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  • When positive charges move in this direction,

  • then per definition, we say the current goes in this

  • direction. When negative charges go in

  • this direction, we also say the current goes in

  • that direction, that's just our convention.

  • If I apply a potential difference over a conductor,

  • then I'm going to create an electric field in that

  • conductor. And the electrons -- there are

  • free electrons in a conductor -- they can move,

  • but the ions cannot move, because they are frozen into

  • the solid, into the crystal. And so when a current flows in

  • a conductor, it's always the electrons that are responsible

  • for the current. The electrons fuel the electric

  • fields, and then the electrons try to make the electric field

  • zero, but they can't succeed, because we keep the potential

  • difference over the conductor. Often, there is a linear

  • relationship between current and the potential,

  • in which case, we talk about Ohm's Law.

  • Now, I will try to derive Ohm's Law

  • in a very crude way, a poor man's version,

  • and not really one hundred percent kosher,

  • it requires quantum mechanics, which is beyond the course --

  • beyond this course -- but I will do a job that still gives us

  • some interesting insight into Ohm's Law.

  • If I start off with a conductor, for instance,

  • copper, at room temperature, three hundred degrees Kelvin,

  • the free electrons in copper have a speed,

  • an average speed of about a million meters per second.

  • So this is the average speed of those free electrons,

  • about a million meters per second.

  • This in all directions. It's a chaotic motion.

  • It's a thermal motion, it's due to the temperature.

  • The time between collisions -- time between the collisions --

  • and this is a collision of the free electron with the atoms --

  • is approximately -- I call it tau --

  • is about three times ten to the minus fourteen seconds.

  • No surprise, because the speed is enormously

  • high. And the number of free

  • electrons in copper per cubic meter, I call that number N,

  • is about ten to the twenty-nine.

  • There's about one free electron for every atom.

  • So we get twen- ten to the twenty-nine free electrons per

  • cubic meter. So now imagine that I apply a

  • potential difference piece of copper -- or any conductor,

  • for that matter -- then the electrons will experience a

  • force which is the charge of the electron, that's my little E

  • times the electric field that I'm creating,

  • because I apply a potential difference.

  • I realize that the force and the electric field are in

  • opposite directions for electrons, but that's a detail,

  • I'm interested in the magnitudes only.

  • And so now these electrons will experience an acceleration,

  • which is the force divided by the mass of the electron,

  • and so they will pick up, eh, speed, between these colli-

  • collisions, which we call the drift velocity,

  • which is A times tau, it's just eight oh one.

  • And so A equals F divided by M. E F is in the A,

  • so we get E times E divided by the mass of the electrons,

  • times tau. And that is the

  • the drift velocity. When the electric field goes

  • up, the drift velocity goes up, so the electrons move faster in

  • the direction opposite to the current.

  • If the time between collisions gets larger, they -- the

  • acceleration lasts longer, so also, they pick up a larger

  • speed, so that's intuitively pleasing.

  • If we take a specific case, and I take, for instance,

  • copper, and I apply over the -- over a

  • wire -- let's say the wire has a length of 10 meters -- I apply a

  • potential difference I call delta V, but I could have said

  • just V -- I apply there a potential difference of ten

  • volts, then the electric field -- inside the conductor,

  • now -- is about one volt per meter.

  • And so I can calculate, now, for that specific case,

  • I can calculate what the drift velocity would be.

  • So the drift velocity of those free electrons would be the

  • charge of the electron, which is one point six times

  • ten to the minus nineteen Coulombs.

  • The E field is one, so I can forget about that.

  • Tau is three times ten to the minus fourteen,

  • as long as I'm room temperature, and the mass of the

  • electron is about ten to the minus thirty kilograms.

  • And so, if I didn't slip up, I found that this is five times

  • ten to the minus three meters per second, which is half a

  • centimeter per second. So imagine, due to the thermal

  • motion, these free electrons move with a million meters per

  • second. But due to this electric field,

  • they only advance along the wire slowly, like a snail,

  • with a speed on average of half a centimeter per second.

  • And that goes very much against your and my own intuition,

  • but this is the way it is. I mean, a turtle would go

  • faster than these electrons. To go along a ten-meter wire

  • would take half hour. Something that you never

  • thought of. That it would take a half hour

  • for these electrons to go along the wire if you apply potential

  • difference of ten volts, copper ten meters long.

  • Now, I want to massage this further,

  • and see whether we can somehow squeeze out Ohm's Law,

  • which is the linear relation between the potential and the

  • current. So let me start off with a wire

  • which has a cross-section A, and it has a length L,

  • and I put a potential difference

  • over the wire, plus here, and minus there,

  • potential V, so I would get a current in

  • this direction, that's our definition of

  • current, going from plus to minus.

  • The electrons, of course, are moving in this

  • direction, with the drift velocity.

  • And so the electric field in here, which is in this

  • direction, that electric field is approximately V divided by L,

  • potential difference divided by distance.

  • In one second, these free electrons will move

  • from left to right over a distance V D meters.

  • So if I make any cross-section through this wire,

  • anywhere, I can calculate how many electrons pass through that

  • cross-section in one second. In one second,

  • the volume that passes through here, the volume is V D times A

  • but the number of free electrons per cubic meter is

  • called N, so this is now the number of free electrons that

  • passes, per second, through any cross-section.

  • And each electron has a charge E, and so this is the current

  • that will flow. The current,

  • of course, is in this direction, but that's a detail.

  • If I now substitute the drift velocity, which we have here,

  • I substitute that in there, but then I find that the

  • current -- I get a E squared, the charge squared,

  • I get N, I get tau, I get downstairs,

  • the mass of the electron, and then I get A times the

  • electric field E. Because I have here,

  • is electric field E. When you look at this here,

  • that really depends only on the properties of by substance,

  • for a given temperature. And we give that a name.

  • We call this sigma, which is called conductivity.

  • Conductivity. If I calculate,

  • for copper, the conductivity, at room temperature,

  • that's very easy, because I've given you what N

  • is, on the blackboard there, ten to the twenty-nine,

  • you know what tau is at room temperature, three times ten to

  • the minus fourteen, so for copper,

  • at room temperature, you will find about ten to the

  • eighth. You will see more values fro

  • sigma later on during this course.

  • This is in SI units. I can massage this a little

  • further, because E is V divided by L,

  • and so I can write now that the current is that sigma times A

  • times V divided by L. I can write it down a little

  • bit differently, I can say V,

  • therefore, equals L divided by sigma A, times I.

  • And now, you're staring at Ohm's Law, whether you like it

  • or not, because this is what we call

  • the resistance, capital R.

  • We often write down rho for one over sigma, and rho is called

  • the resistivity. So either one will do.

  • So you can also write down -- you can write down V equals I R,

  • and this R, then, is either L divided by sigma A,

  • or L times rho -- let me make it a nicer rho -- divided by A.

  • That's the same thing. The units for resistance R is

  • volts per ampere, but we call that ohm.

  • And so the unit for R is ohm. And so if you want to know what

  • the unit for rho and sigma is,

  • that follows immediately from the equations.

  • The unit for rho is then ohm-meters.

  • So we have derived the resistance here in terms of the

  • dimensions -- namely, the length and the

  • cross-section -- but also in terms of the physics on an

  • atomic scale, which, all by itself,

  • is interesting. If you look at the resistance,

  • you see it is proportional with the

  • length of your wire through which you drive a current.

  • Think of this as water trying to go through a pipe.

  • If you make the pipe longer, the resistance goes up,

  • so that's very intuitively pleasing.