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Problem 30 Easy Difficulty

In the circuit of Exercise 29, $ R = 2 \Omega, C = 0.01 F, Q(0) = 0, $ and $ E(t) = 10 \sin 60t. $ Find the charge and the current at time $ t. $

Answer

$$
\begin{array}{l}{Q=\frac{5}{122} \cdot \sin (60 t)-\frac{3}{61} \cdot \cos (60 t)+\frac{3}{61} \cdot e^{-50 t}} \\ {I=\frac{180}{61} \cdot \sin (60 t)+\frac{150}{61} \cdot \cos (60 t)-\frac{150}{61} \cdot e^{-50 t}}\end{array}
$$

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Video Transcript

we have given a circuit with a generator e and a capacitor C with 0.1 Farhad capacitance resistor R, with two warms on e is dependent on time as 10 times sine 60 t We need to calculate I off t and Q off tea with initial condition is charge in the capacity of times zero is zero Okay? To solve this, we need to convert this circuit into differential equation will apply Kit Juckes Law. So kids shops law. It says the total amount of voltage supply is equal to voltage. Drop across the circuit now If you see wilted, drop across the capacity is given by Cuba I see plus voltage drop across the resistor is given by Are I now we can change I in terms off us. We know I is rate off change off charge or floor charge So therefore the off the we know this 10 times sign 60 t It was two Q We don't know. We have toe variable. See, we know it is 0.1 plus our is known. It's too and I is de Cuba deity now since we don't need this too will developer to throughout this so 10 developer to his five times sine 60 t is equals. Two Q over 0.1 times two is 0.2 plus de Cuba deity. Now, right, this equation in standard form will have day Q by D. T plus one hours 0.2 is 50. So this is 50 Q is equals to five times sign 60 de. Now, this is a linear equation, so we can calculate i f s. That is integrating factor as it is to integration of PDT and peace. This and we have que Estes. So it is integration off 50 d t integration of fifties since fifties Constant integration of one is t so 50 teaser integrating factor. Okay, so now our solution will be. Therefore, remember this Q. Let a name is rescued. A. Since this Q is representing charge, but this Q is a variable or a notation in the Web i d X plus P Y plus Q. So don't get confused. Sequenced. Accuse Will I write in Q dash Okay, so our solution is cute times integrating factor which is erased 2. 50 t which is equals to integration off que times This time it is cute. Dash, So cute does that is five times sign 60 d into ah yes, which is a raise to 50 de de de. Okay, so since Vice Constant, it will come out on integration off this to town, we know it is a raise to 50 t. I'll just write a formula here. So formulas integration off areas to a X sign BX the X is it is two x over a square plus B square, a sign BX minus because BX so we'll apply this year, so I'll just copy this down. So five times it is 2. 50 t is 50 so 50 square is 2500 b is 60 so 60 square is 3600 times a sign BX or ace fifties or 50 times sine 60 t minus B that it's 60 cause 60 t and we have plus c. So this answer what we need to find the value off c. So before finding it will simplify this well divided by areas to 50 t so this will become dividing by. This will cancel this term canceled system so we'll have five or 2500 plus 36 hundreds. 6100. Now here we have fifties and 60 so we can take 10 out. So we'll have five times. Sign 60 T minus six times cause 60 D And when we'll divide by it is to 50 t here. It will come in numerator as minus power. So it is to minus 50 t. Okay, so we can cancel this 10 and we can develop by five. So 5105 2005 fight to is that sorry? Okay, so we have Q as one. Work 1 22 five times. Sign 60 D minus six times cause 60 t plus Siri's to erase to minus 50. 53 times. It is to minus 50 t. Okay, so now we need value of C. So condition, given this at Kew discharging the capacitor is zero at T equals to zero. Therefore, zero is equals to one or 22 five times 60 times 00 Sign zero is zero minus 60 times. 66 times cause zero, which is one. And this is C times it is 20 because 15 to 0. Okay, so we have zero is equals to minus six or 1 22 plus c. So therefore see is 6/1 22. So we have our Q as their four. Q is 1 22 five times sine 60 t minus six times cost 60 t plus C times that it's six by 1 22 times it is to minus 50 teach. And that's a solution for Q. But we need I. And we know I is the Cuba de de. So we'll have I z equals toe one by 1. 22 five constant different session of Sinus cause so cause 60 t in two different station off 60 tease 60 minus six times cost. Sorry, cause will be differentiated. So differentiation, of course, is minus sign. This is 60 t and differentiation of 60 days 60 yeah, plus six by 1 22 times different station off it is to excess. It is two x and differentiation of minus 50 ts minus 50. So I becomes if we take out since we have 60 here, so we can take out sixties or 60/1 22. Five times cause 60 T minus, minus, plus off plus six times sine 60 T. Yeah. And here we have minus 15 to 6. So that is equals to it's official minus 56 A thirties or 300 over 1. 22 times. It is to minus 50 t. Now, if you see, we can divide this, by the way, at least so toe checked Visa 262 ones. Similarly. Year $200. Sorry to 1 50 times. And this is 60. Therefore, eyes 30/61. Five times cost 60 T plus six times sine 60 t minus 1 50 Over. This was 61. So 61 it is two minus 50 50. And this a solution for Q, so Yeah, sorry. Hi. So this is cute. And this is Hi. Thank you.