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How to derive electromagnetic wave equations

Written by

Bernardo Palos

in

Electromagnetic wave equations describe how electric and magnetic fields propagate through space as waves. The derivation begins from Maxwell’s equations in free space (vacuum), where there are no charges or currents. Here’s a step-by-step process to derive the wave equations for the electric field E and magnetic field B.

Maxwell’s Equations in Free Space

  1. Gauss’s Law for Electricity

∇⋅E=0nabla cdot mathbf{E} = 0

  1. Gauss’s Law for Magnetism

∇⋅B=0nabla cdot mathbf{B} = 0

  1. Faraday’s Law of Induction

∇×E=−∂B∂tnabla times mathbf{E} = – frac{partial mathbf{B}}{partial t}

  1. Ampère-Maxwell Law

∇×B=μ0ϵ0∂E∂tnabla times mathbf{B} = mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}

Here, μ0mu_0 is the permeability of free space, and ϵ0epsilon_0 is the permittivity of free space.

Step 1: Take the Curl of Faraday’s Law

Start from Faraday’s law:

∇×E=−∂B∂tnabla times mathbf{E} = -frac{partial mathbf{B}}{partial t}

Apply the curl operator ∇×nabla times to both sides:

∇×(∇×E)=−∂∂t(∇×B)nabla times (nabla times mathbf{E}) = – frac{partial}{partial t} (nabla times mathbf{B})

Step 2: Use the Vector Identity for Curl of Curl

The vector identity for any vector field Amathbf{A} is:

∇×(∇×A)=∇(∇⋅A)−∇2Anabla times (nabla times mathbf{A}) = nabla (nabla cdot mathbf{A}) – nabla^2 mathbf{A}

For Emathbf{E}, since ∇⋅E=0nabla cdot mathbf{E} = 0 (from Gauss’s law), this reduces to:

∇×(∇×E)=−∇2Enabla times (nabla times mathbf{E}) = – nabla^2 mathbf{E}

So the left-hand side becomes:

−∇2E=−∂∂t(∇×B)-nabla^2 mathbf{E} = – frac{partial}{partial t} (nabla times mathbf{B})

Step 3: Substitute ∇×Bnabla times mathbf{B} from Ampère-Maxwell Law

From Ampère-Maxwell law:

∇×B=μ0ϵ0∂E∂tnabla times mathbf{B} = mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}

Plugging into the equation gives:

−∇2E=−∂∂t(μ0ϵ0∂E∂t)-nabla^2 mathbf{E} = – frac{partial}{partial t} left( mu_0 epsilon_0 frac{partial mathbf{E}}{partial t} right)

Which simplifies to:

∇2E=μ0ϵ0∂2E∂t2nabla^2 mathbf{E} = mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2}

Step 4: Repeat the Process for the Magnetic Field

Start from Ampère-Maxwell law:

∇×B=μ0ϵ0∂E∂tnabla times mathbf{B} = mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}

Take curl on both sides:

∇×(∇×B)=μ0ϵ0∂∂t(∇×E)nabla times (nabla times mathbf{B}) = mu_0 epsilon_0 frac{partial}{partial t} (nabla times mathbf{E})

Using the same vector identity, and knowing ∇⋅B=0nabla cdot mathbf{B} = 0:

−∇2B=μ0ϵ0∂∂t(−∂B∂t)-nabla^2 mathbf{B} = mu_0 epsilon_0 frac{partial}{partial t} left( -frac{partial mathbf{B}}{partial t} right)

Simplify the right-hand side:

−∇2B=−μ0ϵ0∂2B∂t2-nabla^2 mathbf{B} = – mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2}

Or:

∇2B=μ0ϵ0∂2B∂t2nabla^2 mathbf{B} = mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2}

Final Electromagnetic Wave Equations

The wave equations for electric and magnetic fields are:

∇2E=μ0ϵ0∂2E∂t2nabla^2 mathbf{E} = mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2} ∇2B=μ0ϵ0∂2B∂t2nabla^2 mathbf{B} = mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2}

These are classic wave equations showing that Emathbf{E} and Bmathbf{B} propagate as waves with speed

v=1μ0ϵ0v = frac{1}{sqrt{mu_0 epsilon_0}}

which equals the speed of light in vacuum cc.

Summary

  • Start from Maxwell’s equations in free space.

  • Use vector calculus identities to take curls and substitute fields.

  • Derive second-order differential equations for Emathbf{E} and Bmathbf{B}.

  • Resulting wave equations describe electromagnetic waves traveling at speed cc.

This derivation links the fundamentals of electromagnetism to the existence of light as an electromagnetic wave.

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