Electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light. The oscillations of the two fields are perpendicular to each other and perpendicular to the direction of energy and wave propagation, forming a transverse wave.Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can subsequently interact with other charged particles

Maxwell describe this phenomenon by the following 4 Vector differential equations

${\displaystyle \nabla \cdot E=0}$

${\displaystyle \nabla \times E=-{\frac {1}{T}}E}$

${\displaystyle \nabla \cdot B=0}$

${\displaystyle \nabla \times B=-{\frac {1}{T}}B}$

${\displaystyle T=\mu \epsilon }$

Using partial differential math

${\displaystyle \nabla (\nabla \times E)=\nabla (-{\frac {1}{T}}E)}$

${\displaystyle \nabla ^{2}E=-\omega E}$

${\displaystyle \nabla (\nabla \times B)=\nabla (-{\frac {1}{T}}B)}$

${\displaystyle \nabla ^{2}B=-\omega B}$

We obtain Electromagnetic wave equations below

${\displaystyle \nabla ^{2}E=-\omega E}$

${\displaystyle \nabla ^{2}B=-\omega B}$

Solving the Electromagnetic wave equations above we have the following Electromagnetic wave functions below

${\displaystyle E=ASin\omega t}$

${\displaystyle B=ASin\omega t}$

${\displaystyle \omega =\lambda f={\sqrt {\frac {1}{T}}}=C}$

${\displaystyle T=\mu \epsilon }$