Deriving c

Screen Shot 2017-06-14 at 13.50.39Physics has its occasional eureka moment. James Clerk Maxwell had one such moment in 1862 when he discovered that the equations he had found describing electric and magnetic waves combined to describe light. The speed of light c, this article will demonstrate, can easily be found from Maxwell’s equations. As this derivation is an undergraduate degree exam question each step will be described qualitatively, but a brief glance at the vector operators guide in Resources is recommended.

One of the most profound 19th century discoveries was Faraday’s finding that electric currents passing through wires induce magnetic field loops, as shown below.Screen Shot 2017-06-14 at 15.16.15By 1862 Maxwell had shown that the gradient of the magnetic field H was directly proportional to the curl (or rotationality) of the electric field E. This essentially means that electricity and magnetism are one and the same, and a magnetic field flowing around a wire will equally induce a current through it, as described by:
Screen Shot 2017-06-14 at 15.20.08Where μ0 is the magnetic permeability constant, defined later. Taking the curl of both sides gives:Screen Shot 2017-06-14 at 15.20.14The curl of H (∇ x H) is defined by Ampére’s law as:Screen Shot 2017-06-14 at 15.29.30Where σ is the conductivity of the medium and ε₀ is the electric permittivity constant. Inserting this expression above gives:
Screen Shot 2017-06-14 at 15.29.36Using the following identity the Laplacian can be extracted. However, imposing the vacuum condition means that σ = 0 and the divergence of E (∇ . E) = 0.Screen Shot 2017-06-14 at 15.31.03Thus:Screen Shot 2017-06-14 at 15.31.11What has been derived is a wave equation, describing an electromagnetic wave, where μ0 = 1.26 x 10-6 N A⁻² and ε0 = 8.85 x10-12 F m⁻¹. Notice the similarity to the general equation for a wave:Screen Shot 2017-06-14 at 15.33.02In this equation v is the speed of the wave. The speed of an electromagnetic wave can therefore be found:Screen Shot 2017-06-14 at 15.33.41Screen Shot 2017-06-14 at 15.33.52This gives v = ~ 3 x10⁸ m s⁻¹, the speed of light. Thus, light can be treated as an electromagnetic wave. The century and a half since Maxwell first worked out this derivation has corroborated and built upon it.

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