Maxwell's Equation
| Equation | Integral Form | Differential Form | Physical Meaning |
|---|---|---|---|
| Gauss’s Law (Electric) | \(\displaystyle \oint_{\text{surface}} \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}\) | \(\displaystyle \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}\) | Electric field lines begin/end on charges |
| Gauss’s Law (Magnetism) | \(\displaystyle \oint_{\text{surface}} \vec{B} \cdot d\vec{A} = 0\) | \(\displaystyle \nabla \cdot \vec{B} = 0\) | No magnetic monopoles |
| Faraday’s Law | \(\displaystyle \oint_{\text{loop}} \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int_{\text{surface}} \vec{B} \cdot d\vec{A}\) | \(\displaystyle \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\) | Changing \(\vec{B}\) induces \(\vec{E}\) |
| Ampère’s Law (Maxwell) | \(\displaystyle \oint_{\text{loop}} \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d}{dt} \int_{\text{surface}} \vec{E} \cdot d\vec{A}\) | \(\displaystyle \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}\) | Changing \(\vec{E}\)/currents induce \(\vec{B}\) |
_Last updated: June 06, 2025