We've seen last time that using Biot and Savart's formula that if you have a current going straight into the blackboard perpendicular to the blackboard that we get a magnetic field at a distance R.
The magnetic field tangentially to the circle, B here, B here, and that the strength of that magnetic field equals mu 0 times I divided by 2 pi R.
If you walk around this circle, just walk around, and you carve up this circle in little elements dL, and you calculate the closed circle integral, so the closed circle of B dot dL, so everywhere locally you dot B with dL, the B and dL are in exactly the same direction everywhere, then you would find that this obviously is B times 2 pi R.
But B times 2 pi R equals also mu 0 times I.
This dL here has nothing to do with this dL here.
Don't confuse the two.
This dL is a small amount of length in the wire that goes into the blackboard which carries a current.
This dL is simply your dL when you walk around this current wire.
It doesn't matter at what distance you walk around.
You always get mu 0 times I.
You see it right in front of your eyes because B is inversely proportional to R.
And it was Ampere who first recognized that you don't have to walk around in a circle to get the answer mu 0 I, but that you could walk around in any crooked path as long as it is a closed path, something like this.
And now you have here the local B, which of course is perpendicular to this radius and here you have your local dL and if now you go around, closed circled any path, it doesn't have to be a circle, dot dL that now becomes mu 0 times I, which is known as Ampere's Law, and I then is often given an index enclosed.
It is the current which is enclosed by that path.
It is actually easy to prove this using Biot and Savart's formalism.
This is almost a third Maxwell's equation.
We already had two out of four.
This is almost number three, not quite.
We're going to amend it in the future.
What is ill-defined a little bit in this equation is what we mean by enclosed, and I'm going to define that now so uniquely that there is never any misunderstanding.
If I have a very strange looking closed path that I have chosen, that's the path I walk, I have to attach to that closed loop a surface, an open surface.
You can make it flat.
You're free to choose it.
You can also make it sort of a plastic bag so it's open here.
You can put your hands in here, and here, like a hat.
Any surface is fine, but you must attach to that loop a surface, so here I have some path that you could be walking, and this would be perfectly fine open- open surface.
Could be flat, but it could also be open, so it's like a hat.
And now I can define uniquely what it means by- what it means by this I enclosed, because if now I have a current that goes through this surface and pokes out here, then I have a current penetrating the surface and that is uniquely defined, and if I have another one coming in through the surface, call it I2, this is penetrating that surface.
By convention, if you go clockwise around, we follow the same notation that we had before, in the right-hand corkscrew notation, the connection between magnetic field and current.
If you go around clockwise seen from this side, so you go clockwise, then I1 as I have it here, in your equation would have to be larger than 0.
I2 is then smaller than 0.
But if you decide to go counterclockwise, which is perfectly fine, Ampere's law doesn't at all dictate in which direction you have to march around, then I1 would be negative and then I2 would be positive.
So we follow the right-hand corkscrew notation.
And so if you want to amend now Ampere's Law to do me a favor, but you don't do books a favor because all the books use the word enclosed.
I would like to see this replaced by penetration.
It is the penetration of the surface of the current, that is uniquely defined.
But a current enclosed by a loop is ill-defined.
Because where possible, when we apply Ampere's Law, we will try to find easy passes around circles sometimes, sometimes rectangles, and since you are free to choose the surface that you attach to the loop if you can get away with it you use a flat surface, but you cannot always get away with a flat surface.
So the recipe is as follows.
You choose your closed loop.
Any loop is allowed.
It may not help you very much if you choose the wrong loop.
Any loop is allowed.
You then attach an open surface to that loop.
And I penetrate is now the current that penetrates through that surface, according to this convention.
And the direction of rotation is free to you.
How you go around the path is your choice, but that defines then the sign of the penetrating of the curve, of the- of the current, according to the right-hand corkscrew.
So now we can, for the first time, calculate the magnetic field inside a wire that draws a current using Ampere's Law.
I have here a wire that has a radius capital R and a current is coming to me, I, and let's assume that the current is uniformly throughout the wire, so it has a uniform current density.
And I would like to know what the magnetic field is everywhere.
Cylindrical symmetry, I want to know outside the wire and I want to know inside the wire.
Let's first look at radius which is larger than R, and so here we have the cross-section of that wire, radius R.
The current I is going through this surface.
I now have to choose a closed path.
