The current density `bar(J)` inside a long, solid cylindrical wire of radius `a = 12 mm` is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to `J = (J_(0) r)/(a)`, where `J_(0) = (10^(5))/(4 pi) A//m^(2)`. Find the magnitude of the magnetic field at `r = (a)/(2)` in `mu T`

Explanation

Current in the element `=J(2pir.dr)`
Current enclosed by Amperian loop of radius `(a)/(2)`
`I=underset(0)overset(a//2)int(J_(0)r)/(a), 2pir. dr=(2piJ_(0))/(3a)((a)/(2))^(3)=(piJ_(0)a^(2))/(12)`
Applying Ampere's law
`B.2pi.(a)/(2)=mu_(0).(piJ_(0)a^(2))/(12)rArrB=(mu_(0)J_(0)a)/(12)`
On putting values
`B=10muT`