26.3 Ampère’s law

Ampère’s law considers the electric current, I, enclosed by a magnetic field, B, that forms a closed loop. In post 25.16, we saw how to calculate the magnetic field surrounding an electrical current using the Biot-Savart law. Now we are going to consider the current enclosed by a magnetic field line. We will represent an elemental arc in the loop by the vector δL. According to Ampère’s law there is electrical current, I, enclosed within this magnetic loop given by the definite integral where the P at the foot of the integral sign denotes integration around the whole loop and μ0 is the permeability of free space. I am following convention here by considering that the conductor carrying the current is surrounded by a vacuum (but it doesn’t make much difference if it is surrounded by air – see post 16.25). I’m not going to give a rigorous derivation of equation 1; In post 25.16, I used a simple form of the Biot-Savart law to calculate the magnetic field due to the current in an infinite…