State Ampere's circuital law. By using it derive an expression for magnetic field intensity at a point due to a straight current carrying conductor.

Explanation

It state that the line integral of magnetic field induction B around any closed path in vacuum is equal to `mu _(0)` times the total current enclosed by the closed path i.e.,`ointvec B . vec (dl) = mu _(0) I`
This result is independent of shape and size of closed path.
Proof
Consider a long straight conductor XY, lying in the plane of paper.
Let I be the current flowing in the direction as shown in The magnetic field `vec B ` is produced around the conductor.
The magnitude of magnetic field induction B produced at point
P, at a distnce r from the conductor given by
`B = ( mu _(0) I )/( 2pi r ) " "...(i)`
The direction of `vec B` at every point is aong the tanget to the circle. Clearly, the angle between `vec B and vec (dl)` is always zero. The line integral of `vec B` around the closed path of radius r is given by ,
`oint vec B. vec (dl) = oint B dl co 0 ^(@) = oint B dl implies ointvec B. vec (dl) = B ointdl`
Substituting the value of B from equation (i), we get
`oint vec B. vec (dl) = ( mu _(0) I )/( 2 pi r ) oint dl`
`implies ointvec B. vec (dl) = ( mu _(0) I )/( 2pi r ) xx 2 pi r " "[because oint dl = 2 pi r ]`
`impliesoint vec B. vec (dl) = mu _(0) I" "...(i)`