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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.