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The Maxwell's equations in vacuum, in absence of charges and currents are written and are shown to imply wave equation for the electric and magnetic fields. The plane wave solutions, the electromagnetic waves, are shown to travel with a velocity equal to \(1/\sqrt{\mu_0\epsilon_0}\). The numerical value of this expression equals the velocity of light. This leads to the identification of light as electromagnetic waves.
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Starting with Maxwell's equation we will show that the e.m. fields have travelling wave solutions in absence of charges and currents. In absence of charges and currents we have the equations \begin{eqnarray}\label{EQ01} \vec{\nabla}\cdot\vec{E}&=&0\\\label{EQ02} \vec{\nabla}\times\vec{E}&=&-\frac{\partial\vec{B}}{\partial t}\\\label{EQ03} \vec{\nabla}\cdot\vec{B}&=0&\\\label{EQ04} \vec{\nabla}\times\vec{B}&=&\mu_0\epsilon_0 \frac{\partial\vec{E}}{\partial t} \end{eqnarray} Taking time derivative of {EQ02} and substituting for \(\pp[\vec{E}]{t}\) from {EQ03} we get \begin{eqnarray}\nonumber \PP[\vec{B}]{t} &=& -\nabla \times \pp[\vec{B}]{t} \\\nonumber &=&\frac{1}{\mu_0\epsilon_0} \nabla \times (\nabla \times \vec{B})\\\nonumber &=&\frac{1}{\mu_0\epsilon_0}\Big( \nabla ^2 \vec{B} - \nabla (\nabla \cdot \vec{B})\Big)\\\label{EQ05} &=& \frac{1}{\mu_0\epsilon_0} \nabla ^2 \vec{B} \end{eqnarray} We write the above equation in the form \begin{equation}\label{EQ06} \frac{1}{c^2} \PP[\vec{B}]{t} - \nabla ^2 \vec{B}=0 \end{equation} where \(c=1/\sqrt{\mu_0\epsilon_0}\). Similarly taking time derivative of {EQ04} and making use of {EQ02}, we get \begin{equation}\label{EQ07} \frac{1}{c^2} \PP[\vec{E}]{t} - \nabla ^2 \vec{E}=0 \end{equation} Eqs. {EQ06} and {EQ07} are wave equations for the electromagnetic fields. These electromagnetic waves travel with velocity \(c\) whose value is already known from electrostatics and magnetostatics. \QFY{\input{em-qfy-11001}
References
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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)
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Sec 9.2.1 The wave equation for \(Vec{E}\) and \(\vec{B}\). David Griffiths, Introduction to Electrodynamics , 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).
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4727:Diamond Point
