Question

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.

Answer 1

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.