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 integral of the products of tangential component of magnetic field and the element of length in a closed loop is equal to `mu_0` times the total current passing through the surface.
`oint vecB . Dvecl=mu_0I`
Let I be the current in the wire PQ . Let P be a point at a perpendicular distance r from the wire . Let dl be the element of length of the closed path. The direction of the magnetic field will be tangential to the point P on the element .

Applying Ampere.s circuital law , we have
`oint vecB . dvecl=mu_0I`
i.e., `oint Bdl cos 0^@=mu_0I`
`theta=0^@` between tangent drawn at P to the element of length dl and `vecB`.
Thus , `B oint dl=mu_0I`
or `B=(mu_0I)/(oint dl)=(mu_0I)/(2pir)`
Thus , `B prop 1/r`