The current density `vecj` inside a long, solid, cylindrica wire of radius `a=12mm` is in the direction of the central axix, and its magnitude varies linearly with raidal distance `r` from the axis according to `J=(J_(0)r)/(a)`, where `J_(0)=(10^(5))/(4pi) A//m^(2)`. Find the magnitude of the magnetic field at `r=(9)/(a)` in `muT`.

Answer

`10`

Explanation

Current in the element `=J(2pir.dr)`
Current enclosed by Amperian loop of radius `a/2`
`l=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 rArr B=(mu_(0)J_0a)/12`
On putting values
`B=10muT`