[NOTES/EM-10010]-Wave Equation in Free Space

In absence of any medium and in free space, \(\rho=0, \vec{j}=0\), it is proved that the electric and magnetic fields satisfy wave equation.

It turns out that even for \(\rho=0, \vec{j}=0\) the Maxewll's equations have nontrivial solutions. These are traveling wave solutions for the electric and magnetic fields. To show this, we begin with Maxwell's equations, in absence of charges and currents, {\it i.e.} with \(\rho=0, \vec{j}=0\). \begin{align}\label{EQ21} &\nabla\cdot\vec{E} = 0,\\\label{EQ22} &\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}.\\\label{EQ23} &\nabla\cdot\vec{B} = 0,\\ &\nabla\times\vec{B}=\label{EQ24} \mu_0\epsilon_0 \frac{\partial\vec{E}}{\partial t}. \end{align} Taking curl of EQ22} and using the identity \begin{eqnarray}\nabla\times(\nabla\times \vec{E})&=&\nabla(\nabla\cdot\vec{E})-\nabla^2\vec{E}, \end{eqnarray} and making use of EQ21}, we get \begin{eqnarray} \nabla^2\vec{E} &=& -\pp{t}\big(\nabla\times \vec{B}\big). \label{EQ05} \end{eqnarray} Next using EQ05} to eliminate \(\nabla\times \vec{B}\) from EQ05}, we arrive at \begin{eqnarray}\label{EQ09A} \nabla^2 \vec{E} - \frac{1}{(\mu_0\epsilon_0)} \PP[\vec{E}]{t} &=&0, \end{eqnarray} In a similar manner, starting with curl of EQ24} we will get \begin{eqnarray} \nabla^2 \vec{B} - \frac{1}{(\mu_0\epsilon_0)} \PP[\vec{B}]{t} &=&0 \label{EQ10A} \end{eqnarray} The equations EQ09A} and EQ10A} represent waves equation. \noindent \PlainBox[0.97]{\vspace{1.5mm} Thus we have a remarkable result that the electric and magnetic fields can exist as traveling waves even in absence of charges and currents.\vspace{1.5mm}} In addition EQ21} and EQ23} imply that the waves described by EQ09A} and EQ10A} are transverse. These waves are named  electromagnetic waves. The velocity of waves can be seen to be equal to \(\sqrt{\mu_0\epsilon_0}\), The numerical value of the velocity turns out to be equal to the velocity of light and the electromagnetic waves are identified with the light waves,

References: 

1. Sec 20-1  Waves in Free space: plane waves R. P. Feynman, Robert B. Leighton and Mathew Sands Lectures on Physics, vol-II, B.I. Publications (1964)

2. Sec 9.2.1 The Wave Equation for \(\vec{E}\) and \(\vec{B}\)} Sec 9.2.2 {\bf Monochromatic plane waves}David Griffiths, Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).