Maxwell's Equations
A personal note to myself, about how I understand them.
The other day, I was thinking about how speakers work. Like, how exactly do they convert electric signals into physical vibrations that propagate through air and reach our ears? How does one convert electrical energy into mechanical energy? As I went down the rabbit hole, I realized I completely lost my bearing of electrodynamics, and Maxwell’s laws. So I took a few hours to reacquaint myself and learn them again. I did a better job this time, I think.
Gauss’ Law for electric fields / electrostatics
\[\nabla\cdot\overrightarrow{E}=\frac{\rho}{\epsilon_0}\simeq Q\]The divergence of an electric field is equal to charge density per unit permitivity of the medium (charge creates an electric field).
Gauss’ Law for magnetic fields
\[\nabla\cdot\overrightarrow{B}=0\]There is no magnetic equivalent of an electric charge, nothing can cause a magnetic divergent field.
Faraday’s Law
\[\nabla\times\overrightarrow{E}=\frac{\partial\overrightarrow{B}}{\partial t}\]A changing magnetic field induces an electric field.
Ampere-Maxwell Law
\[\nabla\times\overrightarrow{B}=\mu_0\overrightarrow{J} + \mu_0\epsilon_0\frac{\partial\overrightarrow{E}}{\partial t}\]Magnetic fields can be induced by changing the electric field or by passing a current through a wire (electromagnetic effect).
Laws 3 and 4 is how light waves propagate. The electric field inducing a magnetic field which in turn induces an electric field.
Lorentz Force
\[\overrightarrow{F}=q\left(\overrightarrow{E}+\overrightarrow{v}\times\overrightarrow{B}\right)\]While this is not part of the four Maxwell’s equations, it is very important. This along with the above Maxwell’s equations, completely describe how electromagnetic fields are generated by moving charges, and currents. This stuff, in essence, is the basis of classical electrodynamics.