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[20%] Find the general expression relating the charge q(t) on the capacitor in a circuit containing a capacitance (C 1OpF) , a resistor (R = 10 ko) and voltage sour...

Question

[20%] Find the general expression relating the charge q(t) on the capacitor in a circuit containing a capacitance (C 1OpF) , a resistor (R = 10 ko) and voltage source(E = 10 V), given the initial conditions:if i(t=0) = 0 when t = 0.[Recall; i(t) = 490) [Solve for the resulting differential equation using the "Integration Factor" technique, showing all your workings and assumptions mark all final answers clearly] (Hint: Solve for the equation using the variable labels; then replace with

[20%] Find the general expression relating the charge q(t) on the capacitor in a circuit containing a capacitance (C 1OpF) , a resistor (R = 10 ko) and voltage source (E = 10 V), given the initial conditions: if i(t=0) = 0 when t = 0. [Recall; i(t) = 490) [Solve for the resulting differential equation using the "Integration Factor" technique, showing all your workings and assumptions mark all final answers clearly] (Hint: Solve for the equation using the variable labels; then replace with the values at the end) I0ko 10V 1OpF



Answers

Electrical Charge The equation governing the amount of electrical charge $q$ (in coulombs) of an $R C$ circuit consisting of a resistance $R$ (in ohms), a capacitance $C$ (in farads), an electromotive force $E,$ and no inductance is $$ \frac{d q}{d t}+\frac{1}{R C} q=\frac{E}{R} $$ where $t$ is the time in seconds. Solve the differential equation, assuming $E, R,$ and $C$ are constants and $q=0$ when $t=0$.

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In this question where electrical circuit Q. Is the church on the capacity and are Here's the resistance. He is the footage sauce and T. Is time we want to find Q. In terms of T. So let's integrate the right hand side. Now before we do that I wanted to fit to the form of integrating it about you respect to you. So my result will be it a value plus see a constant. So let's you be key over RC. So do you over D. T. Will be one over R. C. The you will be one over R. C. E. T. So looks like I need a one over R. C. Over here. So let me do this. E over our. My duty is over here. Let me put one over R. C. Now since what we are seeing is over here. I need to balance by putting RC here in front. Mhm. And my veto power of T over our cities like that remains unchanged. Yeah. We can see that the R. And R. Here cancers. So we will have easy evil power of this is you. And this part over here is the you. Now you can see that you fit the this form. So I'll be using this result. Yeah. So it'll be easy. It's the power you plus let me use K. And another C. You'll be sounds so confusing. Now replace back the you with key over R. C. Because this is a function of T. Okay now to find K. We were so integral. Zero Q go zero. My K. Is going to be minus E. C. So putting it back into over here we will have Q. It'll party over R. C is equals to easy E. T over R C minus easy. Now every time will be divided by this. Every time. Yes, we will get Q is equals to easy minus E. C. e to bomb minus T over R C victor writes out the ec, I'll get one minus either power minus t u r c. That's our answer.

You got our sister 400 to the naked eye. So we got the equipment. Six Double. 400 for you. Divided by who? He's here. 10 86 clones. Q prime. You get 0.6 r squared was for our team. For people here. I make it 100,000 plus four Squared three. Yeah, you e he, uh I just want out us keys. Where que zero. You? You wait three Each native, one of my three minus 50 square of 3 70 one thing 1003 Really? You prime Fear will be too. I see one. It's good. We have a post 50 off minus 50 square. See you. Rain is 1000 way, I think. Here So see equal 27 5 20 plus square CTO train sitting into the equation But above you B b c two people to town naked six We're fight to times square above three entering C two into one equations fee One of many people tend to ladies naked Problem plus square Did you take the square root to the city? In the next six? Making 20 was square 37 to square with 3 70 e to the negative off, maybe 100 g plus And in the anchor six. That case we're ready 20 plus where three deputy multiplied by each and negative off each 1000. He was put minus 1000 years e to the negative one. I eat too negative. 1000 1000 plus post 50 square Ruutel 3 70 Divide a bit theory.

Between children magic stored in a capacitor this equal to Q squared. Thank you. Is the charge told in the capacity of over two times depositions off the babysitter. Now we have provided with the initial putting children and see the capacitor. So to find the initial charge, we can use this equation secure. Not really be guys see times. Excuse me. The initial part in Trinity start off started the capital and we need to take over knowledge, substitute the values. What? And I'm solving this. Get killed. Not to be. I'm counting our finest equivalence. So that's still a nation chart stored in the deposition? No, in a RC circuit that is discharging that respect your time. The charge as a function auf die can be given by this a question. What is the time? Q Is the charge in the capacitor at time T on dhe? How here is the capacity of red constant that is equal to our times. See, now we know that calling is the derivative off chart with respect your time. So I will be equal to negative. Thank you over TT. And the sign negative is because Q is decreasing the time This is a This is the question for discharging RC circuit. Therefore, Q is decreasing the time for the cue, buddy. They lead to a negative value. And in order to make that positive, we need the current to be negative off the innovative. This is equal to cannot overto you to depart minus Steve over knows that this is positive. Now I'm from here. We can say that the initial current that is acting goes to seal the current will be equal to kill, not over. So this basically came by substituting t to be close to zero. And now that we have this begin first beginning calculate the time constant yet is equal to the resistance times capacitance sampler, women, all the values. So this gifts the acquisitive time constantly be once did it. And now we can find the initial current using this a question. This one. So we have kick you note from the first fight, and we can collect it now to snow. So millions substitute those two values and get the current to be one. Find seahorse times 10 to the garage minus three at Pierre's. Now in the second part, we have to find the world stage. So we know that cube is equal to kill not times into the power line. Stay over on DVI from fatter. This law can be said to be the DOD stored in the capacitor over compositions. So now we can substitute this over here. This means that we will be equal to kill not into the part dynasty over how you see. So I'm just substituting the cue. Oh, from this expression over here said Nope living in all the values and this gifts the world needs to be equal. Who One point dying stand in the party voice I'm saying that about minus minus t maybe is measured in seconds because our how is second city has to be in seconds in order to make this ratio unit list so that we can apply the exponential function to it. So that's the expression for the voltage. Also, there's one more thing that I forgot to write over years. So we need the current function the current function. So I'II is equal to I Not times you could apologize minus D over now. So now in just one hour. So I noticed this values you. I want someone just gonna substitute this? You did the part minus stick because dollies from 51 seconds and here as well is they should be in seconds. So that's expression for current. Andi In third part, we concluded the voltage across the capacitors. So let me just right here. See? We'll see. You know that, Uh do you know that this is the voltage across the capacitor? Because in the fourth part, we have to find a world particular service. Sister, Now the world is a crystal. A sister is current Times are on DDE. We have already found that skull and over here don't mind up capital on the small characters, their save So the current is expressed over here. So I'm just gonna use this expression place off current over here, minus three MPR. Even if I'm minus t dines one time sanctify six wolves said this will give the artery one point old times and the party rules he did about minus T and he does it thes measured in sick. It's now you have to find the power dissipated Cross That is a stretch Still power is I squared times tush So using the same expression off. AII that we found in part, bees said, scared and needed a polish minus T squares. And that gives you to devote when Mina Studi dimes. It is a stooge, the resistance. And this keeps the power to be one word times E to department student. And obviously he is measured in seconds. So those are the answers to on the parts. So we found first charge the initial charge store or basically an expression off charts. In the second part, we found the expression off current. In the third part, we found VC or did strolled in the capacity of an act that particular time t In the fourth part, we found the world is stored in the resisters time as a function off key. And in the last part, we found the power dissipated at time t across the inner resistor. Thank you.


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