The Wave Equation
Maxwell's Equations & the Electromagnetic Wave Equation
Maxwell's equations are among the most important equations in all of physics. They unify electricity, magnetism, and light into a single mathematical framework, describing how electric and magnetic fields are generated and how they evolve in space and time.
Historically, electricity and magnetism were originally studied as separate phenomena. Experiments by scientists such as Coulomb, Ampère, and Faraday gradually revealed deep connections between them: electric currents produce magnetic fields, and changing magnetic fields induce electric currents. In the 1860s, James Clerk Maxwell synthesized these experimental laws into a single set of equations and introduced a crucial additional term, the displacement current, which made the theory mathematically self-consistent and ultimately led to the prediction of electromagnetic waves.
Maxwell's equations are
$$ \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0} $$ $$ \vec{\nabla} \cdot \vec{B} = 0 $$ $$ \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} $$ $$ \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} $$Here:
- \( \vec{E} \) is the electric field
- \( \vec{B} \) is the magnetic field
- \( \rho \) is the charge density
- \( \vec{J} \) is the current density
- \( \epsilon_0 \) is the vacuum permittivity
- \( \mu_0 \) is the vacuum permeability
Maxwell's equations describe how electric and magnetic fields evolve through space and time. The divergence equations describe sources of the fields, while the curl equations describe how changing electric and magnetic fields generate one another. As we will now show, these equations naturally lead to a wave equation.
In a vacuum, Maxwell's equations are
$$ \vec{\nabla} \cdot \vec{E} = 0 $$ $$ \vec{\nabla} \cdot \vec{B} = 0 $$ $$ \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$ $$ \vec{\nabla} \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} $$These equations describe electric and magnetic fields in a region with no charges and no currents. A key consequence is that changing electric and magnetic fields can sustain each other and propagate as waves.
We want to show that both \( \vec{E} \) and \( \vec{B} \) satisfy the wave equation:
$$ \frac{\partial^2}{\partial t^2} \vec{E} = c^2 \nabla^2 \vec{E} $$ $$ \frac{\partial^2}{\partial t^2} \vec{B} = c^2 \nabla^2 \vec{B} $$This will also allow us to determine the wave speed \( c \).
Wave Equation for the Electric Field
We use the vector identity
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E}) - \nabla^2 \vec{E} $$Since \( \vec{\nabla} \cdot \vec{E} = 0 \) in vacuum, this becomes
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) = -\nabla^2 \vec{E} $$From Faraday's law,
$$ \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$Taking the curl of both sides gives
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) = \vec{\nabla} \times \left(-\frac{\partial \vec{B}}{\partial t}\right) $$ $$ = -\frac{\partial}{\partial t} \left( \vec{\nabla} \times \vec{B} \right) $$Using Ampère-Maxwell's law,
$$ \vec{\nabla} \times \vec{B} = \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} $$we obtain
$$ -\frac{\partial}{\partial t} \left( \vec{\nabla} \times \vec{B} \right) = -\frac{\partial}{\partial t} \left( \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) $$ $$ = -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} $$Therefore,
$$ -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = -\nabla^2 \vec{E} $$Hence,
$$ \mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = \nabla^2 \vec{E} $$or equivalently,
$$ \frac{\partial^2}{\partial t^2} \vec{E} = \frac{1}{\mu_0\epsilon_0} \nabla^2 \vec{E} $$This has the same structure as the wave equation. Comparing with
$$ \frac{\partial^2}{\partial t^2} \vec{E} = c^2 \nabla^2 \vec{E} $$gives
$$ c^2 = \frac{1}{\mu_0\epsilon_0} $$Therefore,
$$ c = \frac{1}{\sqrt{\mu_0\epsilon_0}} $$Wave Equation for the Magnetic Field
The same idea works for the magnetic field. We use the identity
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{B}) = \vec{\nabla}(\vec{\nabla} \cdot \vec{B}) - \nabla^2 \vec{B} $$Since \( \vec{\nabla} \cdot \vec{B} = 0 \), this becomes
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{B}) = -\nabla^2 \vec{B} $$From Ampère-Maxwell's law,
$$ \vec{\nabla} \times \vec{B} = \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} $$Taking the curl of both sides gives
$$ \vec{\nabla} \times (\vec{\nabla} \times \vec{B}) = \vec{\nabla} \times \left( \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) $$ $$ = \mu_0\epsilon_0 \frac{\partial}{\partial t} \left( \vec{\nabla} \times \vec{E} \right) $$Using Faraday's law,
$$ \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$we get
$$ \mu_0\epsilon_0 \frac{\partial}{\partial t} \left( \vec{\nabla} \times \vec{E} \right) = \mu_0\epsilon_0 \frac{\partial}{\partial t} \left( -\frac{\partial \vec{B}}{\partial t} \right) $$ $$ = -\mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} $$Therefore,
$$ -\mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} = -\nabla^2 \vec{B} $$Hence,
$$ \mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} = \nabla^2 \vec{B} $$or equivalently,
$$ \frac{\partial^2}{\partial t^2} \vec{B} = \frac{1}{\mu_0\epsilon_0} \nabla^2 \vec{B} $$So both electric and magnetic fields satisfy wave equations with the same wave speed
$$ c = \frac{1}{\sqrt{\mu_0\epsilon_0}} $$Using experimentally measured values for the vacuum permittivity and permeability,
$$ \epsilon_0 \approx 8.85\times10^{-12} \;\text{F/m} $$ $$ \mu_0 \approx 4\pi\times10^{-7} \;\text{N/A}^2 $$we obtain
$$ c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3.00\times10^8 \;\text{m/s}, $$which is precisely the measured speed of light. This result is one of the most remarkable in the history of physics, Maxwell realized that this could not be a coincidence: the equations predicted that light itself is an electromagnetic wave, consisting of self-propagating oscillating electric and magnetic fields.
This was one of the great unifications in physics. Phenomena that had previously been studied separately, such as electricity, magnetism, radio waves, visible light, and X-rays, were revealed to be manifestations of the same underlying electromagnetic field. Maxwell's prediction was later confirmed experimentally by Heinrich Hertz and became one of the foundations of modern physics.