We use two experimental laws and charge flow continuity to deduce Maxwell's equations. Electromagnetic waves are solutions to Maxwell's equations subject to proper boundary conditions.
2.1 Maxwell's Equations in Free Space
Maxwell integrated all experimental laws of electricity and magnetism, and through a self-consistency check, discovered a term not found previously. This missing term predicts the existence of electromagnetic waves, which was later verified experimentally by Hertz; thus, the integrated, corrected equations are called Maxwell's equations. Now, we all know that light is an electromagnetic wave. To investigate the behavior of light for its applications, e.g., optical lithography, we must first study Maxwell's equations.