Chapter 10

Conservation Laws & Poynting Theorem

10.1 Energy Conservation — Poynting's Theorem

The electromagnetic field carries energy. The total energy density is:

$$u = \frac{1}{2}\left(\epsilon_0 E^2 + \frac{1}{\mu_0}B^2\right)$$

The Poynting vector represents energy flux:

$$\mathbf{S} = \frac{1}{\mu_0}(\mathbf{E} \times \mathbf{B}) \qquad \text{[W/m}^2\text{]}$$

Poynting's theorem (energy conservation):

$$\frac{\partial u}{\partial t} + \nabla\cdot\mathbf{S} = -\mathbf{J}\cdot\mathbf{E}$$

Rate of change of EM energy + energy flux out = negative of work done on charges (Ohmic loss).

The electromagnetic field carries momentum with density:

$$\mathbf{g} = \mu_0\epsilon_0\mathbf{S} = \frac{\mathbf{S}}{c^2} = \epsilon_0(\mathbf{E}\times\mathbf{B})$$

This leads to radiation pressure: light pushes on surfaces it illuminates. For a plane wave, $|\mathbf{g}| = u/c$.

The Maxwell stress tensor $\overleftrightarrow{T}$ encodes the mechanical force per unit area exerted by the EM field. Its components are:

$$T_{ij} = \epsilon_0\left(E_i E_j - \frac{1}{2}\delta_{ij}E^2\right) + \frac{1}{\mu_0}\left(B_i B_j - \frac{1}{2}\delta_{ij}B^2\right)$$

The electromagnetic force on a volume $\mathcal{V}$ is:

$$\mathbf{F} = \oint_S \overleftrightarrow{T}\cdot d\mathbf{a} - \epsilon_0\mu_0\frac{\partial}{\partial t}\int_\mathcal{V}\mathbf{S}\,d\tau$$

Maxwell's equations automatically imply charge conservation:

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0$$

This is a local statement: charge cannot disappear at one point without flowing away as current.