, true times like capacitance times this crab boat off Hell time See which finally, of course, too. The square it off. L divided by two times the square root off. See which then I have off the square root off, Help oversee They looked reactions. In the other hand, is it close to the frequency times they looked at us? We can see close to two times l divided by the square root off health time See And these equals two two times and square it off l O R c e. Then excel miners. That seat easy goes to two times the square root off. They'll oversee minus 1/2 in the square it off L A mercy and these is equal to three. House off this Beirut off l oversee. Then we can plug in these results in our expression for the impedance we get. The easy question is quite a bit off r squared plus mine over for me times they'll oversee. Finally, we can plug in the values given by the problem to get square root off 20 squared plus nine over four times 295 times 10 to the minus street divided by 49.7 times 10 to the minus six. This is approximately 117 0.3 owns And now we can proceed to calculate what is the average power consumed by our circuit. The average power is equals Truth. The remans for voltage squared, divided by the impedance squared times the resistance as we had seen before. Right you're then what? We get 21.5 squared, divided by 117 point treat squared times the resistance which is 20 20 and these is approximately zero point 67 too. What? On the next item, we have the following Sit up now the frequency. Is it close to 1/2 off the resonant frequency which is the cause to one divided by two times the square it off. L time, see? No, the capacity reactant is enclosed one divided by one divided by to the square Root off l time. See Time seat on these. Easy question. Two times the square root off l time See why you open your other hand. Excel is equals to one he wanted by two times the square it off l time see times ill which is equals to 1/2 off the square Root off l oversee And that this time we get Excel My nose XY is equal to 1/2 square Root off L over seats minus two square root off l ever see which is equal to minus street over to express route off L over seats Then the impudence is the cost of the square root off r squared plus mine's three over two times the square root off l oversee squared, which is equal to the square root off r squared plus nine were four times l Oversee. Note that way Already have seen his expression right here. So the value off, the impudence will be precisely the same. And then they value off the average power. We will also be the same. So on the last item, the average power is close to 0.672 What's as well.

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