# Electromagnetic radiation -- Maxwell's rainbow

#### Prerequisites

You've probably heard in your earlier studies that "light is an electromagnetic wave".  You also probably know that light has oscillatory properties since you are likely to have seen "spectral analyses" of light in your chemistry or biology classes where the different colors of light are specified by "wavelengths" or "frequencies." But it's not really obvious what this means or "what's waving". Having some insight into the nature of electromagnetic waves will help you to make sense of a variety of phenomena in chemistry and biology: what "spectra" mean and how light interacts with matter. In addition, a variety of important measurement devices rely on the fact that a wavelength of light is small and that the superposition of two light waves can be extremely sensitive to small shifts in distance (through interference).

We'll briefly review in this page some of the properties of light — what it means to be "electromagnetic radiation". But first, let's tell just of little of the story of how we came to know that.

## How do we know that light is EM radiation?

The idea that light is electromagnetic in nature is fairly new. It was only first suggested by the Scottish physicist James Clerk Maxwell(1831-1879) in a famous paper in 1865 in which he proposed his unified field theory of electricity and magnetism. (Yes, this is the same Maxwell who played a critical role in the building of our understanding of the statistical theory of gases.)

How he came to this is a rather extraordinary story, and one of those examples where building mathematical models and demanding consistency seems to yield unexpected yet experimentally true results. The full story requires a lot of math (vector calculus) and typically takes a full semester physics class to build. But the story is not too difficult to understand even without the math.

It begins with the independent studies of electric and magnetic forces. Electric forces are between charges, while magnetic forces occur, both between bar magnets and electric currents. The electric force law between charges (Coulomb's law) has a constant, Coulomb's constant, $k_C$, that sets the scale of the electric force. The magnetic force law between electric currents has a similar constant, Ampere's constant, $k_A$, that sets the scale of the magnetic force. (At the time, and in physics courses for majors today, the Coulomb constant was written $1/4πε_0$ and the magnetic constant as $μ_0/4π$.)

Then a series of physicists began to write mathematical laws describing the electric and magnetic fields produced by charges and currents. One interesting result was that a changing magnetic field led to an electric field (Faraday's law), the principle that makes electric generators possible. Maxwell then noticed that the electric and magnetic equations lacked a symmetry; changing magnetic fields produced electric fields, but not the other way around. Or could they? By a careful analysis of the charging capacitor (where there are currents going in and coming out but there is a break in the middle), Maxwell demonstrated that indeed a changing electric field produces a magnetic field.

But then he observed that his equations said that a changing electric field led to a changing magnetic field — which led to a changing electric field which .... and so on. By combination and manipulation of his equations, he produced a wave equation: a partial differential equation for the electric and magnetic fields that related how the way they the fields changed in space and time were related. These equations had traveling solutions — solutions of the form $f(x-vt)$ and $f(x+vt)$ (in one dimension) that is familiar from our study of waves on a string. Furthermore, the speed of these propagating signals could be calculated from the force constants  and turned out to agree with the speed of light (which by that time had been fairly well measured).

Wait... what? This was really an extraordinary result! Two measurements that can be done in a freshman laboratory of force strengths alone could be combined to yield the speed of light.

This was very convincing evidence that light must in fact be just what Maxwell said — an electromagnetic oscillation. But the proof of the pudding was that if light were electrical oscillations, then invisible radiation should also be producible by making an oscillating electric spark. This suggestion was taken up by others and 20 years later, Heinrich Hertz (shown on a German stamp from 1994) demonstrated radio waves. The result is radio, TV, WiFi, GPS, and cellular phones — all a result of taking the math seriously!

## What does it mean to say that light is EM radiation?

Just saying that "light is an electromagnetic wave" doesn't help us very much. What's waving? What's moving? We know that electric fields are vector values that are assigned to points in space. They might change, but they don't move. What's going on?

What's happening is similar in spirit to what we saw when we studied transverse waves on an elastic string. In that case, the beads of which the string is made moved perpendicular to the string while the pattern of motion moved along the string. The motion of the beads were transverse to the motion of the pattern — the wave.

In the case of the electric field, the electric field at a point doesn't move, but it points in a direction and it changes in magnitude (and perhaps direction). So here are the key elements of the story:

In an electromagnetic wave, the electric field (and magnetic field) at each point in space changes and the pattern of changes is what propagates.

In an electromagnetic wave, the electric and magnetic fields are perpendicular to the direction in which the pattern of the wave is moving.

The electric field and its effects on matter are typically much greater than the magnetic field and its effects on matter so we often describe an EM wave just in terms of the electric field associated with it. The magnetic fields are important in serving as intermediaries to "bootstrap" the changes in the E field in time into changes in space, but we will not talk about them much since they don't play much of a role in the way light interacts with matter (except in very hot plasmas).

