A long wire has a circular cross-section with radius `a`. The current density in the wire is `J(r) =J_0(a^2-r^2)/a^2` where `r` is the distance from the axis. Calculate the magnetic field at an arbitrary point inside or outside the wire.

Problem: 

A long wire has a circular cross-section with radius `a`. The current density in the wire is `J(r) =J_0(a^2-r^2)/a^2` where `r` is the distance from the axis. Calculate the magnetic field at an arbitrary point inside or outside the wire.

Solution:

For inside the wire`(r<a)`:

`I_("enc")=intvecJ.vecds`

or, `I_"enc"=int_0^rJ_0(a^2-r^2)/a^2 2pirdr`

or, `I_"enc"=2piJ_0[r^2/2-r^4/(4a^4)]`

The magnetic field inside the wire is given by 

`ointvecB_(r<a).vecdr=mu_0I_"enc"`

or, `B_(r<a).2pir=mu_0 2piJ_0[r^2/2-r^4/(4a^4)]`

or, `B_(r<a)=mu_0J_0[r/2-r^3/(4a^2)]`[ANS]

For outside the wire `(r>a)`:

The current enclosed by the Ampere's loop is given by