1. Gauss’ Law:

$$ \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = 4 \pi \iiint_{\Omega} \rho dV $$

2. Gauss’ Law for Magnetism: $$ \iint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$

3. Maxwell-Faraday Equation:

$$ \oint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$

4. Ampère’s circuital law:

$$ \oint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left(\iint_{\Sigma} \mathbf{J} \cdot d\mathbf{S} + \epsilon_0 \frac{d}{dt} \iint_{\Sigma} \mathbf{E} \cdot d\mathbf{S}\right) $$

1. Gauss’ Law:

$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} $$

2. Gauss’ Law for Magnetism:

$$ \nabla \cdot \mathbf{B} = 0 $$

3. Maxwell-Faraday Equation:

$$ \nabla \times \mathbf{E} = \frac{\partial \mathbf{B}}{\partial t} $$

4. Ampère’s circuital law:

$$ \nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right) $$