# Maxwell's Equation¶

$$\large \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \quad \text{: Gauss's Law}$$$$\large \nabla \cdot \vec{B} = 0 \quad \text{: no magnetic monopole exists}$$$$\large \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \quad \text{: Faraday's Law}$$$$\large \nabla \times \vec{B} = \mu_0 \left( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) \quad \text{: Ampare's Law }$$

## In a Static Electromagnetic Field¶

$$\large \frac{\partial \vec{B}}{\partial t} = 0 \quad \Rightarrow \nabla \times \vec{E} = 0$$

## What if a Magnetic Monopole Exists?¶

$$\large \nabla \cdot \vec{B} = 4 \pi \rho_m$$$$\large \nabla \times \vec{E} = - \frac{1}{c} \frac{\partial \vec{B}}{\partial t} - \frac{4\pi}{c} \vec{J_m}$$