Q. 495.0( 1 Vote )

A long, cylindrical wire of radius b carries a current i distributed uniformly over its cross section. Find the magnitude of the magnetic field at a point inside the wire at a distance a from the axis.

Answer :

Given:


Radius of wire = b


Current distributed throughout the cross section = i


Diagram:



Formula used:


Ampere’s circuital law states that the line integral of the magnetic field for a closed surface is μ0 times the current enclosed by the surface.


, where B = magnetic field, dl = line element, μ0 = magnetic permeability of vacuum, I = current enclosed.


Since the current i is uniformly distributed throughout the cross section of the wire,


For an amperian loop of radius a(a<b) from the center of the wire,


I (current enclosed)



Where


i= total current distributed throughout the wire,


b = radius of the wire,


a = distance from the axis at which the magnetic field is to be found.


Hence, from Ampere’s circuital law,



since 2πa = circumference of Amperian loop of radius a, and I = enclosed current = ia2/b2.


Therefore, magnetic field at a distance a from the axis inside the wire,


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