Since we have cylindrical symmetry it is clear that we would choose a circle, with radius little r, so we can be sure that the magnetic field strength is the same everywhere because of reasons of symmetry.
Since the current is coming towards me and I am free to choose in which direction I'm going to march, I know that the magnetic field is in this direction, so I might as well also march in this direction so that my dL's are all in this direction.
I don't have to do that.
I could march the other way around, but if I march counterclockwise then both terms left and right of Ampere's Law will be positive.
I now have to attach an open surface to my path.
Well, this will be, the blackboard will be, that open surface.
And so now I apply Ampere's Law, so I get B times 2 pi little r, because dL and B are in the same direction so it's a trivial integral.
That now equals mu 0 times I, which now penetrates my surface.
Uniquely determined, all these current from this wire that comes to me penetrates my surface, so times I, and so B equals mu 0 times I divided by 2 pi R, and that's the same result that we found last time, when we applied Biot and Savart.
So that's no surprise that you see this.
But now we have a way of finding the magnetic field also inside the wire, so here we have the wire again, the cross-section, current coming out of the blackboard, and now I want a radius which is smaller than capital R and of course my closed path again for reasons for symmetry is going to be a circle with radius R.
And my surface that I attach is a flat surface, and so here I go, B times 2 pi little r equals mu 0 times- ah, now I have to be careful, because now not the full current I is now penetrating my surface, but it is only a fraction that penetrates the surface, and the fraction that penetrates the surface is now little r squared divided by capital R squared times I.
You see, because the total current comes through the radius capital R, but I only have now a circle with radius little r.
And so I lose one r here and so we get a very different result.
You get now that the magnetic field equals mu 0 times I is now linear in little r divided by 2 pi capital R squared.
And this grows linearly with r, whereas this falls off as 1 over r.
And if you substitute in this equation r equals capital R, which then would be the magnetic field right at the surface of the wire, you find exactly the same result here.
Little r becomes a capital R.
If little r becomes a capital R, you lose one capital R, you get the same result.
And so if you make a plot of the magnetic field as a function of little r, then it looks like- like so, so this is little r, this is capital R, and this is the magnetic field strength because we know that it is tangentially to the circles.
It would be straight line and then here it falls off as 1 over r, and the maximum value here is the value that you find there if you substitute little r equals capital R.
I will now show you that we can, using Ampere's Law, also come very close to calculating the magnetic field inside what we call solenoids.
Solenoids is like a slinky current that goes around in a spiral, one loop after another.
I want to remind you that if we had a loop, a nice current loop coming out of the blackboard here, and the current going into the blackboard so there's a circular wire but I only show you the cross-section.
I want to remind you that the magnetic field as we discussed last time, would be clockwise here, would be counterclockwise here.
In the middle, remember, it was like this.
And then in-between it was like so.
That was sort of the magnetic field configuration in the vicinity of a loop through which we have a current going.
But now imagine that you put another loop here, current again coming out of the blackboard, going into the blackboard, and another one, and so on, several.
What do you think is going to happen with these magnetic field lines which now diverge?
They're going to be sucked in here.
This loop also wants the field lines to come through its circle, so to speak, and this one too, and so you're beginning to get a near-constant magnetic field and the more tightly these loops are wound, the more accurately will your magnetic field be approximately constant, and I have some transparencies which will show that in more detail.
Here we have a figure from your book.
You see five windings, a spiral.
If you look from the left, the current is going in clockwise direction, and so the magnetic field is going from this side to that direction.
And when you look here you see that the magnetic field is approximately constant inside, and outside these current loops, outside the solenoids -- we call them solenoids -- magnetic field is extremely low.
And if you start winding these loops very tightly, then you get a configuration looks like this.
You get an almost perfect constant magnetic field inside the solenoids and the magnetic field outside the solenoids is extremely weak.
And now I would like to calculate with you using Ampere's Law what that magnetic field inside such a solenoid would be.
And we have to make a few assumptions.
Let this be my solenoid, and the length of the solenoid is capital L.
A current I is going through like so.
I, and I assume that if I look from the left side that the windings are wound clockwise, so I know that the magnetic field is then in this direction.
I make the assumption that the magnetic field outside the solenoid is approximately 0.
I will show you later with a demonstration that that's a pretty good approximation.
And so the question now is, what is the magnetic field there.
And I assume I have N loops, N windings, capital N.
So now I have to choose a path.
I have to apply Ampere's Law.
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