The reason we were able to treat light as if it were rays is that the wavelength of light is so small (fractions of micrometer) compared to distances we tend to pay attention to. If we look at a ray of light, it's really more appropriate to consider it a "pencil" of light — a long thin cylinder. And even if that cylinder is very thin, when we look at the scale of wavelengths of light, if we consider slices perpendicular to the ray, those transverse slices are very large compared to the wavelength.

Looking at a bit of the electric fields along the ray, in the simplest case (a ray of a single wavelength and a single polarization) the fields along the ray might look something like the figure shown below. We refer to this as the shish-ka-bab model of an electromagnetic wave.

To make sense of what this means, interpret this picture as follows.

• This is a picture of a bit of an EM wave in space propagating in the x direction, frozen in time.
• Along each plane perpendicular to the direction of propagation (the y-z planes shown) the E field is constant in space; that is, it is the same vector everywhere on the plane.
• At different points along the ray, the values on each plane is different (just like the values of the displacements of the beads in a transverse wave on the elastic string are different at different positions). In this case, the variation in space shown is sinusoidal:
$$\overrightarrow{E}(x,y,z,t) = \overrightarrow{E}_0 \sin{(kx-\omega t)}$$

This looks somewhat like our sinusoidal propagating waves along the elastic string — it has the same sine function — but it's a little more complicated to interpret.

Here are some things to watch out for:

1. The electric field is defined at all points in space — therefore it is a function of $x, y, z$.  At each point it can also be different at different times — therefore it's a function of $t$ as well.
2. The electric field is a vector. It has a direction as well as a magnitude. In this case, the vector $\overrightarrow{E}_0$ gives us both the maximum amplitude and the direction (though when sine goes negative, the direction of the field reverses). In the picture shown, the vector $\vec{E}_0$ points in the y direction.
3. The picture shows a fixed time. If we look along any line that is parallel to the ray (the "shaklik" in the shish-ke-bab) at this time, the E field will oscillate as a function of x. Along the planes perpendicular to x (changing y and/or z) the field is a constant (since the equation doesn't depend on y or z).
4. As time runs, the pattern of planes moves in the direction of the blue arrow with the speed of light. At any fixed point in space, the E field will oscillate as the different planes pass it. It's like someone is carrying the sheiks-ka-bob of planes and running with it along the direction of the x axis.

Sometimes the E and B fields in a traveling EM wave are plotted as perpendicular to the direction of propagation and to each other as in the figure below at the right. We have shown the corresponding planes above it. The figure (b) is often misleading (a dangerous bend) since it sort of looks like both the E and B vectors are extending into space above and below the ray.  They are not. They are meant to all lie along the ray but you can't plot an electric field vector anywhere but in space — but an E field vector is not a distance and so it doesn't actually take up space. Each vector only describes the field at the place where its "foot" is — at the base of the vector. And it doesn't show that the field is constant in the transverse direction to the propagation.

## EM radiation is transverse: polarization

Since the E field is transverse to the direction of propagation (shown here as the x direction), it can point in any direction in the y-z plane. It can even spin around in the y-z plane. How the E field points in the plane perpendicular to the direction of motion of the pattern is called the polarization of the EM wave. We can typically describe the polarization by a vector as shown in the equation above.

If the EM wave looks like a single fixed vector times a sinusoidal function, it is called linearly polarized. That means that the E field always points in a single direction. Just like any vector, the polarization vector, $\overrightarrow{E}_0$, can be written as a linear combination of a y-direction vector and a z-direction vector. Sometimes we might have a vector in one direction times a $\sin{(kx-ωt)}$ and a vector in another direction times a $\cos{(kx-ωt)}$ function. This results in a vector that spins in the transverse plane and is called circularly or elliptically polarized

The fact that an EM wave can have different polarizations means that a single light ray can actually carry more information than just its intensity. It can have different amount of different polarization. And since different polarizations of light can interact with matter in different ways, it can carry useful information. For example, that sunlight scattering off the small fluctuations of the air molecules that create the blue of the sky, comes to our eyes polarized. Although we can't detect that (most of us), some animals can and that can provide them with useful directional information, even when the sun is hidden by clouds.

## Frequency and wavelength: Maxwell's rainbow

Just like the sinusoidal waves on our elastic string, sinusoidal electromagnetic waves have frequency and wavelength and satisfy the equations that relate them to the speed:

$$\lambda f = c \quad \quad \lambda = cT$$

where $c$ is the speed of light and $T = 1/f$  is the period. The light we see occupies only a small range of the possible wavelengths — from about 300-650 nm. The rest extends the rainbow to everything from gamma rays to to radio waves and beyond: Maxwell's rainbow.

Joe Redish 4/23/12

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