Ampere’s Circuital Law

Statement. The line integral Ñ„ B. dl for a closed curve is equal to μ0 times the net current i through the area bounded by the curve. That is,
            Ñ„ B. dl = μ₀i,
where μ0 is the permeability constant.

Proof. Consider a long straight conductor carrying a current i perpendicular to the page directed outward. According to Biot-Savart law, the magnitude of the magnetic induction at a distance r from it is given by

    B = μ₀i / 2πr         …. (1)

At each point on the circle, B has constant magnitude B and dl which is always tangential to the path of integration, point in the same direction as B. Thus,

Ñ„ B. dl = Ñ„     B dl = BÑ„ dl = (B) (2πr)
Here, 2πr = Ñ„ dl is the circumference of the circle.
Substituting the value of B from Eq. (1), we get
    Ñ„ B. dl = μ₀i  (2πr) = μ₀i
 _____       2πr

Thus the integral Ñ„ B. dl is μ0 times the current through the area bounded by the circle. This is Ampere’s law.