Reference for the multiple forms of Maxwell’s equations for electromagnetism.
Differential form
\[\begin{align*} \vec{\nabla} \cdot \vec{E} & = \rho \\ \vec{\nabla} \cdot \vec{B} & = 0 \\ \vec{\nabla} \times \vec{E} & = - \frac{\partial \vec{B}}{\partial t} \\ \vec{\nabla} \times \vec{B} & = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} + \mu_0 \vec{j} \\ \end{align*}\]
Integral form
\[\begin{align*} \oint \vec{E} \cdot d\vec{A} & = q \\ \oint \vec{B} \cdot d\vec{A} & = 0 \\ \oint \vec{E} \cdot d\vec{s} & = - \frac{d \Phi_B}{dt} \\ \oint \vec{B} \cdot d\vec{s} & = \mu_0 \epsilon_0 \frac{d \Phi_E}{dt} + \mu_0 j \end{align*}\]
Glossary
| symbol | description |
|---|---|
| \(\vec{E}\) | electric field |
| \(\vec{B}\) | magnetic field |
| \(\Phi_E\) | electric field flux |
| \(\Phi_B\) | magnetic field flux |
| \(q\) | charge |
| \(\rho\) | charge density |
| \(\vec{j}\) | displacement current |
References
[1]
David Halliday, Robert Resnick, and Jearl Walker. 2021. Fundamental of physics (extended 12th ed.). Wiley.
[2]
Leonard Susskind and Art Friedman. 2017. Special relativity and classical field theory: The theoretical minimum. Basic Books.