(i) Due to translatory motion of charge :
In `n` particle each having a charge `q` , pass through a given area in time `t` then `i = (nq)/t`.

If `n` particles each having a charge `q` pass per second per unit area, the current associated with cross-sectional area `A` is `i = nqA`.

If there are `n` particle per unit volume each having a charge `q` and moving with velocity `v`, the current thorough, cross section `A` is `i =nqvA`


(ii) Due to rotatory motion of charge :
If a point charge `q` is moving in a circle of radius `r` with speed `v` (frequency `f` . angular speed `w` and time period `T`) then corresponding currents `i=qf=q/T=(qv)/(2 pi r)=(q omega)/(2 pi)`

`text(Conductor)`

In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors.

`text(Insulator or Dielectrics :)`

Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can't leave their parent atoms or molecules and hence can't move through long distances. Such materials are also called dielectrics.

`text(Semiconductor :)`

In semiconductors, the behavior is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free themselves and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller that in a conductor, its behavior is in between a conductor and an insulator and hence, the name semiconductor. A freed electronic a semiconductor leaves a vacancy (hole) in its normal bound position. These vacancies (holes) also help in conduction.

So in metallic conductors current is due to motion of free electrons, in liquids ie electrolytes current is due to motion of positive and negative ions, in gases current is due to motion of positive ions and electrons, in semiconductor current is due to motion of electron and holes .

In case of flow of charge through a cross-section, current density is defined as a vector having magnitude equal to current per unit area surrounding that point. Remember area is normal to the direction of charge flow (or current passes) through that point. Current density at point `P` is given by

`vec J=(di)/(dA) hat n`

If the cross-sectional area is not normal to the current, the cross-sectional area normal to current in accordance with following figure will be `dA cos q` and so in this situation:

`J=(di)/(dA cos theta)` i.e., `di = J dA cos theta` or `di=vec J * vec(dA) `

` i = int vec J * vec(dA)`

i.e. , in terms of current density, current is the flux of current density.

Note : `q` If current density `vec J` is uniform for a normal cross -section.

`vec A` then : `i=int vec J * vec(ds)=vec j * int vec(ds)` [as `vec J=` constant]

or `i=vec J * vec A=JA cos 0=JA ,=> J=i/A` [as `int vec (dA)=vec A` and `q=0^(circ)`]

Current density `J` is a vector quantity having `S.I.` unit `Amp//m ^2` and dimension. [ `L ^(-2) A`]

According to modern views. a metal consists of a "lattice' of fixed positively charged ions in which billions and billions of free electrons are moving randomly at speed which at room temperature (i.e. `300` `K`) in accordance with kinetic theory of gases is given by

`V_(rms)= sqrt((3 xx(1.38 xx 10^(-23)) xx 300)/(9.1 xx 10^(-31)))=10^5 m//s`

The randomly moving free electrons inside the metal collide with the lattice and follow a zig-zag path as shown in figure (`A`).

However, in absence of any electric field due to this random motion, the number of electrons crossing from left to right is equal to the number of electrons crossing from right to left (otherwise metal will not remain equipotential) so the net current through a cross-section is zero.
When a potential difference is applied across the conductor ..Setting of electric field inside the conductor is done with the speed of light, inside the conductor due to electric force the path of electron in general becomes curved (parabolic) instead of straight lines and electrons drift opposite to the field figure (`B`). Due to this drift the random motion of electrons get modified and there is a net transfer of electrons across a cross-section resulting in current.