Deriving the speed of light from Maxwell’s equations

We are all familiar with the concept of the speed of light. It is the speed beyond which no object may travel. Many seem to associate Einstein for the necessity of this universal constant, and while it is inherent to his theory of special and general theories of relativity, it was not necessarily something he discovered. It is, in fact, a consequence of the Maxwell equations from my first post. I will be deriving the speed of light quantity using the four field equations of electrodynamics, and I will explain how Einstein used this fact to challenge Newtonian relativity in his theory of special relativity (I am not as familiar with general relativity).  The reason for this post is just to demonstrate the origin of a well-known concept; the speed of light.

We start with Maxwell’s equations of electrodynamics

\nabla \cdot \textbf{E}=\frac{\rho}{\epsilon_{0}}, (1)

\nabla \cdot \textbf{B}=0, (2)

\nabla \times \textbf{E}=-\frac{\partial \textbf{B}}{\partial t}, (3)

\nabla \times \textbf{B}=\mu_{0}\textbf{j}+\mu_{0}\epsilon_{0}\frac{\partial \textbf{E}}{\partial t}. (4)

Now, we let \rho =0, which means that the charge density must be zero, and we also let the current density \textbf{j}=0. Moreover, note that the form of the wave equation as

\frac{\partial^{2} u}{\partial t^{2}}=\frac{1}{v^{2}}\nabla^{2}u. (5)

This equation describes the change in position of material in three dimensions (choose whichever coordinate system you like) propagating through some amount of time, with some velocity v.

After making these assumptions, we arrive at

\nabla \cdot \textbf{E}=0, (6)

\nabla \cdot \textbf{B}=0, (7)

\nabla \times \textbf{E}=-\frac{\partial \textbf{B}}{\partial t}, (8)

\nabla \times \textbf{B}=\mu_{0}\epsilon_{0}\frac{\partial \textbf{E}}{\partial t}. (9)

Also note the vector identity\nabla \times (\nabla \times \textbf{A})=\nabla(\nabla\cdot\textbf{A})-\nabla^{2}\textbf{A}. Now, take the curl of Eqs.(8) and (9), and we get

\frac{1}{\mu_{0}\epsilon_{0}}\nabla^{2}\textbf{E}=\frac{\partial^{2}\textbf{E}}{\partial t^{2}}, (10)

and

\frac{1}{\mu_{0}\epsilon_{0}}\nabla^{2}\textbf{B}=\frac{\partial^{2}\textbf{B}}{\partial t^{2}}, (11)

where we have used Eqs. (6), (7), (8), and (9) to simplify the expressions. Eqs. (10) and (11) are the electromagnetic wave equations. Note the form of these equations and how they compare to Eq. (5). They are identical, and upon inspection one can see that the velocity with which light travels is

\frac{1}{c^{2}}=\frac{1}{\mu_{0}\epsilon_{0}} \implies c=\sqrt[]{\mu_{0}\epsilon_{0}}, (12)

where \mu_{0} is the permeability of free space and \epsilon_{0} is the permittivity of free space.

Most waves on Earth require a medium to travel. Sound waves, for example are actually pressure waves that move by collisions of the individual molecules in the air.  For some time, light was thought to require a medium to travel. So it was proposed that since light can travel through the vacuum of space, there must exist a universal medium dubbed “the ether”. This “ether” was sought after most notably in the famous Michelson-Morley experiment, in which an interferometer was constructed to measure the Earth’s velocity through this medium. However, when they failed to find any evidence that the “ether” existed, the new way of thinking was that it didn’t exist. It turned out that light doesn’t need a medium to travel through space. Technically-speaking, space itself acts as the medium through which light travels.

 In Newtonian relativity, it was assumed that time and space were separate constructs and were regarded as absolute. In other words, it was the speed that changed. What this meant is that even as speeds became very large, space and time remained the same. What Einstein did was that he saw the consequence of Maxwell’s equations and regarded this speed as absolute, and allowed space and time (really spacetime) to vary. In Einstein’s theory of special relativity, as one approaches the speed of light, time slows down, and objects become contracted. These phenomena are known as time dilation and length contraction:

\delta t = \frac{\delta t_{0}}{\sqrt[]{1-v^{2}/c^{2}}}, (13)

\delta l = l_{0}\sqrt[]{1-v^{2}/c^{2}}. (14)

These phenomena will be discussed in more detail in a future post. Thus, Maxwell’s formulation of the electrodynamic field equations led Einstein to change the way we perceive the fundamental concepts of space and time.

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