With that, and I hope you discussed in some detail at the workshops, let's consider the following. Let's say that you have.
2 looks to conducting loops as shown there.
They can be of any arbitrary shape and size and so on.
So let me draw one here.
This is Luke 1.
And this is look to arbitrary positioning saves and so on.
This is 1.
This is true and I would say that the in loop one and Ranger current.
What would be the effect of that? Let's just put in the opposite direction.
What would be the effect of that current one?
It will create a magnetic field. OK, we know that the currents are sources of magnetic field, so there will be a magnetic field produced here perhaps, and I will point.
Would be like that like this like they just think of some other Big Daddy complex saying.
And this is B1. This is the magnetic field produced by the current I-1.
I'm.
In in Loop 1 Jeff. And of course this magnetic field. I have grown this lines as if there open lines, but you know that the magnetic field shells around so you after awhile this lines perhaps you will come back and close the door. Don't take them to be open, it's open in the fraction of the schematics on here.
Some of this magnetic field will actually go through the surface.
Let's call that Surface S 2 because it is a surface in the coil.
In the in the conducting loop too. So then there will be a magnetic flux becausw of B1 through.
The 2nd loop through loop 2.
Let's try to calculate this.
And what is that magnetic field B1? I can calculate it everywhere in space. Let's say I want to calculate it here.
At this point.
Maybe it's there some like this.
How do I calculate it? What if?
I have an element.
DL on that loop.
And I take R.
Which is the vector of connects deal with the point where I want to know the magnetic field.
Then new not I won.
Over 4 Pi.
DL, it's called that one DL1. Cross, our.
Over R ^3.
Is the contribution to the magnetic field because of that element?
DL1, and if I want to know the magnetic field through the entire loop, I basically integrate this over the entire closed loop 1.
And this is B1.
This is because of all.
So this can be really really complex to do if you have you know complex geometries of loops and somebody that doesn't matter. I mean I'm not going to do any calculation here, it's just in principle.
If you know the geometry, you know the magnetic field through biosurveillance.
So I know the magnetic field so then.
I don't talk too late the flux.
F.
2.
Which is big cause of the magnetic field B1 crossing the area as to what is this surface integral I need to integrate the magnetic field be one over the entire surface has two, this one.
And that will give me the flux.
In in the conducting Loop 2, because of the magnetic field B1.
Again, this can be very complex, right? If the if B1 is kind of.
We would save you know, complex and so on, and the surface has to, you know it can also be arbitrarily complex.
But in principle.
I would just stick it to computer to make it in principle, then that's it.
And fight 2 timber it and us putting that expression for the magnetic field there fight too.
Is the integral over the Surface S2 of.
No, not. I won over 4 by the integral over the length of one of the L1 cross R / R ^3.
And all of that which will be a vector. I would that take its dot product with these two.
I can rearrange, isn't it?
And they can ride, fight to us.
Well.
It will be the integral and assure that destination is not the.
He's gotten a closer. Let me delete this. It's an open surface, so it's an integral over the open surface as two.
No, not over 4 Pi.
Crawl over the close surface, line L1 of DL1, cross R / R ^3.
I'll take the dot product.
Of all of these, with the has two and all of this.
Is multiplied with, I won.
So again, there is a very complex expression inside. It can be really difficult to do it if the coins have really complex shapes and so on, but in the end it's purely geometrical.
See, there's nothing there other than constants and distances and areas and in control, and that purely geometrical factor. I can write it as M21.
Yeah.
So the flux F2 is this constant.
M2 one times I won.
It tells me that if I run a current.
In conducting loop one I will get a flux in the conducting loop 2 and the flux will be proportional to the current.
So the flux in one will be proportional to the current in the other.
And the constant of proportionality, that thing called M2 one, is what I call mutual inductance. OK, so the mutual inductance.
What is a purely geometrical factor? You can see it here. There's nothing here other than distances and lengths and so on, purely geometrical factor that connects flux in one conducting loop and current in the other.
I could do the opposite. I could run a current in Loop 2 and see what is the flux and look one and.
I would have a similar expression. Then F1 would have been proportional to the current.
In Luke 2 and there would have been another constant, you know another complex expression like this one above which I would call and 1 two.
In fact, it's not very easy to see this from this expression I have written here.
And and I.
I think we're not yet in a position to expand much further. It's not very difficult, but we have not really used vector potentials all that much in this lecture series, as we just stuck to the basics. But if you were to use vector potentials, you could write this mutual inductance.
A simpler form.
And he waited for you here.
You can write it like this. It's just a matter of using the vector potentials and Stokes theorem. But since we have not really.
