Proving the Velocity of Electromagnetic Waves is Equal to the Speed of Light: An Explanation Using Maxwell's Equations

To demonstrate that electromagnetic waves travel at the speed of light, we can derive this relationship using Maxwell's equations. These fundamental laws describe the interactions between electric and magnetic fields, providing a clear pathway to understanding the mechanics of electromagnetic wave propagation.

Maxwell's Equations in Free Space

Maxwell's equations in a vacuum (free space) can be succinctly written as:

Gauss's Law for Electricity

$ abla cdot mathbf{E} frac{rho}{varepsilon_0}$

In free space, $rho 0$, thus:

$ abla cdot mathbf{E} 0$

Gauss's Law for Magnetism

$ abla cdot mathbf{B} 0$

Faraday's Law of Induction

$ abla times mathbf{E} -frac{partial mathbf{B}}{partial t}$

Ampère-Maxwell Law

$ abla times mathbf{B} mu_0 varepsilon_0 frac{partial mathbf{E}}{partial t}$

Deriving the Wave Equation for Electromagnetic Fields

To prove that electromagnetic waves propagate at the speed of light, we derive the wave equation for the electric field $mathbf{E}$ and the magnetic field $mathbf{B}$.

Step 1: Taking the Curl of Faraday's Law

$ abla times abla times mathbf{E} - abla times frac{partial mathbf{B}}{partial t}$

Using the vector identity $ abla times abla times mathbf{A} abla( abla cdot mathbf{A}) - abla^2 mathbf{A}$ and since $ abla cdot mathbf{E} 0$, we have:

$ abla( abla cdot mathbf{E}) 0$ implies $ abla times abla times mathbf{E} - abla^2 mathbf{E}$

Thus, we can rewrite the equation as:

$- abla^2 mathbf{E} - abla times frac{partial mathbf{B}}{partial t}$

Step 2: Substituting Ampère-Maxwell Law

From Ampère's Law, we have:

$ abla times mathbf{B} mu_0 varepsilon_0 frac{partial mathbf{E}}{partial t}$

Substituting this into the previous equation, we get:

$- abla^2 mathbf{E} -mu_0 varepsilon_0 frac{partial^2 mathbf{E}}{partial t^2}$

Rearranging gives us the wave equation:

$ abla^2 mathbf{E} mu_0 varepsilon_0 frac{partial^2 mathbf{E}}{partial t^2}$

Identifying the Wave Speed

The standard form of a wave equation is:

$ abla^2 psi v^2 frac{partial^2 psi}{partial t^2}$

where $v$ is the wave speed. Comparing this with our equation, we see that:

$v^2 frac{1}{mu_0 varepsilon_0}$

Therefore, the speed of electromagnetic waves is:

$v frac{1}{sqrt{mu_0 varepsilon_0}}$

The Speed of Light in Vacuum

The constants $mu_0$ (permeability of free space) and $varepsilon_0$ (permittivity of free space) are related to the speed of light $c$ in vacuum:

$c frac{1}{sqrt{mu_0 varepsilon_0}} approx 3 times 10^8 , text{m/s}$

Conclusion

Thus, we conclude that electromagnetic waves propagate through free space at the speed of light $c$, confirming that the velocity of electromagnetic waves is indeed equal to the velocity of light.