Let us consider an infinitely long conductor XY carrying current I amps from X to Y.
Let 'P' be a point at a distance r from the conductor and on the perpendicular to it.
Let 'dl' be the small length of the circle at P.
Line integral of magnetic-field along the circular path is
\(\phi \vec B.\vec{dl} = \int Bdl cos\theta\) \(\theta = 0 \) \(cos\theta = 1\)
= \(\int Bdl = B \int dl\) \(\int dl = 2\pi r\)
\(\therefore \phi \vec B. \vec{dl} = B \times 2\pi r\) .............(1)
According to ampere circuital law
\(\phi \vec B. \vec {dl} = \mu_0I \) ...................(2)
From eqn. (1) and (2)
B x \(2\pi r = \mu_0I\)
B = \(\frac{\mu_0I}{2\pi r}\)
B = \(\frac{\mu_0}{4\pi}\times\frac{2I}{r}\)
