This is a sequel to post 26.03, so I suggest you read that first.

Maxwell realised that Ampère’s law applied only when an electric current, in a conductor enclosed by a magnetic field loop, was constant. If the current changes, for any reason, in the first picture of post 26.03, then equation 1 (of post 26.03) must be modified.

Maxwell developed this idea by considering a capacitor, consisting of two parallel plates, as described in post 18.19, in a circuit where the current was changing. There is no current flowing between the plates (because the capacitor stores charge), but Maxwell realised that he could extend Ampère’s law by considering a hypothetical current called the displacement current.

When the current in a circuit changes, the charge on the capacitor changes (see post 18.19) so that the electrical field, between the plates, changes (see post 16.25). When the electrical field changes, the associated magnetic field must also change; it may help to think of the example of electromagnetic induction as being a relationship between electric and magnetic fields, as explained in post 25.14. Now let’s think of the electric flux across the area that encloses the volume between the plates of the capacitor. If Q is the total charge within this volume, then according to Gauss’s law (as explained in post 25.11), the electric flux is given by

where ε₀ is the permittivity of free space.

We have seen that the charge on the capacitor can change, so the electric flux must then change. Differentiating both sides of equation 1 with respect to time, t, gives

Mathematically, dQ/dt has the properties of a current (see post 18.24). So the electric flux changes as if there were a hypothetical current (the “displacement current”) flowing within the capacitor, although no current really flows.

From equation 2, the displacement current, I_d, is given by

Maxwell extended Ampère’s law, given by equation 1 of post 26.03, by adding the displacement current to the real current so that

Combining equations 3 and 4 gives the result

Equation 5 is a statement of the Maxwell-Ampère law.

If we want to make it clear that equation 5 is about a relationship between electrical and magnetic fields, we can combine equation 5 with the definition of flux, from the last equation in post 25.9, to give

In equation 6, E is an electrical field and the definite integral is over the whole surface; the unit vector u is perpendicular to an element, δA, of area.

Equation 5 shows explicitly that the Maxwell-Ampère law is a relationship between fields.

Related posts

26.03 Ampère’s law
25.14 Electromagnetic induction and fields
25.09 More about fields
17.24 Fields and vectors