Relation between drift velocity (Vd) and current density(J).

#Relation between drift velocity (Vd) and current density(J).

Considered a current carrying conductor of length (l) and area of cross section (A). The free electron density of this conductor be “n” electrons/metre³.
                       when a potential difference V voltage applied across its end then the electric field E is produced and free electron are move in the direction opposite to the electric field by the drift velocity.

“Drift velocity”: Drift velocity is define as the average velocity with which the electrons get drifted towords positive end of the conductor under the influence of an external electric field. it is denoted by Vd.

Thus, Vd = l/t = (U+V)/2

The Free electron density of a conductor be “n” electrons/metre ³.

i.e 1m³ volume containing = n free electron
Hence,
(AL) m³ Volume containing (N) = n.(AL) electrons

Thus, amount of charge flows through any cross section of a conductor,Q = N.e
Q = n.AL.e

We know that
Current, I = Q/t

Hence, I = (n.AL.e)/t

I = n.e.A(l/t)

{By the definition
Drift velocity,Vd = l/t }

I = n.eA(Vd)

And current density, J = I/A

J = (n.e.A.Vd)/A

J = n.e.Vd

Here the quantity (n.e) is constant term
Hence,  [ J ∝ Vd ]

Thus the current density of a current carrying conductor is directly proportional to the Drift velocity of free electron.

Note: Nature of electrical current:
The electric current has magnitude and direction both but it is a scalar quantity,the electric current is considered as a scalar quantity because of the following reason

1) It obeys law of additions of scalars .

2) If current carrying conductor is holded or folded in different shape and different portion the value of current does not change.