Maxwell’s Equations

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?

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

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