3-7-97
Relevant sections in the book : 24.4 - 24.5
The energy in an electromagnetic wave is tied up in the electric and magnetic fields. In general, the energy per unit volume in an electric field is given by:
In a magnetic field, the energy per unit volume is:
An electromagnetic wave has both electric and magnetic fields, so the total energy density associated with an electromagnetic wave is:
It turns out that for an electromagnetic wave, the energy associated with the electric field is equal to the energy associated with the magnetic field, so the energy density can be written in terms of just one or the other:
This also implies that in an electromagnetic wave, E = cB.
A more common way to handle the energy is to look at how much energy is carried by the wave from one place to another. A good measure of this is the intensity of the wave, which is the power that passes perpendicularly through an area divided by the area. The intensity, S, and the energy density are related by a factor of c:
Generally, it's most useful to use the average power, or average intensity, of the wave. To find the average values, you have to use some average for the electric field E and the magnetic field B. The root mean square averages are used; the relationship between the peak and rms values is:
To talk about the polarization of an electromagnetic wave, it's easiest to look at polarized light. Just remember that whatever applies to light generally applies to other forms of electromagnetic waves, too. So, what is meant by polarized light? It's light in which there's a preferred direction for the electric and magnetic field vectors in the wave. In unpolarized light, there is no preferred direction: the waves come in with electric and magnetic field vectors in random directions. In linearly polarized light, the electric field vectors are all along one line (and so are the magnetic field vectors, because they're perpendicular to the electric field vectors). Most light sources emit unpolarized light, but there are several ways light can be polarized.
One way to polarize light is by reflection. Light reflecting off a surface will tend to be polarized, with the direction of polarization (the way the electric field vectors point) being parallel to the plane of the interface.
Another way to polarize light is by selectively absorbing light with electric field vectors pointing in a particular direction. Certain materials, known as dichroic materials, do this, absorbing light polarized one way but not absorbing light polarized perpendicular to that direction. If the material is thick enough to absorb all the light polarized in one direction, the light emerging from the material will be linearly polarized. Polarizers (such as the lenses of polarizing sunglasses) are made from this kind of material.
If unpolarized light passes through a polarizer, the intensity of the transmitted light will be 1/2 of what it was coming in. If linearly polarized light passes through a polarizer, the intensity of the light transmitted is given by Malus' law:
A third way to polarize light is by scattering. Light scattering off atoms and molecules in the atmosphere is unpolarized if the light keeps traveling in the same direction, is linearly polarized if at scatters in a direction perpendicular to the way it was traveling, and somewhere between linearly polarized and unpolarized if it scatters of at another angle.
There are plenty of materials that affect the polarization of light. Certain materials (such as calcite) exhibit a property known as birefringence. A crystal of birefringent material affects light polarized in a particular direction differently from light polarized at 90 degrees to that direction; it refracts light polarized one way at a different angle than it refracts light polarized the other way. Looking through a birefringent crystal at something, you'd see a double image.
Liquid crystal displays, such as those in digital watches and calculators, also exploit the properties of polarized light. The textbook gives a good description of how these work.
The textbook doesn't really talk about this, but we should look at this because it explains why the sky is blue and why the sun looks red at sunrise and sunset. In a nutshell, it's because molecules in the atmosphere scatter light at the blue end of the visible spectrum much more than light at the red end of the visible spectrum. This is because the scattering of light (i.e., the probability that light will interact with molecules when it passes through the atmosphere) is inversely proportional to the wavelength to the fourth power.
Violet light, with a wavelength of about 400 nm, is almost 10 times as likely to be scattered than red light, which has a wavelength of about 700 nm. At noon, when the Sun is high in the sky, light from the Sun passes through a relatively thin layer of atmosphere, so only a small fraction of the light will be scattered. The Sun looks yellow-white because all the colors are represented almost equally. At sunrise or sunset, on the other hand, light from the Sun has to pass through much more atmosphere to reach our eyes. Along the way, most of the light towards the blue end of the spectrum is scattered in other directions, but much less of the light towards the red end of the spectrum is scattered, making the Sun appear to be orange or red.
So why is the sky blue? Again, let's look at it when the Sun is high in the sky. Some of the light from the Sun traveling towards other parts of the Earth is scattered towards us by the molecules in the atmosphere. Most of this scattered light is light from the blue end of the spectrum, so the sky appears blue.
Why can't this same argument be applied to clouds? Why do they look white, and not blue? It's because of the size of the water droplets in clouds. The droplets are much larger than the molecules in the atmosphere, and they scatter light of all colors equally. This makes them look white.
