Sir , please explain the solved example,I did not understand it properly

As per Amphere law  . Magnetic field summed up in a closed loop equals the current passing through the conductor which is enclosed by the loop.

 

In the problem it is asked to find the magnetic field due to current carrying conductor in the region (1) outside the conductor and (2) inside the conductor.

 

First part is very simple. consider an ampherian circular loop of radius outside the condutor. here the loop radius is grater than condutor radius. the line integral for the closed loop is 2πr if the radius of loop is r ( r>a, a is the conductor radius).

 

Hence B×2πr = μ₀ I 

 

 

In the second part it is required to get magnetic field at a point inside the conductor. If we draw the circular ampherian loop with radial distance equal to the distance from conductor central axis to the point of intrest, then the enclosed current is proportional to loop area. Current is uniformly distributed across the crosssection of the conductor whose cross section is circular and radius is a. 

 

Hence the current density J is given by :-       J = I / (πa²). 

Hence the enclosed current in the ampherian loop of radius r is

Hence magnetic field is