Question

Define current density. Write an expression which connects current density with drift speed.

Answer 1

Think from the basics, current is nothing but amount of charge passing through a cross-sectional area in unit time and current density is nothing but a quantity which expresses how much current is flowing through a unit area.

Formula Used:
The relation between current density and drift velocity
$J=ne{{v}_{d}}$
Where
$J=$ Current density
${{v}_{d}}=$ Drift velocity

Complete step by step answer:
Current density:- The amount of electric current flowing perpendicularly to the unit cross section area.
$J=\dfrac{i}{A}$
Where $J$ is the current density.
$i$ is the current in the conductor.
$A$ is the cross section area.
Relation between current density and drift velocity:- The strength of the electric current in a conductor is measured by the magnitude of electric charge flowing per second through a cross section area of the conductor. If the charge carried by the free electron passing through a section of the wire in $t$ second be $q$, then strength of current in the wire is

$i=\dfrac{q}{t}$ ….(1)
Suppose area of cross section A, and number of free ${{e}^{-}}$ per unit volume of the wire is n.
The number of electrons passing per second through a cross section of the wire will be $nA{{v}_{d}}t$.
In $t$ second, $nA{{v}_{d}}t$ electrons will pass.
Charge on the electrons ${{e}^{-}}$
Then charge passing through any cross section of wire in $t$ second will be.
$q=\left( nA{{v}_{d}}t \right)\times e$
Substituting the value of $q$ in equation (1)
$i=\dfrac{neA{{v}_{d}}t}{t}$
$i=neA{{v}_{d}}$ ……………(2)
We know that
$J=\dfrac{i}{A}$
$J=\dfrac{neA{{v}_{d}}}{A}$
$J=ne{{v}_{d}}$
This is the relation between current density and drift velocity and written in vector form
$\overrightarrow{J}=-ne\overrightarrow{{{v}_{d}}}$
$\left( - \right)$ sign signifies that electron move in the direction $-\overrightarrow{J.}$

Note: Current flows in the direction of motion of a positive charge but since electrons are negatively charged, current flow and direction of motion of electrons is opposite to each other.