A long straight wire of a circular cross-section of radius 'a' carries a steady current I. The current is uniformly distributed across the cross-section. Apply Ampere's circuital law to calculate the magnetic field at a point at distance 'r' in the region for (i) `rlta` and (ii) `rgta`.

Explanation

(i)For `rlta`
`ointvecB.vec(dl)=mu_(0)I_("enclosed")`
`I_("enclosed")/(pia^(2))=I/(pir^(2))`
`I_("enclosed")=Ir^(2)/a^(2)`
`vecB.vec(dl)=Bdl=(because cos theta=1)`
`becauseointBdl=mu_(0)lr^(2)/a^(2)`
`B(2pir)=mu_(0)Ir^(2)/a^(2)`
`B=mu_(0)/(2pi)I/a^(2) r`
(ii)For `rgta`
From Ampere's circuital law
`oint vecB.vecdl=mu_(0)I_("enclosed")`
`vecB.vec(dl)=Bdl cos theta`
`theta=0`
`vecB.vec(dl)=Bdl`
`I_("enclosed")=I`
`ointBdl=mu_(0)I`
`Bointdl=mu_(0)I`
`B(2pir)=mu_(0)I`
`B=(mu_(0)I)/(2pir)`