Electron’s conductivity & mobility

From the mathematical form of Ohm’s law, we know that

Here

, is called the current density, l is the conductor length,  is the density and A is the cross sectional area.

If E be the uniform electric field in the conductor then we have

Or, 

Or, 

Here , is called the conductivity.

This conductivity is generated due to what is called electrons mobility.

Mobility is defined as

 ………………………………………………… (1)

Here µ is called the electron’s mobility, Vd is the drift velocity and E is the applied electric field.

Now the question may arise that what is this drift velocity?

While being in motion due to the applied electric field, an electron make collisions with the heavier ions or atoms. After these collisions the speed of the electron remain unchanged but they come out in random directions, thus making their average velocity zero. If a bulk material contains N number of electrons, then

Here vi is the velocity of the ith   electron(i=1,2,3,……). If a be the acceleration of the electron, then, the force on the electron is

Where m is the mass and e, is the charge of the electron respectively. After any collision, the electron is accelerated before the next collision takes place. So, the velocity just after the collision and that sometimes after will be different.

Let us suppose that vi is the velocity of the electron after collision and Vi be that after time t before the next collision, then form the equation of kinematics, we can write

Considering velocities from all directions, i.e., average velocity then

But 

Hence,

The time gap between two collisions is not constant. But if we take an average of all these times, then it is called mean free time which is denoted by ,

Hence, drift velocity

From equation (1), mobility is

If there is a cylindrical conductor of length Δx of cross sectional area A. The electron’s drift velocity is Vd and n be the volume density of electrons inside the conductor. The amount of charge transported through the volume of the conductor in time Δt is

Here the sign of electronic charge is neglected.

Putting the value of Vd, we get,

From the definition of current density j, we have

Hence, we get

Again, we know

Hence, conductivity

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February 7, 2018

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