00:01
Hi there, so for this problem, we have the diagram of an rcl circuit with a resistance that is equal to 10 oms, and the capacitance is equal to 1 over 100 ferrets, that is the same as 10 to the minus 2 ferrets.
00:28
The inductance for this circuit is also given, and that is 1 over 2 henrys.
00:37
And the voltage for this is also given, that is 12 volts.
00:44
Now the switch is close at a times the equals to 0, so we need to assume that the initial current at that time equals to 0 is equal to 0.
00:58
And the initial charge on the capacitor is also given, and that is equal to zero.
01:08
So we need to find the current, the expression for the current that depends on the time.
01:17
So that is going to be the solution of the following equation, that is the differential.
01:26
The second derivative of the current with respect to the time, and this, plus, plus the resistance over the inductance times the derivative of the current with respect to time.
01:40
And this plus 1 over the inductance times the capacitance times the current.
01:47
And this we set this equals to zero.
01:51
And according to kirchhoff's law, we have another condition in that one is that the derivative of the current with respect to time evaluated at the time equals.
02:04
Equals to zero is equal to the voltage divided by the inductance.
02:10
So in this case, we start with this expression right here.
02:17
And now we just simply substitute the values that we are given for this problem.
02:22
So we will have the second derivative of the current with respect to time.
02:29
We substitute into here the resistance over the index.
02:38
So as you can see, that will give us a value of 20.
02:42
We're going to put this without first, without the units.
02:46
So we have 20 times derivative of the current with respect the time.
02:52
And the other term that we have in here is the inductance times the capacitance, the inverse of that.
03:03
And if we put those values in here, we will obtain the following.
03:07
We will obtain 200 times the current, and that is equal to 0.
03:17
Now, to solve this differential equation, we're going to set the following.
03:21
We're going to set that m square is equal to the second derivative of the current with respect to time, and m is equal to simply the derivative of the current with respect to time.
03:34
So from this, we're going to have the following equation, m squared plus 20 times m and this plus 20 plus 200 sorry and this is equal to 0.
03:51
Now for this problem we can use the quadratic equation to obtain the values that we can obtain for m.
04:00
So from the quadratic equation we will have the following.
04:03
That is minus 20 plus minus the square root of 20.
04:09
D to the squared and this minus 4 times 200 and this divided by 2.
04:19
Now as you can see from this we're going to obtain a complex number because it is going to be minus 10 plus minus 10 times the complex number i or the magic number number i so from this we can say that the solution for this it has an exponential of minus 10 times the time that this corresponds to this and this this in complex number will give us a part that is constitute of two terms with the cosine in the sign function so we are going to have a constant that we call the 1 cosine of 10 times the time, and this plus a constant c2 times the sign of 10 times the time.
05:24
Now in here, to obtain the constant c1 and c2, we need to use the conditions given from the beginning of this problem.
05:35
Now, the first one that we are going to use is that when the current, when at the time equals to 0, the current is equal to 0.
05:46
So if we substitute in here the time equals to 0, as you can know, as you know, the sign of 0 is 0...