Ampere’s circuital law was discovered by André-Marie Ampere in the year 1826 A.D. This law states that, “The line integral of magnetic field B along any closed path in free space is equal to the vacuum permeability (μo) times the total current enclosed by the surface.” Mathematically,
Let us consider a straight conductor carrying current ‘I’. Then the magnetic field around the conductor are concentric circles.
Let ‘P’ be a point at a distance ‘r’ from the conductor. From point P, let us draw a circle of radius ‘r’. Then, the magnitude of magnetic field ‘B’ at any point on the circle is
B = μoI/2πr
And the direction of B is along the tangent to the circle at that point. If we consider an element of length ‘dl’ on the circle, then the angle between ‘dl’ and B is θ = 0o. Now, the line integral of magnetic field B along the closed path isHence, Ampere’s circuital law is proved.