Poynting's theorem and continuity equation

Question

I have recently come to realise that many of the most fundamental theorems can be reduced to a continuity equation. Doing some research on the topic of said equations, I have found out they have stronger implications than just the conservation of a given magnitude, since they also impose locality. I am specially interested in Poynting's theorem:

$$-\frac{\partial U}{\partial t}=\int_V\vec{J}\cdot \vec{E}dV + \oint \vec{S}\cdot d\vec{A}$$

Using Gauss' theorem to transform the surface integral into a volume integral and taking a region of space in which there is no free charge, we get:

$$\frac{du}{dt}+\vec{\nabla}\cdot \vec{S}=0$$

where $u$ is the electromagnetic energy density, which fullfils: $U=\int_V u\ dV$

What are the implications of this? My best guess is the electromagnetic energy density in a given region diminishes when EM energy escapes said region in the form of radiation, represented by the divergence of Poynting's vector. Is there a general (physical) interpretation for what happens to the system for two general magnitudes $f,\vec{G}$ such that they fulfill the continuity equation?

$$\frac{\partial f}{\partial t}+\vec{\nabla}\cdot\vec{G}=0$$

I have also found that, regardless of which theorem I choose to reduce to this form, I always end up with a density on the time derivative and a flux on the gradient. How so?

Answer 1

Continuity equations like the ones you are concerned with always arise whenever one attempts to mathematically formulate a notion of the local conservation of some quantity. The physical interpretation of the divergence is that the quantity specifies the degree to which a point in space is a "source" or "sink" of some lines of flux. For example, the divergence is often seen defined as: $$\nabla\vec G(\vec a)=\lim\limits_{r\rightarrow 0}{3\over 4\pi r^3}\int\int_{\mathcal{B}(r,\vec a)}\vec F\cdot \mathbf{\hat n}\;dA,\quad\vec x\in\mathbb{R^3}.$$ Where the surface of integration is the boundary of a ball about the point $\vec a$.Thus, it makes sense to reason that if we define some kind of "current" $\vec J(\vec x,t)=\rho(\vec x,t)\vec v(\vec x,t)$, where $\rho(\vec x,t)$ is some kind of density and $\vec v(\vec x,t)$ a vector field that specifies the "flow", then if the "substance" in question is conserved, the rate of change of the substance specified by the density in some region $R$ is given by: $${d\over dt}\int\int\int_R\rho(\vec x,t)\;d^3x=-\int\int_{\partial R}\vec J(\vec x,t)\cdot\mathbf{\hat n}\;dA.$$ So if we take the region $R$ to be a ball of radius $r$ centered on the point $\vec x$, then using the above definition of divergence we get in the limit, the differential form of the conservation law: $${\partial\rho\over\partial t}+\nabla\cdot\vec J=0.$$ Note we have written the conservation law for any kind of conserved quantity, it could charge, mass, energy, or other quantity. This kind of reasoning can allow one to derive some interesting PDEs of mathematical physics, for example, by using Newton's law of cooling, or its more general incarnation, Fick's law along with the above conservation principle, then on can derive as a consequence the diffusion operator.

So, you will see this same mathematics all over physics wherever there are local concepts of conservation, however, there is a point of contact with a powerful formalism of physics, namely Noether's theorem.

Noether's theorem is a statement to the effect that for any given physical system, there are conserved quantities associated with the symmetries of the system. So for each symmetry of the physical system one will be able to derive an associated conservation law, as we know from above, that will amount to a continuity equation, with an associated "Noether current"(flux) and "Noether charge"(density) as each of the quantities in the continuity equation are colloquially known within the general context of field theory.