State Ampere's circuital law . Using it, derive the expression for magnetic field at a point due to a long current carrying conductor .

Explanation

The line intergral of `vec B . d vec I` around any closed path enclosing a surface is `mu_(0)` times the total steady current passing through the surface.
Consider a straight conductor of infinite length in the plane of the paper.
Let I be the current through it from X to Y.
Let P be a point at a perpendicular distance r from the straight wire.
Consider a circle of radius r around the wire passing through P as an amperian loop.
The magnetic field `vec B` at P due to the current in wire is tangential to the circle.
Hence the magnetic field `vec B` is parallel to the line element `d vec I` at P.

`oint vec B . d vec I = oint B dI cos 0 = B oint dI = B (2pi r)`
According to Ampere.s circuital law, `oint vec B . d vec I = mu_(0) I`
`B (2 pi r) = mu_(0) I`
`B = (mu_(0) I)/ (2 pi r)`
`B = (mu_(0))/ (4 pi r) (2 I)/ (r)`