3 The Ampère–Maxwell law
3.1 Limitations of Ampère's law
In order to analyse the limitations of Ampère's law, and suggest ways of overcoming them, we need to use some properties of divergence. For ease of reference, these properties are given below:
Some properties of divergence
The divergence of any curl is equal to zero:
A constant k can be taken outside a divergence:
A time derivative can be taken outside a divergence:
You can take these properties on trust if you wish, but it is easy enough to prove them by expanding both sides in Cartesian coordinates.
Prove Equation 7.5.
Expanding the left-hand side of Equation 7.5 gives
which vanishes because mixed partial derivatives do not depend on the order of partial differentiation.
Now let's examine the differential version of Ampère's law, which states that
The limitations of this law are revealed by taking the divergence of both sides. This gives
The divergence of any curl is equal to zero so, using Equation 7.6 and the equation of continuity, we have
We therefore see that Ampère's law requires the charge density to remain constant. Put another way:
Ampère's law fails when the charge density changes in time.