The fact that the speed of light is constant regardless of your reference frame is a direct consequence of Maxwell's Equations. Einstein used this peculiar feature of light as the basis for special relativity. So how does the speed of light appear from Maxwell's Equations?
Starting with Maxwell's equation for the curl of an electric field:
Take the curl of both sides,
using the vector identity:
and Maxwell's equation for an Electric Field in a Vacuum (no charge):
We see that the Electric Field Wave equation is:
where,
This differential equation is in the form of the wave equation:
This differential equation describes a sinusoidal wave propagating in the xdirection with velocity V.
With solutions:
So the solution to the Electric Field Wave Equation is:
E=E_{0}Sin(kx±ωt)
Where ω (the angular frequency) is V/K and K (the wavenumber) is 2π/λ, here V is the velocity of the wave and λ is it's wavelength.
So taking another look at the Electric Field wave equation:
We see that the Electric Field is a sinusodial wave propagating at speed c. With a similiar method we can derive a Magnetic Field wave equation which describes the magnetic field as a sinusoidal wave propagating with speed c. So light, which is an Electromagnetic Wave, propagates with speed c.
Special Thanks to Wikipedia for the Equations.

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