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[NOTES/EM-11003]-Electromagnetic waves in vacuum

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The plane wave solution of wave equation for free fields in vacuum is obtained. It is proved that the electric and magnetic fields are mutually perpendicular and both ar perpendicular to the direction of propagation. It shown that the amplitudes of the electric and magnetic fields obey the relation \(|\vec{E}_0|=c|\vec{B}_0|\).


 

1. Solution in vacuum

Assuming that there is no charge or current density present, we set $\rho=0$ $\vec{j}=0$. In this case the free Maxwell's equations take the form: \begin{align*} &\bar{\nabla}\cdot\bar{E}=0& \bar{\nabla}\cdot\bar{B}=0& \\ \bar{\nabla}\times\bar{E}=&-~\frac{\partial\vec{B}}{\partial t}, &\bar{\nabla}\times\bar{B}=&\mu_0\epsilon_0~\frac{\partial\vec{E}}{\partial t}.& \end{align*} Taking curl of the third equation, we get \begin{align*} \bar{\nabla}\times(\bar{\nabla}\times\bar{E}) = &-~\frac{\partial}{\partial t}(\vec{\nabla}\times\vec{B}) =-\mu_0\epsilon_0~\frac{\partial^2\vec{E}}{\partial t^2}\\ \bar{\nabla}(\bar{\nabla}\cdot\bar{E})-\nabla^2\vec{E} = & - \mu_0\epsilon_0~\frac{\partial^2\bar{E}}{\partial t^2} \end{align*} since \(\nabla\cdot\vec{E}=0\), we get the wave equation. \begin{align*} \nabla^2\vec{E}-&\mu_0\epsilon_0~\frac{\partial^2\vec{E}}{\partial t^2} = 0 \end{align*} Proceeding in a similar manner, we can now derive wave equation for the magnetic field. \begin{align*} \bar{\nabla}\times(\bar{\nabla}\times\bar{B}) =& \mu_0\epsilon_0\vec{\nabla}\times\left(\frac{\partial\vec{E}} {\partial t^2}\right)\\ \nabla(\bar{\nabla}\cdot\bar{B})-\nabla^2\bar{B}=&-\mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2} \end{align*} or $$ \nabla^2\vec{B}-\mu_0\epsilon_0~\frac{\partial^2\vec{B}} {\partial t^2}=0 $$ Therefore, even if $\rho=0$, $\vec{j}=0$ we have equations $$ \square~\bar{E} = 0 \qquad\qquad \square~\vec{B}=0 $$ which have propagating wave solutions.

2.Propagating wave solutions

Assuming plane wave form for the electric field $$ \vec{E} = \vec{E}_0~\exp(i\vec{k}\cdot\vec{r}-i\omega t) $$ we get the relation $$ \vec{k}^2-(\mu_0\epsilon_0)\omega^2 = 0 $$ Here \(k, \omega,c\) are the wave vector, the frequency and the velocity of the electromagnetic waves, where the velocity is given by $$c=\frac{1}{\sqrt{\mu_0\epsilon_0}}.$$ and $\lambda=2\pi/k\qquad\qquad \nu=\omega/2\pi$\\ The magnetic field $\vec{B}$ and the potentials have similar solutions. \noindent In general \begin{align*} \phi = & Re[\phi_0\exp(-\vec{k}\cdot\vec{r}-i\omega t)]\\ \phi = & A \cos(\vec{k}\cdot\vec{r}-i\omega t)+B\sin(\vec{k}\cdot\vec{r}-i\omega t) \end{align*} where $\phi_0=$ is a complex constant. The solutions for the fields can be written down as \begin{eqnarray} \vec{E}&=&\vec{E}_0\exp(i\vec{k}\cdot\vec{r}-i\omega t)\\ \vec{B}&=&\vec{B}_0\exp(i\vec{k}\cdot\vec{r}-i\omega t) \end{eqnarray} and \(\vec{E}_0, \vec{B}\) are complex constants. The Maxwell's equations \begin{equation} {\nabla}\cdot\vec{E}=0, \quad \text{and}\quad \bar{\nabla}\cdot\vec{B}=0, \end{equation} imply that $\vec{E}$ and $\vec{B}$ are $\perp$ to the direction of propagation, i.e. \begin{equation} \vec{k}\cdot\bar{E}=0 \qquad \vec{k}\cdot\vec{B}_0=0. \end{equation} \noindent The Maxwell's equation \(\bar{\nabla}\times\bar{E} = - \frac{\partial\vec{B}}{\partial t} \) implies \begin{align*} \vec{k}\times\vec{E}_0 = & \omega \vec{B}_0 \end{align*} This equation is equivalent to two results that $\vec{B}$ is $\perp~\vec{E}$ and \(|\vec{k}||\vec{E}_0| = \omega |\vec{B}_0|\Rightarrow B_0= ~\frac{1}{c}|\vec{E}_0|\) To summarize we have the results that

  • $\vec{E}, \vec{B}$ are mutually $\perp$ and also $\perp$ to the direction of propagation.
  • $\vec{k},\vec{E},\vec{B}$, form a set of right handed orthonormal vectors.

References

  1. Sec 20-1  Waves in free space , plane wavesR. 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  Monochromatic plane waves. David Griffiths, Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).  

 

 

 

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