Electromagnetic Waves - Maple Programming Help

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Electromagnetic Waves

"We can scarily avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena"
- James Clerk Maxwell quoted from James Clerk Maxwell, A Biography by Ivan Tolstoy

Main Concept

Electricity and magnetism appeared to be completely different phenomena until there was growing evidence in the late 1800s that suggested they were deeply connected. The cumulation of which was a treatise written in 1873 by the Scottish physicist James Clerk Maxwell, which combined all the theories into four mathematically elegant equations. In the case of no source charges and no current, Maxwell's equations reduce to the three-dimensional wave equation for each Cartesian component of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$. The speed of propagation of these waves is , which is precisely the speed of light! This suggests that light is in fact just an electromagnetic wave!

The animation shown later illustrates the propagation of an electromagnetic wave, in the $\stackrel{\mathbf{ˆ}}{\mathbf{z}}$ direction, given by the following equations for electric and magnetic fields:

 Let there be light! A changing magnetic field generates an electric field, and a changing electric field generates a magnetic field. This is precisely what electromagnetic radiation is. A source charge simultaneously emits changing electric and magnetic fields which then continually feedback each other. This is how radiation can propagate through empty space.   The electromagnetic spectrum consists of all the possible wavelengths for electromagnetic radiation. Waves with long wavelengths have low frequencies and hence carry low energy. For example, radio waves typically have a wavelength between 1 mm and 100 km! The visible spectrum of light is only is a from about 400 to 700 nanometers. Electromagnetic waves with short wavelengths have high frequencies and are dangerous because of their incredibly high energy, for example, X-rays and gamma rays.  (Wavelength image from Universe by Freedman and Kaufmann.)



 Monochromatic waves It follows from Maxwell's equations that electromagnetic waves are transverse waves, that is, the direction of propagation (energy transfer) is perpendicular to the direction of the oscillation (directions of $\mathbf{E}$ and $\mathbf{B}$ fields). The direction of propagation is calculated from cross product of the electric and magnetic fields ($\mathbf{E}×\mathbf{B}$), resulting in a direction perpendicular to both $\mathbf{E}$ and $\mathbf{B}$ fields. For example, $\stackrel{ˆ}{\mathbf{x}}×\stackrel{ˆ}{\mathbf{y}}\mathbf{=}\stackrel{\mathbf{ˆ}}{\mathbf{z}}$. Sinusoidal waves are very convenient mathematically and furthermore, any wave can be written as a linear combination of sinusoidal waves (via a Fourier transform), making them especially useful to physicists. Since a sinusoidal wave has only one frequency, it is a monochromatic wave. This is why the following animation shows a sinusoidal wave; it has become the paradigm for a monochromatic plane wave.



In this animation, the blue part is the electric field and the red part is the magnetic field.



  

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