00:01
The equivalent to a mass on a spring is what's called a tank circuit.
00:07
In electronics, it consists of a capacitor that is usually starting out charged up, and it is connected to an inductor.
00:19
And what happens is electrical energy gets exchanged between the capacitor and the amount 1 -half q squared over c, so that is kind of similar to the displacement of a mass on a spring.
00:39
But it is exchanged between the capacitor and the inductor, which is a lot like the kinetic energy of a mass on a spring.
00:52
So the current in the circuit will switch back and forth, and we can think about it as the capacitor is discharging through the inductor.
01:03
Which stores the energy in its magnetic field.
01:09
And then the capacitor gets charged up through that energy, but charged up the opposite way.
01:22
One full oscillation in the charge, so there is a natural frequency of oscillation, and that is, well, we'll show what that looks like in a little bit, but the oscillation frequency is given as one over the inductance times the capacitance.
01:44
So just to show what this looks like on a graph, the charge as a function of time on the capacitor will assume that it starts out with the right plate positive and as high as possible.
02:02
But that charge basically goes through a cosine type of discharge and then returns to its full initial value, much like a mass on a spring.
02:25
So there is a period that's given by 2 pi over a mega knot for one full cycle.
02:34
The current in the inductor on the other hand, you might say, is there a current through the capacitor? well, there is kind of, we won't get into that.
02:48
But the current in the inductor is simply dq by dt is the current, and our charge looks like whatever q0 is times cosine of the oscillation frequency times time.
03:11
And if we, of course, take the derivative of that, it is minus q0, omega -naut, sine of omega -0, so the current in the inductor looks like a negative sign at the same time values.
03:36
Let's see, so it'll kind of go down and then it'll go up, and it has a peak of q0 omega -0.
03:46
So that's basically how a tank circuit works, and we can kind of take an example with two capacitance and an inductor.
03:56
Given.
03:58
And we will assume that the capacitor is charged up with a 12 -volt battery.
04:12
So we can actually figure out the q -0.
04:18
Q -0 is just c times that v, or in this case, 6 times 10 to the minus 5 faraday's times 12 volts will give us the charge in coulums.
04:38
So 7 .2 times 10 to the minus 4 quillums.
04:47
So that will get us going.
04:49
The other thing that we, of course, need to get going is the oscillation frequency.
04:57
It is simply 1 over the square root of 1 .5.
05:01
Henry is the proper unit for the inductance.
05:06
And 6 times 10 to the minus 5 faraday.
05:10
That should give us radians per second.
05:16
And working that out, we get 105 .4.
05:24
We can even figure out our time unit based on that.
05:37
Period can be obtained from that angular frequency, and we get 5 .96 times 10 to the minus to 60 milliseconds.
05:49
So that kind of scales our graph for us.
05:56
Now we can use our equations for charge as a function of time then, and current as a function of time, to get these quantities at any later time.
06:14
So here's a case.
06:15
What about t equals 0 .02 to 3 seconds? i will show where that is in terms of the cycle.
06:29
So it is way less than a cycle and about a third of the cycle in.
06:39
So we have these marked off in quarters...