Deriving the Wave Equation for Electromagnetic Waves in Free Space Using Maxwell’s Equations

Introduction

The derivation of the wave equation for electromagnetic waves in free space using Maxwellrsquo;s equations is a fundamental concept in physics. This article will guide you through the process of understanding how these equations lead to the wave equation and why

Maxwell’s Equations in Differential Form

Maxwell’s equations are a set of four interrelated differential equations that describe the behavior of electric and magnetic fields. Here are the four equations, starting with the differential form appropriate for free space (vacuum):

Guass’s Law for Electricity: nabla cdot mathbf{E} frac{rho}{epsilon_0} Guass’s Law for Magnetism: nabla cdot mathbf{B} 0 Faraday’s Law of Induction: nabla times mathbf{E} -frac{partial mathbf{B}}{partial t} Ampèrersquo;s Law with Maxwellrsquo;s Addition: nabla times mathbf{B} mu_0 mathbf{J} mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}

In free space, we assume that there are no charges ((rho 0)) and no currents ((mathbf{J} 0)).

Deriving the Wave Equation

Taking the Curl of Faraday’s Law

From Faraday’s Law:

$$ abla times mathbf{E} -frac{partial mathbf{B}}{partial t} $$

Take the curl of both sides:

$$ abla times ( abla times mathbf{E}) - abla times left(frac{partial mathbf{B}}{partial t}right) $$

Using the vector identity:

$$ abla times ( abla times mathbf{A}) abla ( abla cdot mathbf{A}) - abla^2 mathbf{A} $$

We simplify to:

$$ abla times ( abla times mathbf{E}) - abla^2 mathbf{E} $$

Thus:

$$ - abla^2 mathbf{E} - abla times left(frac{partial mathbf{B}}{partial t}right) $$ Substituting Ampèrersquo;s Law

From Ampèrersquo;s Law for free space (assuming no currents and no polarization):

$$ abla times mathbf{B} mu_0 epsilon_0 frac{partial mathbf{E}}{partial t} $$

Substitute this into the equation derived in Step 1:

$$ - abla^2 mathbf{E} -mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2} $$

Simplifying gives:

$$ abla^2 mathbf{E} mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2} $$ Similar Derivation for the Magnetic Field

From Ampèrersquo;s Law, take the curl:

$$ abla times mathbf{B} mu_0 epsilon_0 frac{partial mathbf{E}}{partial t} $$

Using Faradayrsquo;s Law and the vector identity:

$$ abla times ( abla times mathbf{B}) mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2} $$

Leading to:

$$ abla^2 mathbf{B} mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2} $$ Wave Equation for Electromagnetic Waves

The equations for the electric and magnetic fields derived are in the form of the wave equation:

$$ abla^2 psi frac{1}{v^2} frac{partial^2 psi}{partial t^2} $$

Comparing these, we identify the speed of the electromagnetic wave:

$$ v frac{1}{sqrt{mu_0 epsilon_0}} $$

This calculated speed is remarkably close to the experimentally measured speed of light, which initially led to the conclusion that light is an electromagnetic wave.

Conclusion

Through the manipulation of Maxwell’s equations and the application of vector calculus, we have derived the wave equations for the electric and magnetic fields. These equations describe the propagation of electromagnetic waves, including light, through space.