26.5 Differential form of the Maxwell-Ampère law

This post is a sequel to post 26.4 so I suggest you read that first. In post 26.4, I expressed the Maxwell-Ampère law in terms of two definite integrals. In this post I am going to express it in a differential form. This is similar to what I did in post 25.12 where I expressed Gauss’s law in a differential form. The differential equation of the Maxwell-Ampère law is given by equation 1 above. In this equation, the left-hand side is the cross-product of the operator del with the magnetic field, B. Here μ0 is the permeability of free space, J is the current density (described near the end of post 25.17), ε0 is the permittivity of free space and ∂E/∂t is the derivative of the electric field E with respect to time, t. Equation 2 is the Maxwell-Ampère law described in post 26.4. The definite integral on the left-hand side is a definite integral over the length, L, of a closed loop formed by a magnetic field line. The definite integral on the right-hand side is over the area, A, enclosed by…