> [!Sources]- > >- https://maxwells-equations.com > - [Intuitive Guide to Maxwell's Equations](https://github.com/photonlines/Intuitive-Guide-to-Maxwells-Equations) > - [[Divergence]] and [[Curl]] - 3blue1brown video](https://www.youtube.com/watch?v=rB83DpBJQsE) > - https://www.fiberoptics4sale.com/blogs/electromagnetic-optics/a-plain-explanation-of-maxwells-equations ## 1. Gauss's Law for Electric Fields $\vec ∇ •\vec E =\frac{\rho}{\epsilon_0} $ - $\rho$ = charge density - $\epsilon_0$ = electrical permittivity of free space - $\vec ∇ •\vec E $ = [[divergence]] of the electric field ### [[Integral]] form ![[Maxwell 1.png]] ## 2. Gauss's Law for [[Magnetic Fields]] $\vec ∇• \vec B = 0 $ - $\vec ∇ •\vec B $ = [[divergence]] of the magnetic field The [[divergence]] of a magnetic field is always zero - there are no magnetic unipoles - Magnetic field lines always form continuous loops ### [[Integral]] form ![[Maxwell 2.png]] ## 3. Faraday's Law **A changing magnetic field induces a circulating magnetic field in the direction which opposes the magnetic flux** $\vec ∇ \times \vec E = - \frac {\partial\vec B}{\partial t} $ - $\vec ∇ \times \vec E $ = [[curl]] of the electric field - $ \frac {\partial\vec B}{\partial t}$ = The rate of change of the magnetic field with respect to time ![[Maxwell 3 - Faraday .png|150]] ![[Maxwell 3 - Faraday 2.png|400]] ### Integral form ![[Maxwell 3.png|400]] ![[Maxwell 3 - 2.png|300]] = ![[Maxwell 3-3.png|400]] ## 4. Ampere-Maxwell Law A changing electric field, and/or an electric current, induce a magnetic field $∇ \times \vec B = \mu_0 (J + \epsilon_0 \frac {\partial \vec E}{\partial t}) $ - $∇ \times \vec B $ = [[curl]] of the magnetic field - $\mu_0$ = magnetic permeability of free space - $J$ = electric current density - The amount of charge moving through a surface over time - ![[Maxwell 4 - charge carrying wire.png|350]] - $\epsilon_0$ = electric permittivity of free space - $\frac{\partial \vec E}{\partial t}$ = rate of change of the electric field with respect to time ![[Maxwell 4.png|grid]] ![[Maxwell 4 - right hand rule.png|grid]] ![[Maxwell 4 - wire.png|400]] ### [[Integral]] Form ![[Maxwell 4 - integral.png|500]] ![[Maxwell 4 - path integral.png|200]] = ![[Maxwell 4 - current.png|200]] + ![[Maxwell 4 - surface integral.png|300]]