Recall that \((\nabla\cdot) = \operatorname{div}\) and that \((\nabla\times) = \operatorname{curl}.\) These equations are foundational to the classical theory of electromagnetism. \[ \nabla \cdot \bm{E} = \frac{\rho}{\varepsilon_0} \qquad \qquad \nabla \cdot \bm{B} = \bm{0} \qquad \qquad \nabla \times \bm{E} = -\frac{\partial \bm{B}}{\partial t} \qquad \qquad \nabla \times \bm{B} = \mu_0\biggl(\bm{J} + \varepsilon_0\frac{\partial \bm{E}}{\partial t}\biggr) \] where \(\bm{E}\) is the electric field, \(\bm{B}\) is the magnetic field, \(\rho\) is the electric charge density, \(\bm{J}\) is the current density, \(\varepsilon_0\) is the vacuum permittivity, and \(\mu_0\) is the vacuum permeability. Each of these equations has its own name. In order,
- Gauss’s Law — Electric charge creates an electric field. (Electrons create electric fields)
- Gauss’s Law for Magnetism — Total magnetic field through a volume needs to add up to 0. Ie, if you cut a bar magnet in half, you now have two bar magnets with a North and South pole each. (Not one north pole magnet and one south pole magnet)
- Faraday’s Law — A changing magnetic field creates voltage. Ie, like those shake flashlights where they have a magnet inside and then the flash light turns on.
- Ampère-Maxwell Law — Current creates magnetic field. Ie think of electromagnets like those things that stick to metal in a dump or an MRI machine.