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.

Notice that you have A downstairs.

That means if the pipe is wider, larger cross-section,

it's also easier for the current to flow,

it's easier for the water to flow.

So that's also quite pleasing. Ohm's Law, also,

often holds for insulators, which are not conductors,

even though I have derived it here for conductors,

which have these free electrons.

And so now, I want to make a comparison between very good

conductors, and very good insulators.

So I'll start off with a -- a chunk of material ,

cross-sectional area A -- let's take it one millimeter by one

millimeter -- so A is ten to the minus six square meters.

So here I have a chunk of material, and the length of that

material L is one meter. Put a potential difference over

there, plus here, and minus here.

Current will start to flow in this direction,

electrons will flow in this direction.

The question now is, what is the resistance of this

chunk of material? Well, very easy.

You take these equations, you know L and A,

so if I tell you what sigma is, then you can immediately

calculate what the resistance is.

So let's take, first, a good conductor.

Silver and gold and copper are very

good conductors. They would have values of

sigma, ten to the eight, we just calculated for copper,

you've seen in front of your own eyes.

So that means rho would be ten to the minus eight,

it's one over sigma. And so in this particular case,

since A is ten to the minus six, the resistance R is simply

ten to the sixth times rho.

Because L is one meter. So it's very easy -- resistance

here, R, is ten to the minus two ohms.

One-hundredth of an ohm. For this material if it were

copper. Let's now take a very good

insulator. Glass is an example.

Quartz, porcelain, very good insulators.

Now, sigma, the conductivity, is extremely low.

They vary somewhere from ten to the minus twelve through ten to

the minus sixteen. So rho, now,

the resistivity, is something like ten to the

twelve to twelve to the plus sixteens, and if I take ten to

the fourteen, just I grab -- I have to grab a

number -- then you'll find that R, now, is ten to the twenty

ohms. A one with twenty zeros.

That's an enormous resistance. So you see the difference --

twenty-two orders of magnitude difference between a good

conductor and a good insulator. And if I make this potential

difference over the wire, if I make that one volt,

and if I apply Ohm's Law, V equals I R,

then I can also calculate the current that is going to flow.

If I R is one, then the current here is

hundred amperes, and the current here is ten

to the minus twenty amperes, an insignificant current,

ten to the minus twenty amperes.

I first want to demonstrate to you that Ohm's Law sometimes

holds, I will do a demonstration,

whereby you have a voltage supply -- put a V in here --

and we change the voltage in a matter of a few seconds from

zero to four volts. This is the plus side,

this is the minus side, I have connected it here to a

resistor which is fifty ohms -- we use this symbol for a