A long straight wire of a circular cross section (radius `a`) carrying steady current. Current is uniformly distributed in the wire. Calculate magnetic field inside the region `(r lt a)` in the wire.

Explanation

AMPERE's cicuital law:The line integral `oint vecB.vec(dl)` on a closed curve of any shape is equal to `mu_(0)`(permeability of free space) times the net current I through the area bounded by the curve.
`oint vecB.vec(dl)mu_(0)Sigma I`
Line integral is independent of the shape of path and position of wire with in it. The statement `oint vecB.vec(dl)=0` does not necessarily mean that `vecB=0` everywhere along the path but only that no net current is passing through the path.
Sign of current:The current due to which `vecB` is produced in the same sense as `vec(dl)`(i.e. `vecB.vec(dl)` positive will be taken positive and the current which produces `vecB` in the same opposite to `vecdl` will be negative. Solid infinite current carrying cylinder:Assume current is unifromly distributed on the whole cross section area.
current density `J=I/(piR^(2))`
Case: `r ne R`
take an amperian loop inside the cylinder.By symmetry it should be a circle whose centre is on the axis of cylinder and its axis also coincides with the cylinder axis on the loop.
`oint vecB.vec(dl)=ointB.dl=Boint dl=B`.
`2pir=mu_(0)I/(piR^(2))pir^(2)`
`B=(mu_(0)Ir)/(2piR^(2))=(mu_(0)Jr)/2 rArr vecB=(mu_(0)(vecJxxvecr))/2`