Electromagnetic Waves
3 min read
To see that propagation is really a wave disturbance, take y-component of Eqn. (3) and x-component of Eqn. (4)
![](https://raw.githubusercontent.com/xibsked/menka/master/books/physics-for-engineers-2/5e28cfa0d261e2d465d04f09032d611d1.png)
To get the wave equation for
, take the derivative of eqn. (5) with respect to
and substitute in eqn. (6) and interchange the space and time derivatives,
![$E_x$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img638.png)
![$z$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img132.png)
![\begin{displaymath}\frac{\partial^2 E_x}{\partial z^2} = \mu_0\epsilon_0 \frac{... ...z}\right)= \mu_0\epsilon_0 \frac{\partial^2 E_x}{\partial t^2}\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img650.png)
Similarly, we can show, We get
![\begin{displaymath} \frac{\partial^2 B_y}{\partial z^2} = \mu_0\epsilon_0 \frac{\partial^2 B_y}{\partial t^2} \end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img651.png)
Each of the above equations represents a wave disturbance propagating in the z-direction with a speed
![\begin{displaymath}c = \frac{1}{\sqrt{\mu_0\epsilon_0}}\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img652.png)
On substituting numerical values, the speed of electromagnetic waves in vacuum is
m/sec.
![$3\times 10^{8}$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img653.png)
Consider plane harmonic waves of angular frequency
and wavlength
. We can express the waves as
![$\omega$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img184.png)
![$\lambda=2\pi/k$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img654.png)
![\begin{eqnarray*} E_x &=& E_0\sin(kz-\omega t)\\ B_y &=& B_0\sin(kz-\omega t) \end{eqnarray*}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img655.png)
The amplitudes
an
are not independent as they must satisfy eqns. (5) and (6) :
![$E_0$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img656.png)
![$B_0$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img657.png)
![\begin{eqnarray*} \frac{\partial E_x}{\partial z} &=& E_0 k \cos(kz-\omega t)\\ \frac{\partial B_y}{\partial t} &=& -B_0 \omega \cos(kz-\omega t) \end{eqnarray*}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img658.png)
Using Eqn. (5) we get
![\begin{displaymath}E_0k = B_0\omega\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img659.png)
The ratio of the electric field amplitude to the magnetic field amplitude is given by
![\begin{displaymath}\frac{E_0}{B_0}=\frac{\omega}{k} = c\end{displaymath}](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img660.png)
Fields
and
are in phase, reaching their maximum and minimum values at the same time. The electric field oscillates in the x-z plane and the magnetic field oscillates in the y-z plane. This corresponds to a polarized wave . Conventionally, the plane in which the electric field oscillates is defined as the plane of polarization. In this case it is x-z plane. The figure shows a harmonic plane wave propagating in the z-direction. Note that
and the direction of propagation
form a right handed triad.
![$\vec E$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img13.png)
![$\vec B$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img16.png)
![$\vec E, \vec B$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img661.png)
![$\hat k$](http://www.cdeep.iitb.ac.in/nptel/Core Science/Engineering Physics 2/Slides/Module-3/Lec-20/img/img217.png)
![](https://raw.githubusercontent.com/xibsked/menka/master/books/physics-for-engineers-2/0d6199df0c134b33de6122fc52ede30f1.png)