1.1 Electromagnetics & Maxwell's Equations
Maxwell's equations describe how electric and magnetic fields propagate and interact with matter. In differential form:
- Faraday's Law of Induction:
$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$
- Ampere's Circuital Law:
$$\nabla \times \vec{H} = \vec{J}_f + \frac{\partial \vec{D}}{\partial t}$$
- Gauss's Law for Electricity:
$$\nabla \cdot \vec{D} = \rho_f$$
- Gauss's Law for Magnetism:
$$\nabla \cdot \vec{B} = 0$$
1.2 Electromagnetic Potentials
To solve Maxwell's equations, we define vector potential $\vec{A}$ and scalar potential $\phi$:
- Magnetic Vector Potential:
$$\vec{B} = \nabla \times \vec{A}$$
- Electric Field relation:
$$\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}$$
- Constitutive Relations:
$$\vec{B} = \mu \vec{H}, \quad \vec{D} = \epsilon \vec{E}, \quad \vec{J}_c = \sigma \vec{E}$$
### 1.3 Poynting's Theorem & Wave Equations
- Poynting's Theorem (Power Flow):
$$-\nabla \cdot \vec{S} = \vec{J}_c \cdot \vec{E} + \vec{E} \cdot \frac{\partial \vec{D}}{\partial t} + \vec{H} \cdot \frac{\partial \vec{B}}{\partial t}$$
where $\vec{S} = \vec{E} \times \vec{H}$ is the Poynting vector representing power density.
- Wave Equations under Lorentz Gauge ($\nabla \cdot \vec{A} = -\mu\epsilon \frac{\partial \phi}{\partial t}$):
- For Vector Potential:
$$\nabla^2 \vec{A} - \mu\epsilon \frac{\partial^2 \vec{A}}{\partial t^2} = -\mu \vec{J}_f$$
- For Scalar Potential:
$$\nabla^2 \phi - \mu\epsilon \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho_f}{\epsilon}$$