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Problem 62 Hard Difficulty

The circuit in Figure $P 18.62$ contains two resistors, $R_{1}=2.0$ $\mathrm{k} \Omega$ and $R_{2}=3.0 \mathrm{k} \Omega,$ and two capacitors, $C_{1}=2.0 \mu \mathrm{F}$ and $C_{2}=$
$3.0 \mu \mathrm{F},$ connected to a battery with emf $\mathcal{E}=120 \mathrm{V}$ . If there are no charges on the capacitors before switch $\mathrm{S}$ is closed, determine the charges $q_{1}$ and $q_{2}$ on capacitors $C_{1}$ and $C_{2},$ respectively, as functions of time, after the switch is closed. Hint: First reconstruct the circuit so that it becomes a simple $R C$ circuit containing a single resistor and single capacitor in series, connected to the battery, and then determine the total charge $q$ stored in the circuit.

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Video Transcript

in this. Um, we need to calculate the charge on first as well as the second capacitor. Oh, we're going to forego cities off steps here. Notice was right that sista p in an r c circuit groot off the charge in I took you in circuit is small. Kill isn't going to kill zero because any should charge one minus is to minus t upon how that is Our secret is a time constant. It's market as aggression one. No, we need to find the charges given and cute doing capacity or see within See to respect Terry to know Billy Boyd says their districts which are shown in the circuit, are in pattern combination. So therefore, will first concrete battle combination So which is given by the question as Arvin Artoo upon on went, Bless I do. That's right. The values two kilo multiplied by its weeklong and treacle home last Uchino. So here we get the value asked, Well, under domes. This is the value of father combination that we haven't done so for this show on So capacity does it's even and c do are in battle in I repeat the capacitors even and see two are also imbalance, so we'll take the equivalent as C p. So when capacitors are in battle, their combination is getting Eric. So this, even Placido See even value is to include senators to minus six France Bless three microphone that is seeing good editors to minus six. So therefore CPSC will do five into tennis to minus experience. This is the capacitor equivalent value that we did. No, that's more to the second beach to step. Be now to calculate time constant That is so sure right here that how is required to our intimacy. But in this case, it will be our pee into a city that is equivalent values that we have already got in step away, which is equal to 1200 warms multiplied by five Micro Fred's So here. By my application, we get six into tennis two minus three seconds, which is a value for time constant that we get here Now We need to make use of equation one so right here using Equation one here, which is small. Q is equal to kill monkey, played by one minus. It is to minus T upon toe, but the six, including is to minus tree. But at the same time small Q is equal to key one plus killed too. So therefore, let's put this immigration while what we get as key one plus Que tu is equal to a few times. One minus is to minus t upon six in group analyst to minus three. Let's mark this as immigration, too, which we are going to use further. No steps See, since Stephen and C do are in Palin that you already know so by the formula that is really easy going to you want a bonus even and at the same time reconsider for second said That is cute too. Aw, phone CDO Therefore, can we say Cuban upon see one is equal to cute to a policy, too. That's wide. Therefore we get it. Ask you to music will do C two upon us. Even monkey pride back You won. That's for the value of C two, as three include Dennis to minus six on this even is to include entities to minus six multiplied by Cuban that we have So therefore, finally we get que tu is equal to one foreign five times Cuba. It's mark this as decoration. Three now many to substitute the value of cute too. Any questions? Do right here. Substitute the value off que tu in equation too, because we need to get the value 40 You are so therefore, what we get here as q one plus 1.5 times kyu won because you're a subsidiary in the value of Cuba, for that is equal to Q times one minus etess to minus t upon six include entities to minus three. No, that's all this father. Therefore, 2.5 times Cuban, we simply air that is equal to Q times one minus eat is to minus D. It is minus minus T upon six into generous to minus three. So therefore, finally begin it ask you one is equal. Joe CEO born full time skew and one minus It is, too. My honesty A Bone six in group and is to my industry. Let's mark this as equation for no, we'll do this step, Dina, since we know that cute too is 1.5 times Kyu won should put this value of key one from a question three. So therefore, 1.5 times we get the value off the one from Equation three. So what? Aggression? Three says as it is going to be boring for Q. Again, we have one more packet here that is one minus. Eat is to minus T upon 16 to Dennis to minus three. Therefore, finally, we get on six times Q multiplied by one minus. It is to minus T upon six in good. And it is to my ministry it's mark this as a gration five. No. Thus, what we know already is Q is equal to C B into absolute. Q is equal. Do see be mean to have slowed by using the same formula? So value CB that were already God is find Reynolds to minus six. Finance and value of Absalon is given as 1 20 world that is a battery wordage. So it is 16 to Dennis to minus four Leones next markers. No, what we're going to do is we'll substitute the value of Q in the question for in fire. So right here, substituting value off cue in the Immigrations four and five. So what we get here as Q one first, which is 0.4 times Q multiplied about one minus. It is to dynasty upon six included, and it is two minus three. So therefore 2.4 in good earnest to minus four Gloans and the Bad Amigos we're here remains the same. So my Nestea a bone six in Cook and his to mine industry now fork you too. The question is born six times Q. And one minus eat is to minus D upon six in progenitors two minus three. So what we're going to do is subsidiary the value of Qs six indoor endless to minus four groups. So 0.6 into six into tennis to minus four columns, and the bracket remains the same six into Dennis. To my mystery, it's calculated this and will come as 3.6 into generous Your Highness, for good arms multiplied by one minus. It is too dynasty upon six imprudence too minus tree. So this particular aggression is charge one Thrace capacity. We're right here, John. No, for his capacity. And this particular question is like car John, I think incapacity. This is what we can, right?

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