4 Maxwell's equations
We have reached a major milestone. All four of Maxwell's equations are now in place. This is an appropriate place to review their meaning and significance. We concentrate here on the differential versions, which are as follows:
Name the above equations.
In the order presented, the equations are called: Gauss's law, the no-monopole law, Faraday's law and the Ampère–Maxwell law. It would be a real advantage to remember them. This may come naturally, after sufficient use.
Maxwell's equations are of great generality. They apply to all charge and current densities, whether static or time-dependent. Together, they describe the dynamical behaviour of the electromagnetic field. Each of Maxwell's equations is a local equation, relating field quantities at each point in space and at each instant in time, so all trace of instantaneous action at a distance has been eliminated. The revolutionary nature of this description was recognised by Einstein, who wrote:
‘The formulation of [Maxwell's] equations is the most important event in physics since Newton's time, not only because of their wealth of content, but also because they form a pattern for a new type of law … Maxwell's equations are laws representing the structure of the field … All space is the scene of these laws and not, as for mechanical laws, only points in which matter or charges are present.’
Maxwell's equations are partial differential equations. They link the spatial and temporal rates of change of electric and magnetic fields, and they show how these rates of change are related to the sources of the fields – charge and current densities. The spatial rates of change of the fields are neatly bundled up as div E, div B, curl E and curl B – divergences and curls. This, in itself, is an immense simplification. Each field has three components, which can be partially differentiated with respect to three coordinates, so there are 18 first-order spatial partial derivatives of the electric and magnetic fields. The divergences and curls collect these partial derivatives together, focusing attention on only eight quantities of interest (a scalar for each divergence and three components for each curl). Moreover, divergences and curls have clear physical interpretations, telling us how the fields spread out and circulate at each point.
Where do the electric and magnetic fields come from? The modern answer is that they come from the terms in Maxwell's equations that describe matter – the charge and current densities, ρ and J. To be explicit about this, we can re-order and rearrange Maxwell's equations so that the two source terms appear on the right-hand sides of the first two equations:
In regions where there are no charges or currents, all four equations have zero on the right-hand sides. They then tell us the conditions that electric fields and magnetic fields must satisfy in empty space. These conditions describe the internal structure and dynamics of the electromagnetic field. We will discuss this dynamics in the next section, and you will see that it allows the propagation of wave-like disturbances – electromagnetic waves.
In regions where there are charges and currents, the first two equations have an additional role. They tell us how the electromagnetic field is coupled to matter, and the left-hand sides of these equations describe the response of the electromagnetic field to the local charge and current densities. The last two equations do not have this role, so Maxwell's equations are asymmetrical. The absence of source terms in the last two equations arises because magnetic monopoles, and monopole currents, are assumed to be non-existent.
When Maxwell introduced his equations, he expected them to apply in a special frame of reference – the frame of the stationary ether. This is not the modern view. We now believe equations apply in all inertial frames of reference – that is, all frames in which free particles move uniformly, with no acceleration. Maxwell's equations are also unaffected by time-reversal and by reflections in space.
Only one caveat need be mentioned. Maxwell's equations do not apply in non-inertial frames. In a rotating frame of reference, for example, the electric flux over a closed surface can be non-zero even though the surface encloses no net charge – a clear violation of Gauss's law. This should not alarm you. Most laws of physics, including the laws of conservation of energy and momentum, are restricted to inertial frames of reference, and Maxwell's equations are no exception.
Show that Maxwell's equations are unchanged by the operation of time-reversal, which changes t → −t, J → −J and B → −B, but leaves ρ and E unchanged.
Applying the transformation rules for time-reversal given in the question does not affect Gauss's law. The remaining Maxwell equations transform as follows:
In each case, the transformed equation can be rearranged to recover the original Maxwell equation, so Maxwell's equations are unchanged by time-reversal.
Show that the equation of continuity is contained within the Ampère–Maxwell law and Gauss's law.
Taking the divergence of the Ampère–Maxwell law (Equation 7.10) gives
The left-hand side is equal to zero (from Equation 7.5). Interchanging the divergence and time derivative on the right-hand side and cancelling the factor μ0, then gives
Using Gauss's law, div E = ρ/ε0, we finally obtain
which is the equation of continuity. Maxwell wrote down the equation of continuity alongside his other equations, but it is not counted as one of his four laws of electromagnetism because it is a consequence of two of the other laws.