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`

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`