Figure shows a long straight wire of circular crosssection (radius a) carrying steady current I. The current I is uniformly distributed across this crosssection. Calculate the magnetic field in the region `r lt a` and `r gt a`.

Explanation

(a) Taking the case, when `r lt a`. Let the point `P_1` be inside the wire at a distance r from the axis of wire. Consider an Amperian loop labelled 1, which is a circle of radius r concentric with crosssection such that point `P_1` lies on this loop.
Current enclosed by the loop,
`I^'=(I)/(pia^2)xxpir^2=(Ir^2)/(a^2)`
Let B be the magnetic field induction at `P_1`.
Line integral of `vecB` over the Amperian loop is
`=ointvecB.dvecl=B(2pir)`
According to Ampere's circuital law,
`ointvecB.dvecl=mu_0xx`current threading the closed path
`=(mu_0Ir^2)/(a^2)`
`:. B2pir=mu_0xx(Ir^2)/(a^2)` or `B=(mu_0Ir)/(2pia^2)`
It means, `Bpropr` (When `r lt a`)
(b) Taking the case when `r gt a`. Take the point P outside the wire at a distance `r` from the axis of wire. Consider an Amperian loop labelled 2 which is a circle concentric with crosssection of wire such that point P lies on this loop. Current enclosed by this loop is L.
Let B be the magnetic field induction at P. Line integral of `vecB` over the Amperian loop
`=ointvecB.dvecl=B(2pir)`
According to Ampere's circuital law
`ointvecB.dvecl=mu_0xx` current threading the closed
`loop=mu_0I`
`:. B2pir=mu_0I` or `B=(mu_0I)/(2pir)`
It means, `Bprop1/r` (When `r gt a`)