00:01
Okay, we have a circuit.
00:01
We're given this information.
00:04
This is what the circuit looks like.
00:07
The equation for the charge on the capacitor looks like this.
00:14
It's a second order differential equation for q.
00:22
Okay.
00:23
So we need to find the time constant and the frequency.
00:31
And the easiest way to do that is to assume of a solution that looks like this, where we have e to the i omega t.
00:41
Omega could be complex.
00:44
The reason to do it this way is you can do it with signs and coscience and exponentials, but it's a lot easier to make mistakes taking the derivatives and then trying to solve it.
00:54
This way is pretty straightforward because it's easy to take those derivatives.
01:09
So the thing in parentheses is our eigenvalue equation.
01:13
We have to solve for omega.
01:17
It's just a quadratic.
01:24
Looks like that.
01:27
So the i .r over l term, that's actually our decreasing exponential because we've got i omega in our solution.
01:38
So the first term is going to be the decreasing exponential.
01:43
The second term is going to give us the frequency.
01:48
Actually, it's the angular frequency.
02:02
So the thing under the square root is four.
02:05
So we get omega is 60i plus or minus 20.
02:26
So i can write my q like this.
02:41
Remember omega is 20.
02:43
So my general form of my charge on the capacitor looks like this as a function of time.
02:50
Q1 and q2 are right now unknown constants.
02:59
Current, we have to calculate by taking the derivative...