00:01
Here we'll look at the equation that governs a capacitor and an inductor in a circuit together.
00:08
So this is something known as a tank circuit, which we'll see might call that a little bit.
00:14
But we have a situation of conservation of energy means that the voltage across both components has to be equal.
00:25
And so the voltage across the capacitor is given by the electrical relationship q over c.
00:36
And across the inductor, we won't get into the physics behind this, but it is minus the inductance times the first derivative of the current with respect to time.
00:50
And those must be equal to each other.
00:53
So that is going to give us our differential equation that governs that circuit.
01:01
And then we'll have to think through what quantities we want in there.
01:09
So we're wanting to find the charge on the capacitor as a function of time.
01:14
So just a reminder that a capacitor has two plates, a positive, and a negative plate.
01:21
And what we mean by the charge is the absolute value of either.
01:27
Charge on either plate.
01:30
And when the top is positive, you'll get current coming out of the top plate.
01:35
If it switches, you'll have current coming out of the bottom plate.
01:40
But the other electrical definition we need is the current is defined to be the time rate of charge flow.
01:51
And so our differential equation is q over lc is equal to negative.
01:59
Well, let's put the negative on with the q and the lc, the second derivative of q with respect to time.
02:09
So that is a second order linear differential equation.
02:13
We can figure out that it will be governed by a characteristic equation if we set q equal to the exponent of e to the lambda times time.
02:31
Take a second derivative with respect to time.
02:38
That lambda will come out twice.
02:44
And of course, the characteristic equation then is lambda squared plus 1 over lc equals 0.
02:55
And that has two roots, both imaginary.
02:59
Lambda is plus and minus i over 1 over the square root of lc.
03:08
And the imaginary, yeah, that's a different i than the current.
03:12
Let's give it a different symbol.
03:15
That means the square root of minus one...