The average velocity gained by the free electrons of a conductor in the opposite direction of the externally applied electric field is called Drift Velocity. The parameter is called drift velocity of electrons.

### The relation between Electric current and Drift velocity:

## Derivation of OHM’S Law:

Current in terms of drift velocity (v_{d}) is I = enAv_{d}

Number of electrons in length l of the conductor = n × volume of the conductor = n Al

Total charge contained in length l of the conductor is q = en Al

All the electrons which enter the conductor at the right end will pass through the conductor at the left end in time,

This equation relates the current I with the drift velocity v_{d}

Current density ‘j’ is given by

**In vector form,**

The above equation is valid for both positive and negative values of q.

### Deduction of Ohm’s Law:

when a potential difference V is applied across a conductor of length l, the drift velocity in terms of V is given by

If the area of cross section of the conductor is A and the number of electrons per unit volume of the electron density of the conductor is n, then the current through the conductor will be

At a fixed temperature, the quantities m, l, n, e, τ and A, all have constant values for a given conductor.

Therefore,

This prove Ohm’s law for a conductor and here

is the resistance of the conductor.

### Resistivity in terms of electron density and relaxation time:

The resistance R of a conductor of length l, the area of cross-section A and resistivity ρ is given by

where τ is the relaxation time. Comparing the above two equations, we get

Constant value e = 1.6 × 10^{-19}C.

Obviously, ρ is independent of the dimension of the conductor but depends on its two parameters:

- The number of free electrons per unit volume or electron density of the conductor.
- The relaxation time τ, the average time between two successive collisions of an electron.

**Units:**

Drift velocity v_{d} is in ms^{-1} , free-electron density in m^{3}, cross-sectional area A in m^{2}, current density j in Am^{-2}. All resistance in Ω.

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