Electrodynamics

Thus far we’ve spent a considerable amount of time studying statics, focusing on fields that are time-independent. We will now begin turning our attention to dynamics, where we allow the fields to depend explicitly on time. This leads us into the subject of electrodynamics, which we will study in depth for the rest of the course. To begin our study of this topic we will discuss a few important concepts that will lead us in the next chapter to Maxwell’s equations, the crown jewel of electromagnetism.

• State quasistatic field equations
• Introduce the displacement current to maintain charge conservation

\[ \begin{align*} \nabla \cdot \mathbf{E} &= 4\pi \rho \ , \\ \nabla \times \mathbf{E} &= \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t} \ , \\ \nabla \cdot \mathbf{B} &= 0 \ , \\ \nabla \times \mathbf{B} &= \frac{4\pi}{c} \mathbf{J} - \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} \ . \\ \end{align*} \]

\[ \begin{align*} \mathbf{E} &= -\nabla\phi + \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} \ , \\ \mathbf{B} &= \nabla \cdot \mathbf{A} \ . \end{align*} \]

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \ . \]