I could talk to you about Maxwell’s equations for hours, but I only have 10 minutes to spare before I’m late.
Maxwell’s equations are the fundamental equations for EM. There is not another set of equations more useful in EM than Maxwell’s.
From the equations we’ve come to understand electromagnetism, and they pave the way to the existence of electromagnetic waves.
In Maxwell’s time physicists knew that electricity and magnetism were related somehow, yet no one could find the correlating parts and piece it together. For year, Maxwell studied EM until he finally came to show that all electromagnetic phenomena could be related through a few equations. With the equations he then went on to prove the existence of EM waves and then go on even further and show that light is a form of EM wave.
The four main Maxwell equations are Gauss, Faraday, and Ampere.
Gauss’ law of electricity is primarily used to show that the electric flux of an enclosed surface is equal to the entire amount of electric field in the surface (or the total charge within that volume). The main extension of the equation thus gives us that electric charge creates electric field.
Gauss’ law of magnetism shows that the total magnetic flux of an enclosed surface is equal to zero. This shows us that magnets have paired poles, and that there is no (as of yet) such thing as a magnetic monopole.
Faraday’s law (one of the two neatest equations in the quartet) shows us that a changing magnetic field creates an electric field. So now we have that electric fields can be created by electric charges (as per Gauss’ law) and some sort of change in a magnetic field.
Ampere’s law (which in the “Maxwellian” system is altered from Ampere’s original form) shows the opposite of Faraday’s law in that a changing electric field creates a magnetic field. And where electric fields have a little bit of choice for how it will be created, magnetic fields can only be created by changing electric fields due to the nonexistence of monopoles.
So, I mean, this is it. This is the pinnacle of physics! Okay, maybe not, but Maxwell’s equations are a contender for the most beautiful equations known to man. There is no other equation(s) in physics that excites me as much as Maxwell did when I saw his initial proofs of the equations.
We’ve taken two entities - electricity and magnetism - and connected them! We can create electric fields from magnetic fields, and magnetic fields from electric fields! We’ve so far disproved the existence of monopoles, we’ve proved that light is an EM wave, we’ve proved the existence of EM waves through Maxwell’s equations!!
This is it! THIS IS SCIENCE!! (now i’m late)