Introduction and Dependence of Conductivity on Charge Density

We mostly use copper wires instead of aluminium wires for household purposes.

It is due to high conductivity of copper than the aluminium

Let's understand the concept of conductivity

In every metal some of the electron moves freely and that is why they are called the free electrons or conduction electrons.

Here we see the motion of a free electron inside a conductor when there is no electric field.

Here, the electron has no average displacement, and so its average velocity is zero.

However, when the battery is attached to this conductor, an electric field will set up into it, and the electron will drift toward the positive terminal.

Now the average velocity will not be zero because there is some net displacement.

An electron in the presence of electric field

In the presence of an uniform electric field, the electrons will feel a force and thus will be accelerated.

At, time , the electric field, , and there is no net displacement in the electron, and thus, at , the initial velocity can be taken as .

At the moment we attach the battery to the conductor, an electric field will set up into it, due to which the electrons will drift towards the positive terminal with a drift velocity .

If we apply the first kinematic equation to the motion of the electron

As we know the relation between drift velocity and electric field

and with the help of the expressions of the drift velocity we can derive the formula of the current:

The resistance can be represented in terms of resistivity and the length and area of the resistance.

After putting values of the current, resistance and the voltage obtained, in Ohm’s law equation, we get

Now we know the formula of resistance. Let’s rearrange the equation of the current.

Since, the conductivity is inversely proportional to the resistivity, the equation of the conductivity can be given as

Hence, the conductivity is directly proportional to the charge density.

And the current density in terms of conductivity will be

Let's see the reason behind using the copper wire over an aluminium wire

Because all the good conductor like gold, silver and copper all of them have large number of free electron per unit volume.

That’s why the conduction is more and the resistivity is less, and which one have higher conductivity is called as better conductor.

Another factor is relaxation time, if relaxation time is greater than the resistivity is smaller.

In the beginning, we have said we use copper wire in household, and the reason is that the conductivity of copper is larger than that of aluminum.

Revision

When there is no electric field the conduction electron moves random but have no net displacement, so its average velocity is zero.

In the presence of electric field electron start to move in the opposite direction of electric field with some drift velocity.

The resistivity of the conductor is inversely proportional to charge density

The conductivity of a metal is directly proportional to its charge density.

The End.