Electromagnetic Energy Conservation

Consider the fourth Maxwell equation:

$$\nabla\times{\bf B} = \mu_0\, {\bf j} +\epsilon_0\,\mu_0 \frac{\partial {\bf E}} {\partial t}.$$ (100)

Forming the scalar product with the electric field, and rearranging, we obtain

$$- {\bf E} \cdot {\bf j} = - \frac{ {\bf E}\cdot \nabla\times{\bf B}}{\mu_0} +\epsilon_0 \,{\bf E}\cdot \frac{\partial {\bf E}}{\partial t},$$ (101)

which can be rewritten

$$- {\bf E} \cdot {\bf j} = - \frac{ {\bf E}\cdot \nabla\times{\bf ... ...u_0} +\frac{\partial}{\partial t}\!\left(\frac{\epsilon_0 \,E^{\,2}}{2}\right).$$ (102)

Now,

$$\nabla\cdot({\bf E}\times{\bf B}) \equiv {\bf B}\cdot\nabla\times{\bf E} - {\bf E} \cdot \nabla\times{\bf B},$$ (103)

so

$$- {\bf E} \cdot {\bf j} = \nabla \cdot \left(\frac{{\bf E}\times{... ..._0} + \frac{\partial}{\partial t}\!\left(\frac{\epsilon_0\, E^{\,2}}{2}\right).$$ (104)

Making use of third Maxwell equation,

$$\nabla\times{\bf E} = - \frac{\partial {\bf B}}{\partial t},$$ (105)

we obtain

$$- {\bf E} \cdot {\bf j} = \nabla \cdot \left(\frac{{\bf E}\times{... ...l t} + \frac{\partial}{\partial t}\!\left(\frac{\epsilon_0\,E^{\,2}}{2}\right),$$ (106)

which can be rewritten

$$- {\bf E} \cdot {\bf j} = \nabla \cdot \left(\frac{{\bf E}\times{... ...ial t}\!\left( \frac{\epsilon_0\,E^{\,2}}{2} +\frac{B^{\,2}}{2\,\mu_0} \right).$$ (107)

Thus, we get

$$\frac{\partial U}{\partial t} + \nabla\cdot{\bf u} = -{\bf E}\cdot {\bf j},$$ (108)

where $U$ and ${\bf u}$ are specified in Equations (109) and (110), respectively.

By comparison with Equation (7), we can recognize the previous expression as some sort of conservation equation. Here, $U$ is the density of the conserved quantity, ${\bf u}$ is the flux of the conserved quantity, and $-{\bf E}\cdot{\bf j}$ is the rate at which the conserved quantity is created per unit volume. However, ${\bf E}\cdot{\bf j}$ is the rate per unit volume at which electric charges gain energy via interaction with electromagnetic fields. Hence, $-{\bf E}\cdot{\bf j}$ is the rate per unit volume at which electromagnetic fields gain energy via interaction with charges. It follows that Equation (108) is a conservation equation for electromagnetic energy. Thus.

$$U = \frac{\epsilon_0\,E^{\,2}}{2} + \frac{B^{\,2}}{2\,\mu_0}$$ (109)

can be interpreted as the electromagnetic energy density, and

$${\bf u} = \frac{{\bf E}\times{\bf B}}{\mu_0}$$ (110)

as the electromagnetic energy flux. The latter quantity is usually called the Poynting flux, after its discoverer.