Kelistrikan dan Kemagnetan
Topics covered:
Biot-Savart Law
Gauss' Law for Magnetic Fields
Revisit the "Leyden Jar"
High-Voltage Power Lines
Instructor/speaker: Prof. Walter Lewin
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Well, we have a current going through a wire, like so.
And we look at the magnetic field in the vicinity of this wire , then we know from experiment that if you put pieces of magnetite around the wire that they line up in a circle.
Put around like this.
If that circle has a radius R, then the magnetic fields, that's an experimental fact, is proportional with the current I and is inversely proportional with the radius of that circle.
By convention, the direction of the magnetic field is given by the right-hand corkscrew, rotate this way, the current goes up.
You've seen before, with electric charges, when you have a wire which is uniformly distributed say with positive charge, you've also seen that electric fields in the vicinity of that straight wire falls off as 1 over R, whereas the direction is different than the magnetic field but it also falls off as 1 over R, and the reason is that electric monopoles, individual charges, the electric field falls off as 1 over R squared.
And so when you integrate that out over a straight wire you get the 1 over R field.
So by analogy, it would be very plausible that if you took magnetic monopoles that the magnetic field would also fall off as 1 over R squared, but magnetic monopoles as far as we know don't exist.
In principle they could exist, but we've never seen one, and if any one of you ever find one, that would certainly be a Nobel Prize.
It's by no means impossible.
And so the simple fact that the magnetic field around a current wire falls off as 1 over R, sort of suggests that if you carve this wire up in little elements dL, that each one of those elements contributes to the magnetic field in an inverse R-squared law, and by integrating out over the whole wire you'd then get the 1 over R fields.
And this behind the idea of the formalism by Biot and Savart, who introduced the idea that if you have a little current element dL, and the current is in this direction, and you want to know what the magnetic field is, This is small contribution dB to that little current element, and the distance is R, and the unit vector from the element dL to the point where you want to know the magnetic field is R roof.
Then the idea is that dB, it's a little bit of curr- little bit of magnetic fields.
In this case it would be in the blackboard because of the right-hand corkscrew rule.
The current is in this direction, so these little elements would contribute to magnetic fields in this direction perpendicular to the blackboard.
Is some constant, proportional to the current no doubt, and then is proportional to the length of that little element dL, if it's longer then the magnetic field is larger, and in order to get the direction right perpendicular to the blackboard you take the cross product with the unit vector R.
The unit vector R has length 1 so you only do that in order to get the direction right.
And this, and that inversely proportional to R squared.
That's of course key.
And this is, the formalism by Biot-Savart and you can do experiments and measure the magnetic field in the vicinity of wires and this formalism works, so you then calculate the individual contributions of all these little elements dL and then you do an integration and this formalism works.
You can then also measure what C is, in SI units, C is 10 to the -7.
But we write for C something quite peculiar.
We write for C mu 0 divided by 4 pi, and we call this mu 0 the permeability of free space.
You've seen earlier with Coulomb's law that this constant 9 times 10 to the 9th, we call that 1 over 4 pi epsilon 0.
What is in the name?
And so here we call this mu 0 divided by 4 pi.
So now you can apply Biot-Savart's Law and you can go to a straight wire and you have a current I, and suppose you want to know what the magnetic field at that location P is at a distance capital R, and so what you now have to do, is you carve this up, in an infinite number of small elements dL, and this distance is R, and the unit vector is then like so, and you calculate the small amount of magnetic field due to this little element and you integrate this over the whole wire.
It's mathematics.
You've done it.
You've done it before, where we had uniformly electric charge on the wire.
So I'm not going to do this again for you.
It's a very straightforward piece of mathematics.
The magnetic field by the way, in this case, would come out of the blackboard.
Because of the right-hand corkscrew rule.
And what you find when you do this, we will find that B equals mu 0 times I, divided by 2 pi R, this being R, and so you indeed see that the inverse 1 over R comes out.
And so if you, for instance, take a radius of 0.1 meters, 10 centimeters, and you have a current through the wire of about 100 amperes, then you would end up with a B field.
You use this equation, 2 times 10 to the -4 tesla.
That is about 2 gauss.
100 amperes.
10 centimeter distance is only 2 gauss.
Think about it.
The Earth's magnetic field is half a gauss.
So if you go 1 meter away from the wire, so we have a magnetic field which is 10 times lower, look, it goes with 1 over R, then the magnetic field of the Earth already dominates substantially.
So you need very high currents, actually, when you do these experiments.
It's nice to see that out of Biot-Savart's formalism the 1 over R pops out, but of course you must realize that Biot-Savart knew that the magnetic field falls off as 1 over R.
That was an experimental fact.
So the fact that it falls out is logical, because it was cooked into that formalism.
If you think about it, it all goes back to Newton.
Newton was the one who first suggested that the gravitational field falls off as 1 over R squared.
And then later a logical ex10sion was that the electric fields would fall off as 1 over R squared and out of that came the idea that the fields of magnetic monopole, if they only existed, would fall off as 1 over R squared, and that's all behind this and so the person who really deserves most of the credit for all this in my book is Newton.
