How do you prove Ohm's law using the classical free electron theory?

Let's see, Ohm's law is basically the idea that the current in a conductor is proportional to the voltage across it.

You know, V=IR and all that jazz.

But how do we explain this from a microscopic point of view? What's going on inside that metal when we apply an electric field?

That's where the classical free electron theory comes in.

This thing says that a metal is made of a bunch of positive ions in a lattice and a bunch of free electrons that can move around like gas molecules.

The electrons are constantly colliding with each other and with the ions, but we don't care about the electron-electron collisions because they don't change the net momentum or current.

The electron-ion collisions, on the other hand, are important because they make the electrons lose energy and change direction.

So, when we apply an electric field to the metal, the electrons feel a force that makes them accelerate in the opposite direction of the field.

But they don't go very far before they hit an ion and bounce off randomly.

This means that they have a very small average drift velocity in the direction of the current, which is proportional to the electric field.

The current density, which is the current per unit area, is then given by J=−nevd, where n is the number of free electrons per unit volume, e is the charge of an electron, and vd is the drift velocity.

Now, we can relate the drift velocity to the electric field by using some simple physics.

The force on an electron is $F=-eE$ , where E is the electric field.

The acceleration of an electron is a=F/m, where m is the mass of an electron.

The time between collisions is t, which is also called the relaxation time. The drift velocity is then vd=at=−eEt/m/.

Putting it all together, we get J=ne2Et/m, which shows that the current density is proportional to the electric field.

This is the local form of Ohm's law.

To get the global form, we just integrate over the length of the wire and divide by its cross-sectional area.

We getV=IR, where R=m/(ne2tA), which is called the resistance.

……….

And there you have it. We just proved Ohm's law using the classical free electron theory.

Of course, this theory is not perfect and it has some limitations, but it's good enough for most purposes.