8.1-8.2. Conservation of Energy and Momentum

In this chapter, we will study the conservation of energy and momentum in classical electrodynamics.

When we think of charges moving according to the Lorentz force law, it doesn't seem that energy or momentum is conserved; Of course it is not, because we are only looking at the energy stored in the charge distribution. However, energy and momentum is also latent in the fields. Combining the portions in the fields, we see that energy and momentum are indeed conserved. Moreover, if there is no 'matter' portion (i.e. if we consider an empty space), the energy and momentum themselves obey the continuity equation, meaning they are also 'locally' conserved.

As we will see, those conservation laws are expressed elegantly with two quantities: the Poynting vector and Maxwell's stress tensor.

Conservation of Energy - Poynting's Theorem

The energy density stored in electromagnetic fields is

$\displaystyle u={1\over2}(\epsilon_0 E^2+{1\over\mu_0}B^2)$

Here we introduce the energy flux density, or the Poynting vector:

$\displaystyle \mathbf{S}={1\over\mu_0}(\mathbf{E}\times\mathbf{B})$

Then in a volume $\mathcal{V}$, we have an identity derived from Maxwell's equations:

$\displaystyle \int_{\mathcal{V}}\mathbf{(J\cdot E)} d\tau$ $\displaystyle = -{d\over dt}\int_{\mathcal{V}}{u}d\tau-\oint_S {\mathbf{S}\cdot d\mathbf{a}}$ .

This is called the Poynting's theorem. It means that the change in electromagnetic energy stored in the charge distribution (the left-hand side) is equivalent to the change in the 'field energy', plus the inflow/outflow of the energy across the surface.

In an electromagnetic system, energy is transferred back and forth between the field and matter, while the total amount of energy remains conserved. Here, the Poynting vector represents the transfer of field energy from a region to another.

Poynting's theorem is derived by applying Maxwell's equations (Ampere-Maxwell law and Faraday's law) and vector product rules to the left hand side.

Conservation of Momentum

Like energy, momentum is attributable to both the field and matter. In a volume $\mathcal{V}$, the electromagnetic force applied to a volume in transferred to mechanical momentum of the charges, and the momentum inherient in the fields. The change in mechanical momentum $\mathbf{p}_{mech}$ is calculated according to the Lorentz force law. Meanwhile, the field momentum is expressed with the Poynting vector. We introduce the field momentum density $\mathbf{g}$ as follows:

$\mathbf{g}=\mu_0\epsilon_0\mathbf{S}$

If the total momentum is conserved, we will have an equation in the form

$\displaystyle {d\mathbf{p}_{mech}\over dt} + \int_{\mathcal{V}} {\partial \mathbf{g}\over \partial t}d\tau=$ (in/outflow of momentum across the surface).

What is the inflow/outflow of momentum here? It is the total electromagnetic force applied to the volume. We get it by integrating the electromagnetic force per unit area, over the surface. That is, we integrate the electromagnetic stress over the surface. The stress is expressed with Maxwell's stress tensor $\mathbf{\stackrel\leftrightarrow{T}}$ which is defined as:

$\displaystyle T_{ij}=\epsilon_0 ( E_iE_j - {1\over2}\delta_{ij}E^2 ) + {1\over\mu_0}( B_iB_j - {1\over2}\delta_{ij} B^2 )$

Now, the conservation law is written as:

$\displaystyle {d\mathbf{p}_{mech}\over dt} + \int_{\mathcal{V}} {\partial \mathbf{g}\over \partial t}d\tau=$ $\displaystyle \oint_{S} {\mathbf{\stackrel\leftrightarrow{T}}} \cdot d\mathbf{a}$

Or, we can write:

$\displaystyle \mathbf{F} = \oint_S {\mathbf{\stackrel{\leftrightarrow}{T}}}\cdot{d\mathbf{a}} - \epsilon_0\mu_0 {d\over dt} \int_{\mathcal{V}} \mathbf{S} d\tau$

This equation is derived by integrating the Lorentz force of all charges in the volume, and applying Maxwell's equations and vector product rules.

The Continuity Equation

The continuity equation of electric charge reads

$\displaystyle {\partial\rho\over\partial t}=-\nabla\cdot\mathbf{J}$

This states the local conservation of charge. The 'continuity equations' analogous to this holds also for electromagnetic energy and momentum.

The Poynting's theorem can be written in differential form as follows:

$\displaystyle \mathbf{J\cdot E}=-{\partial u\over \partial t}-\nabla\cdot\mathbf{S}$

Now, if $\mathbf{J}=\mathbf{0}$, it becomes the continuity equation for field energy.

$\displaystyle {\partial u\over \partial t}=-\nabla\cdot\mathbf{S}$

Assuming the mechanical momentum is always zero, the conservation of momentum can also be written in differential form. This becomes the continuity equation for field momentum.

$\displaystyle {\partial\mathbf{g}\over\partial t}=\nabla\cdot{\stackrel\leftrightarrow{\mathbf{T}}}$

These two continuity equations suggest that electromagnetic energy and momentum are locally conserved when there is no charge around. However, when there are charges around, neither field energy nor field momentum are conserved in general. Those can be exchanged with mechanical energy and momentum, and only their total amount is conserved.

References

1. David J. Griffith. <Introduction to Electrodynamics>. 5th ed. Chapters 8.1-8.2.
2. https://en.wikipedia.org/wiki/Maxwell_stress_tensor
3. https://en.wikipedia.org/wiki/Poynting%27s_theorem