00:01
The most general way to handle any circuit, including ones with inductors, capacitors, etc., in them, is to use what are called kirchhoff's laws for circuits.
00:15
And basically, there are just two of them.
00:19
They have as their basis, fundamental physics.
00:24
One of these laws says that you basically have charge conservation, that the sum of currents going into a junction.
00:40
So we'll write that with a summation sign, i for current, and we'll say in has got to equal the sum of currents out because otherwise you would have charge being lost.
00:55
And this is called the junction rule.
00:59
And what we mean by a junction is a place where the wire doesn't just go straight, but there's an intersection, so that's probably a good way to think about a junction.
01:11
It's an intersection where the current has a choice in two directions, at least, to flow.
01:20
The other one is energy conservation, and it has to do with the fact that the electric field is conservative, so that if you go all the way around a loop and you add up the voltage gains, so i'll write that voltage gains, they have to equal the voltage drops, and that is called the looper rule.
01:54
It's basically saying that there is no change in potential between one point and that same point.
02:03
When you return.
02:06
So here we've got one junction.
02:09
Actually, there are two junctions, but they are complementary.
02:12
They involve the same three currents.
02:15
So if we look at the junction rule for the circuit above, we have that i -1 goes in to the junction at the top, and i -3 and i -2 come out.
02:29
So our rule just says that that i -1 has to be the sum of i3 and i2 to conserve charge.
02:41
Now, typically what you do is you look at your circuit, and you should have as many independent equations as there are currents.
02:52
So we are after basically three equations that we can work with, and we may be able to get them down to two, but you're guaranteed, hopefully, that there are three independent equations.
03:08
So let's now look at the loop rule.
03:10
How many loops are there? there are actually three loops, and what to call them.
03:19
There's the two loops made by the two sides of the circuit, so we'll call that one i, roman number one, and then there's a second loop for the separate side, but there is also a third loop that goes all the way around the exterior.
03:44
What typically happens when you have the situation is that only two of those loop equations will result in independent equations, and one of those will kind of be repetitious and not give you much more information.
04:00
Just like that second junction doesn't give you any new information.
04:05
But let's go ahead and write down all three of the loop rules, just so we know what they mean.
04:12
For loop one, what you typically have to do is pick a place to start.
04:18
I usually pick one terminal of the battery, and i'm assuming that the gain goes up, the voltage gains going up the battery at point a to point b.
04:34
So there's your voltage gain.
04:41
We're going to walk around the loop in a clockwise sense.
04:46
Now, the inductor is the l.
04:51
That is an inductor.
04:53
And it will obey faraday's law that the voltage across, the emf across that inductor is l -d -i through it, and that is i1 by d t and now we have to think through is that a positive or a negative term and if the i is gaining the inductor will fight against the current gain and therefore the voltage will drop so yes we have a drop on the right -hand side if i is increasing.
05:43
I2, the voltage always drops going in the direction of current flow through a resistor.
05:52
So that is a definite drop.
05:56
And we have written down voltage gains.
06:00
I like to keep it organized with the gains on the left and the drops on the right.
06:10
So we're good.
06:12
Now, loop two, what do we do? looking at the capacitor, the voltage drop on a capacitor is q over c.
06:28
And that could be either a gain or drop, depending on which way you walk around the loop.
06:34
So again, we are going to walk around that clockwise.
06:38
And by the way, the current is drawn i3.
06:42
The top of the plate is positive.
06:44
The bottom is negative.
06:47
And so we have a voltage drop over the capacitor in the amount of q over c.
07:01
So we're starting, say, on the positive plate of the capacitor, we'll call that point a and going to the opposite plate walking in the clockwise direction.
07:12
And then when we get to the resistor, notice that we are going if we walk around, we are going opposite the current, and so we have a voltage gain in the amount of r1, i2.
07:32
And then finally, we're going with the current i3, and so we have a drop over r2.
07:48
Okay, and that is loop 2.
07:52
And just for completeness, let's go ahead.
07:54
And look at loop 3.
07:57
Again, for loop 3, i will start at point a on the bottom of the battery.
08:04
We're assuming that's a negative terminal and a positive terminal up at top, and that's what's making the current go the way it does in i -1.
08:15
But again, we have a gain with the battery, and again, we have a drop through the inductor.
08:32
And again, we have a drop across the capacitor.
08:39
So we'll put that on the drop side.
08:48
And one thing i do want to make a note of is we could use the transitive property on loops 1 and 3.
09:10
Notice that both of those have a common left -hand side.
09:16
And so we're going to set e equal to e.
09:21
And fill in the two sides, l -d -i -1 -id -t plus r -1 -i -2 is equal to l -d -i -2 plus l -d -i -t plus q over c...