Abstract
So far we have discussed the free space macroscopic electromagnetic fields and the field equations: the Maxwell equations that describe how the fields evolve in space and time. Charges are included in this description as macroscopic charge density and current density. However point charges carry energy and momentum, the Maxwell equations have wave solutions and we might expect this to imply that the fields carry energy and momentum also. We now formalize the concept of energy and momentum of the electromagnetic fields.
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Notes
- 1.
The Maxwell equations and conservation equations for free space and charges discussed here are linear. Our approach then generalizes to media in which these equations remain linear. This is the case if the fields induced in the medium are linearly proportional to those in the surrounding free space. See the revision problems for examples.
- 2.
Throughout this book we will use the index notation i, j to mean all values between i and j inclusive.
- 3.
Liouville’s theorem expresses conservation of probability density along a trajectory in phase space. In a system with no sources or sinks of particles, we can follow the phase space trajectory \(\mathbf{r}(t), \mathbf{v}(t)\) of any particle and along that trajectory the probability of finding the particle in elemental phase space volume \(d\mathbf{r}d\mathbf{v}\) is constant. See advanced problem 2.
- 4.
In linear media, that is, where the response from the bound charges in the medium is linearly proportional to the applied field, \(\mathbf{B}=\mu _r\mu _0\mathbf{H}\), \(\mathbf{D}=\epsilon _r\epsilon _0\mathbf{E}\) and \(\mathbf{S}=\mathbf{E}\wedge \mathbf{H}\) and \(U=\frac{1}{2}[ \mathbf{E}\cdot \mathbf{D} + \mathbf{B}\cdot \mathbf{H}]\) then conservation of energy is still given by (2.33). In nonlinear media (2.33) is no longer valid. See revision problem 8.
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Chapman, S. (2021). Field Energy and Momentum. In: Core Electrodynamics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-66818-1_2
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DOI: https://doi.org/10.1007/978-3-030-66818-1_2
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