Find the charge on the capacitor in an LRC-series circuit...

Find the charge on the capacitor in an LRC-series circuit when $L = \frac{1}{2} h$, $R = 10 \Omega$, $C = 0.01 f$, $E(t) = 150 V$, $q(0) = 1 C$, and $i(0) = 0 A$. q(t) = What is the charge on the capacitor after a long time?

Transcript

00:01 So it is given here l is equal to 1 by 2h.

00:10 Resistance we have given here r is equal to 10 ohm and we have given here the c is equal to 0 .01f and the function of that is equal to we have here 150 volt and at 0 q gives a value that is 1 and at i, i at 0 gives a value that is 0.

00:49 Now we need to find the charge on the capacitor an lrc series circuit.

00:56 So from here first we need to write lrc series circuit is that is l d square q upon dt square plus r dq upon dt plus 1 by c q is equal to we have here function of et.

01:32 So after putting all values we will get from here that is 1 by 2 q double dash plus 10 q dash plus 100 q is equal to 150.

01:53 So after simplify this we will get from here that is y q double dash plus 20 q dash plus 200 q is equal to we need to write here 300.

02:13 So this is we have given an equation.

02:18 So first we need to find the auxiliary equation this differentiated equation.

02:22 So auxiliary equation become m square plus 20m plus 200 is equal to 0.

02:34 So from here we have to write value of m after simplify this equation value of m become minus 10 plus minus 10 iota.

02:49 So from here our coefficient factor qc is equal to e to the power minus 10t multiply with c1 cos 10t plus c2 sin 10t.

03:12 Now so as you can see the right part of this equation number first is 300 which is a constant part.

03:20 So we need to find here the particular solution qp is equal to let's say a.

03:26 So from here we need to write that is the differentiation of first differentiation of this particular part is equal to 0 and second differentiation of this particular part is also is equal to 0.

03:42 So from here if you put the value of particular function pq and we need to put all these value into equation number first.

03:49 So our equation number first becomes 0 plus 0 plus 200 a is equal to 300.

04:00 So from here a value of a is equal to 3 by 2.

04:07 Now from here we can say that our general solution of qt is equal to qc plus qt qp.

04:25 So that is equal to we have here e to the power minus of 10t multiply with c1 cos 10t plus c2 sin 10t plus particular part that is 3 by 2.

04:48 Now we need to differentiate we have given here that is we know that it is equal to q dash t...

Question

Find the charge on the capacitor in an LRC-series circuit when $L = \frac{1}{2} h$, $R = 10 \Omega$, $C = 0.01 f$, $E(t) = 150 V$, $q(0) = 1 C$, and $i(0) = 0 A$. q(t) = What is the charge on the capacitor after a long time?

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