Suppose that the current density in a wire of radius a varius with r according to `Kr^2` where K is a constant and r is the distance from the axis of the wire. Find the magnetic field at a point at distance r form the axis when (a) rlta and (b) rgta.

Explanation

Choose a circular path centred on the conductor's axis and apply
Ampere's law.
(a) To find the current through the area enclosed by the path
`dI=lambdadA=(Kr^2)(2pirdr)`
`:. I=Kint_0^r 2pir^3dr=(piKr^4)/2`
Since `intvecB.dvecl=mu_0I implies B2pir=mu_0(piKr^4)/2`
`implies B=(mu_0Kr^3)/4`
(b) If `rgta,` then net current through the Amperian loop is
`I'=int_0^a Kr^2 2pirdr=(piKa^4)/2`
Therefore, ` B=(mu_0 Ka^4)/(4r)`