Lets try to combine the vector field $\vec E$ and $\vec B$ which have 6 components into a scalar and vector field $\phi$ and $\vec A$ respectively which have four degrees of freedom in total which are the degrees of freedom of electromagnetism.

$\phi$ is the electric potential

$A$ is the magnetic vector potential

🦪 Testing the gauge condition:

🧞 Applying Our new terms into the sourced Maxwell equations ( Gauss’ law and the Ampere-Maxwell law)

  1. Lets start with Gauss’ law

  2. Amperes law

To simplify these expressions we define a new operator

💎 Conclusion: we have shown that by introducing the potential $\phi$ and $\vec A$ Maxwell equations become two sourced wave equations. The scalar potential $\phi$ is sources by the charge density $\rho$ and the vector potential $\vec A$ is sourced by the current density $\vec j$. Notice that the speed of these waves is $c$

🚧 NE Lorentz transformations