Electrodynamics References

Maxwell’s Equations

In General:

\[\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{1}{\epsilon_0} \rho \\ \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \\ \nabla \cdot \mathbf{B} &= 0 \\ \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}\]

In Matter:

\[\begin{aligned} \nabla \cdot \mathbf{D} &= \rho_f \\ \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \\ \nabla \cdot \mathbf{B} &= 0 \\ \\ \nabla \times \mathbf{H} &= \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t} \end{aligned}\]

Auxillary Fields:

\[\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\] \[\mathbf{H} = \frac{1}{\mu_0} \mathbf{B} - \mathbf{M}\]

In Linear Media:

\[\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}, \quad \mathbf{D} = \epsilon \mathbf{E}\] \[\mathbf{M} = \chi_m \mathbf{H}, \quad \mathbf{H} = \frac{1}{\mu} \mathbf{B}\]

Potentials

\[\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}, \quad \mathbf{B} = \nabla \times \mathbf{A}\]

Lorentz Force Law

\[\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})\]

Energy, Momentum, and Power

Energy: \(\quad U = \frac{1}{2} \int \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) d\tau\)

Momentum: \(\quad \mathbf{P} = \epsilon_0 \int (\mathbf{E} \times \mathbf{B}) \, d\tau\)

Poynting Vector (Power per Area): \(\quad \mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})\)

Larmor: \(\quad P = \frac{\mu_0}{6 \pi c} q^2 a^2\)