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The speed of light from Maxwell

PHYS 301 · Maxwell's Equations

Maxwell's equations predict electromagnetic waves traveling at a speed determined by electric and magnetic constants. This lesson derives and interprets $c=1/\sqrt{\mu_0\epsilon_0}$.

Key equations

\nabla\cdot\vec{E}=0\nabla\cdot\vec{B}=0\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}\nabla\times\vec{B}=\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}\nabla\times(\nabla\times\vec{E})=-\frac{\partial}{\partial t}(\nabla\times\vec{B})\nabla\times(\nabla\times\vec{E})=\nabla(\nabla\cdot\vec{E})-\nabla^2\vec{E}\nabla^2\vec{E}=\mu_0\epsilon_0\frac{\partial^2\vec{E}}{\partial t^2}\nabla^2\psi=\frac{1}{v^2}\frac{\partial^2\psi}{\partial t^2}v=\frac{1}{\sqrt{\mu_0\epsilon_0}}c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\nabla^2\vec{B}=\mu_0\epsilon_0\frac{\partial^2\vec{B}}{\partial t^2}v=\frac{c}{n}

Learning objectives

  • Write Maxwell's equations in vacuum.
  • Derive the electromagnetic wave equation for $\vec{E}$.
  • Identify wave speed from the wave equation.
  • Explain why Maxwell identified light as an electromagnetic wave.
  • Connect the fixed speed of light to later developments in relativity.

A surprising prediction

One of Maxwell's greatest achievements was showing that electric and magnetic laws imply waves. Even more remarkably, the predicted wave speed matched the measured speed of light. This revealed that light is an electromagnetic wave.

The speed emerges from two constants: the permittivity of free space epsilon0epsilon_0 and the permeability of free space mu0mu_0.

Maxwell's equations in empty space

In vacuum with no charges and no currents, Maxwell's equations become

ablacdot ec{E}=0$$

ablacdot ec{B}=0$$

abla imes ec{E}=- rac{partial ec{B}}{partial t}$$

abla imes ec{B}=mu_0epsilon_0 rac{partial ec{E}}{partial t}$$

The last two equations show the coupling: changing ec{B} creates curl in ec{E}, and changing ec{E} creates curl in ec{B}.

Deriving the wave equation

Take the curl of Faraday's law:

abla imes( abla imes ec{E})=- rac{partial}{partial t}( abla imes ec{B})$$ Use the vector identity

abla imes( abla imes ec{E})= abla( ablacdot ec{E})- abla^2 ec{E}$$

In vacuum, ablacdot ec{E}=0, so the left side becomes

abla^2 ec{E}$$ Using the Maxwell-Ampère law,

abla imes ec{B}=mu_0epsilon_0 rac{partial ec{E}}{partial t}$$

we obtain

abla^2 ec{E}=mu_0epsilon_0 rac{partial^2 ec{E}}{partial t^2}$$ This is a wave equation. ## Identifying the speed The standard wave equation has form

abla^2psi= rac{1}{v^2} rac{partial^2psi}{partial t^2}$$

Comparing with the electromagnetic wave equation gives

rac{1}{v^2}=mu_0epsilon_0

so

v= rac{1}{sqrt{mu_0epsilon_0}}

This speed is the speed of light in vacuum:

c= rac{1}{sqrt{mu_0epsilon_0}}

Magnetic field wave equation

A similar derivation gives

abla^2 ec{B}=mu_0epsilon_0 rac{partial^2 ec{B}}{partial t^2}$$ Thus both electric and magnetic fields propagate together at speed $c$. ## Physical meaning The constants $epsilon_0$ and $mu_0$ originally appeared in electrostatics and magnetostatics. Maxwell showed that they also determine the speed of electromagnetic waves. The measured value of $$ rac{1}{sqrt{mu_0epsilon_0}}$$ agreed with the known speed of light, unifying optics with electromagnetism. ## Light in materials In a material, electromagnetic waves often travel more slowly than in vacuum. The speed is $$v= rac{c}{n}$$ where $n$ is index of refraction. Material polarization and magnetization affect wave propagation. This leads to refraction, dispersion, and optical phenomena. ## Connection to relativity Maxwell's equations predict a fixed speed $c$ in vacuum. This created tension with Galilean relativity and eventually led Einstein to special relativity, where $c$ is the same for all inertial observers. ## The big idea Maxwell's equations imply wave equations for electric and magnetic fields. The wave speed is determined by $mu_0$ and $epsilon_0$, giving $c=1/sqrt{mu_0epsilon_0}$. This result identifies light as an electromagnetic wave and connects electromagnetism to modern relativity.

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