3.2 Generalising Ampère's law
We need to generalise Ampère's law beyond the confines of static charge densities. Let's try adding an extra (and at this stage unknown) vector field, K to the right-hand side of the differential form of Ampère's law. The modified equation then reads
What can be said about the term K? Taking the divergence of both sides of the modified equation, and using the fact that the divergence of any curl is equal to zero, we obtain
So, using the equation of continuity (Equation 7.4), we have
Now Gauss's law tells us that
so, using Equation 7.7 to interchange the order of the time and space derivatives,
We conclude that
The simplest solution to this equation is
but there are other solutions as well. In fact, the most general solution is
where X is any smooth vector field. You can easily verify that this satisfies Equation 7.8, because taking the divergence of both sides gives
and the last term vanishes, being the divergence of a curl.
This is as far as mathematical analysis can take us. It is not too surprising that we still have a choice to make. You should not expect to derive a fundamental law of physics from other laws. As a general rule, however, it is sensible to adopt the simplest law that is consistent with all the known facts. That is what Maxwell did, and we shall follow his lead. We assume that curl X = 0 in Equation 7.9, and replace Ampère's law by
This equation is called the Ampère–Maxwell law and the additional term, ε0μ0∂E/∂t, is called the Maxwell term. Some authors refer to the Maxwell term as μ0 times the displacement current density. We will not use this terminology in this course, but an optional appendix (best read after this section) describes the curious logic behind it (see Section 6). The Ampère–Maxwell law is the last of Maxwell's four equations of electromagnetism. It is believed to be true in all situations, both static and dynamic.
The above argument for the Ampère–Maxwell law is driven by theory. If we believe the law of conservation of charge (as we do), our mathematical analysis shows that Ampère's law must be modified for time-dependent situations. The simplest modification, consistent with Gauss's law and the equation of continuity, is then given by the Ampère–Maxwell law. Physics walks forward on the two legs of theory and experiment. Sometimes experiment strides ahead and reveals new facts which cry out for theoretical interpretation. The Ampère–Maxwell law is an early example of the opposite process – a law that emerged from a theoretical argument, and cried out for experimental confirmation.
In Maxwell's day, there was no direct experimental evidence requiring a modification to Ampère's law. The Maxwell term ε0μ0∂E/∂t is usually very small in comparison with the term associated with the current density, μ0J. For example, if a mains-frequency current is uniformly distributed throughout a copper wire, the Maxwell term in the wire is only about 5 × 10−17 as large as μ0J. On this basis, it is tempting to dismiss the Maxwell term as a practical irrelevance, but this would be a serious error of judgement. Although small, the Maxwell term can exist in empty space, where no real currents exist, and there it plays a vital role in sustaining the propagation of electromagnetic waves, as you will soon see. Ultimately, the existence of these waves provides the best evidence for the whole of Maxwell's theory, including the Maxwell term and the Ampère–Maxwell law.
Equation 7.10 is the differential version of the Ampère–Maxwell law. Show that the corresponding integral version is
where C is a closed loop and S is any open surface that has C as its perimeter. The sense of positive progression around C and the orientation of S are related by the right-hand grip rule.
Taking the surface integral of both sides of Equation 7.10 over an open surface S gives
Using the curl theorem on the left-hand side we obtain
where the sense of positive progression around C and the orientation of S are related by the right-hand grip rule. This is the required integral version of the Ampère–Maxwell law.