Employed vector potential, not much and I'm just keeping it. I don't expect you to know this as well, but in this case the mutual inductance could be written like this and now it looks much more symmetric and from there you can actually quite readily see that.
If it goes to interchange.
In this is one and two. I will still have the same integral.
OK.
And that tells you basically that M21 and M12 are the same.
Now we can see here. Just Change index index indices one and two still get the same in different.
And for that reason that mutual inductance that I have written here, I'll just drop the indices and tightly.
And just call it M.
So that's the mutual inductance.
Uh.
We show, I think in in in workshop then.
Again, purely a geometrical factor, but you see here there is some something remarkable here.
It doesn't matter like how complex is the shape of this conductors and conducting loops. I have one here with a weird shape. I have another one in some other distance away. Again, some very weird sake.
The fact that M2 one is the same as M12 means the following basically.
If I if I rank in one.
A current I.
I get into.
Efflux fight Tool which is M times I if not I run the same current I.
On Loop 2 on Loop 1, I will get exactly the same flux.
So you know.
So F1 is the same as F2, so under current in one, yet the flux and tool run the same current in two yet exactly the same flux in one, no matter how complex and different and weird the shapes and distances of these loops. And that's that's quite. I think you're remarkably.
So the inductance mutual inductance here is is measured inside the unit is the Henry so.
Henry 1 Henry is basically uebers.
Although untested flux is given investors.
Current in a purse. 1 Henry is 1 November. Other one day.
So we saw what the inductance does. The neutral inductance in this case connects the flux in one loop with the current in the other.
OK.
Now let's consider what happens if the current changes.
Let's go back to the schematic here.
So let's say that if this current I-1 actually change with time, is it, dear? I want other DT which is other than zero or it's it's it's a variable current?
Well, this is a current that changes with time.
Will create a magnetic field but also change with time.
OK.
So that B1 is now a function.
Of time as well as of position.
And then because B1 change with time, the flux through this surface.
Also change with time and there is less or more magnetic field going through it because you'd be one changes with time.
And we know from Faraday's law.
But if you have a closed loop.
And that there is.
A changing magnetic flux through that loop. Then there will be a near meth that will be induced on the loop.
Check.
Remember, Faraday's law.
OK, so let's let's delete here.
I like you.
So if I have.
A magnetic field.
In coil one in conducting loop one but change with time that will induce an EMF in look too. So there will be near math in two which will be minus the rate of change of flux.
In Luke 2, but we have seen.
Not ******* fight two is just the mutual inductance.
Times current one.
Times I want it's the current that changes, not geometrical aspects of the problem and doesn't change only with time only. I want change with time so I can write this as minus M.
DIY.
Times DT.
The opposite, and vice versa I can now.
Around the current on coil two that change with time that will induce an EMF in Luke 1. So in loop one also through the same line of argument.
The EMF could have been minus MD I2 DT so I see another way now that M is useful. We said before that M the mutual inductance connects the flux in one.
Loop with the current in the other. Now we see that the same constant also connects the EMF in one loop with the rate of change of current in the other.
OK.
So two different expressions here.
What M is useful connect.
Flux in one with current in the other, or EMF in one with rate of change of current in the other loop. Other there always two loops here to conducting loops which are involved. And yeah, this is the mutual inductance of the system of two callings tulips.
So you see it then, but because you can run a current in one coil, changes with time.
And producing EMF in another coil can actually transfer energy from one coil to another. These two coils. These two conducting loops are now magnetically coupled.
So you can you can transfer images from one to another basically wirelessly at the end. This principle of mutual inductance is behind. I'm sure you'll know about this.
Inductive charging.
I'm sure not nowadays, and when you view my charger phones using this idea, but you also in the future, we will also charging our cars in pretty much the same way but.
Through through through inductance.
Would you?
So far I talked about mutual inductance that again it is worth remembering connects flux and in one and current in another loop or EMF in one and rate of change of current another loop. But now let's not consider a system of two loops, but let's just consider a single loop. I now have a single.
Conducting look like that for the things adjusted one.
I cannot make current.
Here.
And that, you know, will create a magnetic field.
Not sure.
Through itself
so there will be a flux through itself.
Because of its own parent.
But you like Fi?
Is proportional to I asked before.
See here.
5 two is proportional to I-1 here I don't have oil one and coil two and just have one, but things are still.
I said there would be a flux through itself because of its own current and there will be a constant of proportionality which will be the same kind of quantity. Ask the mutual inductance. It's still in the units of Henry and so on, but because now it does not involve.
Two conducting loops, but only one. I will be calling this self inductance.
But it is the same kind of physical quantity as the mutual inductance changes the same unit social and does connect again the flux and the current, but now not the flux in one loop in the current in the other. But you're the flux and the current in a single loop and that's the self inductance.
