# Establish a relation between electric current and drift velocity.

OR

Prove that the current density of a metallic conductor is directly proportional to the drift speed of electrons.

#### Relation between electric current and drift velocity.

Consider a uniform metallic wire XY of length 1 and cross-sectional area A. A potential difference V is applied across the ends X and Y of the wire.This causes an electric field at each point of the wire of strength.

$$E=\dfrac{V}{l}$$........(i)

Due to this electric field, the electrons gain a drift velocity vd opposite to direction of electric field.IF q be the charge through the cross-section of wire in t seconds, then

Current in wire $$I=\dfrac{q}{t}$$..........(i)

The distance traversed by each electron in time t=average velocity x time =vdt

If we consider two planes P and Q at a distance vd in a conductor, then the total charge flowing in time t will be equal to the total charge on the electrons present within the cylinder PQ.

The volume of this cylinder =cross sectional area x height

$$A vdt$$

If n is the number of free electrons in the wire per unit volume,then the number of free electrons in the cylinder$$=n(Avd t)

$$

If charge on each electron is $$-c(c=1.6 \times 10^{-19}C)$$,then the total charge flowing through a cross section of the wire.

$$q=(nA_v d t)(-e)=-neA_vdt$$.........(iii)

$$\therefore $$ Current flowing in the wire,

$$I=\dfrac{q}{t}=\dfrac{-v}{t}$$

i.e , current$$ I=-ne A_vd$$.......(iv)

This is the relation between current and drift velocity.Negative sign shows that the direction of current is opposite to the drift velocity

Numericaly $$I=-neA \tau d$$...............(v)

$$\therefore$$ Cureent dendity , $$J=\dfrac{t}P{l}{A-}

$$\therefgfore)$$ Curent density $$J=\dfrac{l}{A}.

$$\Rightarrow J \dfrac{I}{A}=d$$

$$\Rightarrow J \alpha D$$

This is current density of maetallic conductor is directly proportional to the drfyr velocity.