Using Ampere's circuital theorem, calculate the magnetic field due to an infinitely long wire carrying current I.

Explanation

Magnetic field due to a long current carrying wire : Consider an infinitely long wire XY carrying current I as shown in Fig. Suppose that P is the point at a distance r from the conductor, where the magnetic field due to the current carrying wire is to be determined. Draw a circle of radius r passing through point P, such that the wire XY is normal to the plane of the circle and passes through its centre.

Let `vecB` is magnetic field at point P due to the current carrying wire and `vec(AB) = vec(dl)` is small element of the circle drawn around the wire.
Since the current threading the circular path is I, according to Ampere.s circuital law, we have
`oint vecB. vec(dl) = mu_0 I`
As `vecB` acts along the tangent and has the same direction at that of `vec(dl)`.
So the angle between the vector `vecB` and `vec(dl)` is `0^@.` Therefore,
`oint vecB. vec(dl) = oint Bdl cos 0^@ = oint Bdl = B oint dl`
But `oint dl=` Circumference of the circular path of radius` r = 2pi r`
Therefore `oint vecB. vec(dl) = B xx 2pi r `
or `B = (mu_0 I)/(2pi r) = (mu_0)/(4pi) . (2I)/r`
It gives the strength of magnetic field due to an infinitely long current carrying wire at a point at a distance r from it.