Poynting's Theorem simplified? How can I apply Poynting's theorem to any system that has a magnetic field & electric field, to state conservation of energy?
How does Poynting's theorem state conservation of energy in EM? I struggled to understand with the differential equations.
Example:
Work being done by the Lorentz force on a conductor by a supplied power source?
 A: Here is a version of Poynting's theorem, appropriate for media with $\epsilon_r=1$, $\mu_r=1$ (for simplicity).
$$\nabla \cdot {\bf S} + \frac{1}{2}\frac{\partial}{\partial t} \left( \epsilon_0 E^2 + \frac{B^2}{\mu_0}\right) + {\bf E} \cdot {\bf J} = 0,$$
where ${\bf S}$ is the Poynting vector (${\bf E} \times {\bf B}/\mu_0$ or ${\bf E} \times {\bf H}$).
The first term, the divergence of the Poynting vector, is the flux per unit volume of ${\bf S}$, which is positive if the flux of ${\bf S}$ is outWard. As the Poynting vector has units of Watts per square metre, this flux is measured in Watts per unit volume (i.e. rate of change of energy per unit volume).
Terms 2 and 3 represent: the rate of change of energy per unit volume in the electromagnetic fields and the rate at which work is done on charges by the electric field (the magnetic field does no work because the magnetic force is perpendicular to the velocity). 
Integrating over a volume, the first term can be replaced using Gauss's theorem to give
$$ \oint {\bf S} \cdot d{\bf A} + \int \frac{1}{2}\frac{\partial}{\partial t} \left( \epsilon_0 E^2 + \frac{B^2}{\mu_0}\right) + {\bf E} \cdot {\bf J}\ dV = 0$$. where the first integral now clearly represents the total flux of ${\bf S}$ into or out of a volume.
So let's consider some scenarios.


*

*No work is done by the electromagnetic fields but the amount of energy per punit volume in the EM fields is decreasing. Poynting's theorem tells us that the energy must be leaving in the form of a positive Poynting vector flux.

*The energy per unit volume remains constant, but work is being done by the fields on the charges. i.e. ${\bf E}\cdot {\bf J}$ is positive. Poynting's theorem tells us that this work is supplied by a flux of ${\bf S}$ inwards.

*Let's say that the work being done by the fields is positive, but as a result the energy per unit volume in the fields is decreasing at the same rate. Poynting's theorem tells us there is no flux of ${\bf S}$ into or out of the considered volume.
A: (add my comment as an answer)
Poynting's theorem for EM is the expression for a continuity equation (and compatibility condition) that describes how energy is transfered in local terms (and how energy is conserved) utilising distributions. 
Similar contituity equations are formulated in many areas especially in waves (including quantum mechanics aka wave mechanics). 
Mathematically it is a form of Stoke's Theorem, which in simpler terms states that what goes in, minus what goes out, equals what remains inside
In the case of Pynting Theorem (as stated in wikipedia link), what is inside is the rate of energy transfer (per unit volume), what goes out is the energy flux leaving the region and what goes in is the rate of work done on a charge distribution
(one can switch the "labels" of what is out, in or inside and it will still be correct as long as it is consistent)
