Maxwell Equations (ME) essentially describe in a tremendous simple way how globally the electromagnetic field behaves in a general medium. As we'll show, the electric and the magnetic field are not independent and that’s the unforgetable discovery of Maxwell itself.
Even if they were not stated all at once (one is due to Gauss, one to Faraday, one to Maxwell itself…) they represent how the electromagnetic field behaves. The huge discovery of Maxwell was that Electric field and Magnetic field are the two faces of the same coin (actually was later proved that in the relativity framework one is specular to the other).
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Essentially it’s stating that the net quantity of the electric flux leaving a sample volume is proportional to the charge inside the volume.
Gauss Law for magentism
Essentially it is stating the impossibility of creating a magnetic monopole; the total magnetic flux through a closed surface is zero.
The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses, i.e., every time the magnetic field change there is the creation of a electric field.
Essentially this law is a consequence of the total energy conservation law! Every time there is a variation of the magnetic field (aka, a variation of energy in the medium), immediately a current is generated in order to keep the magnetic field constant!(following the rule of the right-hand notation). That’s why this law is also known as the Faraday’s law of induction.
Ampère-Maxwell circuital law
The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) that the loop encloses.
The expression for ME can be condensated introducing the 4-vector A in the following way:
See Calculation in picture>
This last formulation is especially suitable for taking into consideration Lorentz transformations and gauge invariance properties.
In particular, combining the four equations we have the analytic expression of the electromagnetic waves expanding in vacuum or in a particular medium.
From the Maxwell equations we can also figure out what the propagation speed of an EM wave is, i.e., v=1μ0ϵ0−−−√=c
> speed of light.