Then consider what happens when that current changes with time.
Its own magnetic field will change with time. The flux through itself will change with time, so that changing current will induce an EMF on itself.
And now again, the EMF as above.
Will it be related to the rate of change? Negative rate of change of the flux?
Only I changes. So basically he.
In DIDTR, again related and L is the constant of proportionality like above. See these expressions. But again, doesn't Apple together to neighboring loops. It's all about a single loop.
Again, this is the self inductance. This is actually quite interesting and the fact that it's changing.
Current induces a near meth on itself and do remember they love it.
This minus sign here and discussion we had back at the night was like.
Let's just say.
Some like that or 8 about lenses law.
And how it is a EMS.
Induced by a changing flux.
Is a kind of inertial effect that resist a change for that reason. Here this EMF.
The self induced EMF is called back EMF.
Back EMF.
The net effect of that back EMF is to try to undo the change in in current. So if you run a current and then let's say, do increase it.
This will change the magnetic field will change, the flux will produce an EMF.
The effect of that EMF would be produced a current in the opposite direction, so you increase it in the 1st place. Now have a current in the opposite direction that tries to undo. This effect price would decrease the original current.
That's why it's called Buck EMF. It resists saying this in the current.
And it's a it's a time to fly off any nurse elephant.
OK.
Um?
I think we're close to the end here. I think I have described what I want to say about the inductance and self inductance and so on.
Yeah, I'm sure you all know a lot about resistance and how they resist the flow of current.
In this lecture series, ******** talked about capacitors. Capacitors is basically an element that has capacitance, and capacitance is an ability of the body to stock charts.
There's another kind of electrical element which is called inductor and an inductor, as we expect, based on what we described is an element that resists changes.
Resist changes in the.
In the current, because you know there is this back EMF which is induced.
And so it has inductance. Also an element that has inductance. This resistance to the change of electric current to the change of letter cutting is called an inductor away. To think about inductors. Guys like this solenoid coil went around many many times like this and that's not the first time we have seen that we have started a very similar element but in a lecture 6 or so when we started on Purslow and back then we did.
Derive the magnetic field.
Of that solenoid, and now we can use that to actually try to derive the self inductance of the solenoid.
And I've been.
The self inductance L is basically what connects the current flowing here.
Through the magnetic flux.
Through a single winding.
But if you have any windings.
Then you can write this like this.
Right?
And this one here is called magnetic flux linkage.
And that's here. Then what L gives you for the entire inductor connects the magnetic flux linkage with the current that flows through through the coin.
If you have.
N.
Wine things.
Per unit length and lower case winding sprinklings thanks have a length of X so then that not uppercase and the total number of windings is the number of windings per unit length times the overall length and the magnetic field is the is short. The flux of the magnetic field is B times the area.
The constructional area here of that coil.
Over I, so that's all we know what B is from above. If we substitute there, we have an X times new, not an.
I * A.
Over I.
But I gets cancelled and you have that L.
Is N squared new not.
X Times Day I think.
And if not, would you want to calculate? Here is the inductance per unit length.
Haha then.
Of the solenoid it is N ^2.
Times, not times say again, it's it's a geometrical quantity.
OK, so that's how we typically represent, and you're not going along the similar lines. Calculate this inductance and for many other configurations.
We typically denote this in in a circuit.
As shown there, I mean still turn it away. Client draw a solenoid so that's now this.
If you run the current I then that current I changes along it's a.
Edges here there will be a voltage which is given by this expression, right? So that this is the back EMF. It is minus the self inductance times the rate of change of.
Of current, so that's the third electrical element you know. I mean, you also know the capacitor.
Remember that if you have a capacitance, CC is Q / V, so V.
Is secure over? See, that's the the voltage.
Here at the edges of that element and you also know the.
Resistor and if you run a current hi there.
Fee are the voltage drop along the edges here of the resistor is I times are so these three things here. Together the voltage drops along the edges of an inductor, capacitor and resistor and we put together allows you now to analyze many many electrical systems in detail and in the rest part of this lecture. Here there is some details about our LLC and other solutions, some quite interesting aspects of.
More more applied physics if you want and also in the next lecture. There is some also very interesting material. I'm hoping that at some point you know you can take an interest and go and have a look at some of that, but for us.
The material for this lecture actually stops here.
And this is also the end of all the examiner examinable material, OK?
Do make me a favor and you at least glance what is there in the next lecture and the other half of this one because it's quite interesting, but I'm not going to examine you.
In the final exam on that material, and in all the previous years as well, this was stopped.
So let's leave it here. It's exactly 3.
Please collect your questions and come back to me before next Monday and next Monday noon. We will have our division.
OK so thanks for connecting and talk to you soon.