Using Biot-Savart, we can calculate now quite easily the magnetic field at the center of a current loop.
Let this be a wire circle and let the current go in this direction, and I would ask you what is the magnetic field right at the center.
Well, the magnetic field right at the center of course is pointing upwards.
Each little element along the line here, dL, each little element will contribute a little bit magnetic field at that point right in this direction.
And if this radius is R, with Biot-Savart now, we can calculate quite easily the total field that you would get at this location, because that total field is then the integral of dB vectorially over the entire wire so the entire loop...
So if you go there, so you would get your mu 0, divided by 4 pi, you get your current and you get your 1 over R squared, and now you have to do an integral over that dL cross R.
Well, R is of course always perpendicular to dL.
Any element dL that you choose, the unit vector R is exactly perpendicular to the element dL, because that's characteristic of a circle.
And so the sine of the angle between dL and R is 1, and so all we have to do is do an integral over dL, which is the integral of the circle, which is the circumference of the circle, and that is 2 pi R.
And so now you find, you lose a pi, you lose an R, so you find mu 0 times I divided by 2R.
Just to show you an example, how in this case how easy it is to use Biot-Savart and calculate the magnetic field right at the center.
If you were asked what the magnetic field was here or there, that would be also relatively easy.
You've done that.
I've given you a problem earlier where we had point charges uniformly distributed on a wire and I asked you what the electric field was here.
So that can also be done now with magnetic fields.
If I ever asked you what the magnetic fields would be here, that of course is an impossibility to do that with Biot-Savart, practically an impossibility.
I wouldn't know how to do that.
But in principle it could be done and certainly with a computer you can do it.
So we can go to our same situation, we can take 100 amperes for I and you can take R 0.1 meters and then the B field, the strength of the B field right at the center of this loop that I found is then 6 times 10 to the -4 tesla.
And that would be 6 gauss.
It's clear that if you want to put in some field lines, magnetic field lines, as a result of this current going around in a circle, that the- through the center there would be a field line like so.
If you're very close to the wire here, which goes into the blackboard, I want you to see this three-dimensionally, then the magnetic field would go like this, clockwise.
Here the current comes to you, so it would be counterclockwise.
If the magnetic field line is here like so, and here it is curled up, then clearly I expect them to be here, sort of like so, and like so, and like so.
This is the kind of magnetic field line configuration that I would expect, then, in the vicinity of such a current loop.
And I want to show this to you in a little bit more detail.
I have here a transparency, and you see there on the right side, current goes into the paper and here it comes out of the paper.
That is a circular loop.
And you see here the field line configuration.
It's not too different from what I have on the blackboard there.
Very close to the wires, of course, you get circles because the 1 over R dominates there.
It's so close to the wire that the 1 over R relationship makes it come out like circles and here too, but then if you're farther away you get configurations like I have there.
When you're very far away from a current loop, the magnetic field configuration is very similar to that of an electric dipole.
I can show you that in the following way.
Let's first look at the electric dipole that you see up there.
This is a positive charge, this is a negative charge.
Don't look anywhere near the charges.
Don't look in between the charges.
Look far away.
Here you see electric field lines and you see them here.
Now look at your current loop here.
The current is going into the paper here, coming out of the paper.
There is a loop.
And look, you see the same configuration, field lines, field lines.
This goes like so.
This one goes like so.
Here, the electric field lines coming in, magnetic field lines are coming in.
Electric field lines are going out.
Magnetic field lines are going out.
They look very similar.
Pengembangan Perkuliahan
1. Buatlah sebuah Esai mengenai materi perkuliahan ini
2. Buatlah sebuah kelompok berjumlah 5 orang untuk menganalisis materi perkuliahan ini
3. Lakukan Penelitian Sederhana dengan kelompok tersebut
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5. Kembangkan produk tersebut dengan senantiasa meningkatkan kualitasnyaVisualizations: Instructors: Course Co-Administrators: Technical Instructors: Course Material: The TEAL project is supported by The Alex and Brit d'Arbeloff Fund for Excellence in MIT Education, MIT iCampus, the Davis Educational Foundation, the National Science Foundation, the Class of 1960 Endowment for Innovation in Education, the Class of 1951 Fund for Excellence in Education, the Class of 1955 Fund for Excellence in Teaching, and the Helena Foundation. Many people have contributed to the development of the course materials. (PDF)Staff
Prof. John Belcher
Dr. Peter Dourmashkin
Prof. Bruce Knuteson
Prof. Gunther Roland
Prof. Bolek Wyslouch
Dr. Brian Wecht
Prof. Eric Katsavounidis
Prof. Robert Simcoe
Prof. Joseph Formaggio
Dr. Peter Dourmashkin
Prof. Robert Redwine
Andy Neely
Matthew Strafuss
Dr. Peter Dourmashkin
Prof. Eric Hudson
Dr. Sen-Ben LiaoAcknowledgements
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