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An $R C$ circuit consists of a resistor $R=32 \Omega,$ a capacitor $C=25 \mu \mathrm{F},$ and an ac generator with an rms voltage of 120 $\mathrm{V}$ .(a) At what frequency will the rms current in this circuit be 2.9 $\mathrm{A}$ ?For this frequency, what are (b) the rms voltage across the resistor, $V_{\mathrm{rms}, R},$ and $(\mathrm{c})$ the rms voltage across the capacitor, $V_{\mathrm{mss}} c^{2}(\mathrm{d})$ Show that $V_{\operatorname{mss}, R}+V_{\text { rms }, C}>120 \mathrm{V}$ but that $\sqrt{V_{\mathrm{rms}, R}^{2}+V_{\mathrm{rms}, c}^{2}}=120 \mathrm{V}$

a) see workb) 93 $\mathrm{V}$c) 76 $\mathrm{V}$

Physics 102 Electricity and Magnetism

Chapter 24

Alternating-Current Circuits

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Electromagnetic Induction

Alternating Current

Cornell University

University of Michigan - Ann Arbor

Hope College

McMaster University

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the first tight and we can use all those law to relate the voltage with the current as follows. The re mass is a coastal y You're a mess times impedance then they are in mass current. Is it close to the ari mass politics divided by the impudence? We want the currents to be cost 2.9 amperes. So 2.9, we will be close to 120 divided by the impedance. Therefore impedance is there any cost to 120 divided by 2.9, But they beat us in. The situation is equal to the square root off R squared, plus the capacity of reactors squared from these easy close to 120 divided by 2.9. We also know what is their value off the resistance. So we can blogging in this equation to get square root off. 32 squared plus XY squared square root is it close to 120 divided by 2.9 square in both sides and sending the storm to the other side. We get XY squared, he's equals to 120 invited by 2.9 square miners, 32 squared and finally we get that XY is equals. True, the square root off 120 divided by 2.9 squared, minus 32 squared. Not that we're not considering their negative square root because there are no negative react. Ince's now remember that the capacity reactors isn't close to one divided by two pi times the frequency times the capacitance. We want to know what is the value off the frequency then we saw for the frequency needs equals to 120 divided by 2.9 squared minus 32 squared Solving for the frequency as easy as sending this with the other side and leads to decide So we got frequency is equals two to pie times the capacitance Times Square Eat off 120 divided by 2.9 squared, minus 32 square under one. Finally, we can plug in their value off the capacitance to get their value for the frequency. What the frequencies in close to two pi times the capacitance, which is 25 times 10 to the minus six because it's, um, Negro for it. Times the square root off 120 divided by 2.9 squared, minus 32 squared. And these is approximately zero point 24 kilo hertz. For the next item, we have to complicate what is There are immense voltage across the resistor governments voltage across the resistor is he calls too. The r E m s current kinds. The resistance of the resistor, which is equals two 2.9 times started to which gives us a narrow master politician off 92 0.8 votes, which can be approximately 2 93 worlds on the next time. Then we have to complete what is the R. M s voltage across the capacitor and these is equals to the R. M s current times the capacity reactors, because he goes to 2.9 times one divided by two pi times 0.24 times 10 to the third time's. The capacitance witches equals to 25 times 10 to the minus six. And this is approximately 76 votes. No, on the last item. It's very easy to see that V. R. M s across your resistance plus Vieira mass, of course, like a pass it or is it close to 93 plus 76 which is it goes to 100 and 69 votes, which is bigger than 120 votes. But on the other hand, a square root off via a mass squared across the resistance plans Vieira Mass. Of course, they could pass it towards Queer is equal to the square root off 93 Squared plus 76 squared, which is approximately 120 votes.

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