###### By Joshua N. Winn, Princeton University

## In classical physics—that is, not quantum physics—electromagnetic radiation takes the form of waves. You get waves whenever there’s some physical quantity that varies smoothly throughout space, and time—like the height of the surface of a pond, or the pressure of the air in a room—that, when it’s disturbed, produces an oscillating pattern.

### Wavelengths

No matter what kind of wave there is, one can define the wavelength, *lambda*, as the distance between maximum values of whatever’s waving, say, between the crests of a water wave. And, we define the frequency, *nu*, as the rate at which the pattern oscillates. When we multiply wavelength and frequency, we get the wave’s phase velocity; the speed with which the pattern moves, which, for electromagnetic radiation, is the speed of light, *c*. So, *lambda* times* nu* equals *c*.

Now with light, it’s a lot less obvious what is waving. It’s not like the air or anything material; it’s the more abstract oscillations of electric and magnetic fields. The electric field is a vector whose magnitude wobbles back and forth, while the magnetic field does the same thing but tilted by 90°, and the whole pattern moves in the direction perpendicular to both vectors.

The wavelengths of visible light range from 0.4 to 0.7 microns. The corresponding frequency, *nu*, is just under 10 to the 15th cycles per second, or Hertz. But that’s just a tiny slice of the whole spectrum. Toward longer wavelengths, there’s infrared radiation, and when the wavelength exceeds a millimeter or so, we start calling them ‘microwaves’ and then ‘radio waves’. Likewise, on the short-wavelength end, we have ultraviolet radiation, and those that are shorter than about 10 nanometers are called ‘X-rays’ and then ‘gamma rays’. All these names and boundaries, they’re arbitrary. They’re based on the different technologies that humans use to study electromagnetic radiation. Nature, herself, makes no sharp distinctions.

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### Charged Particles

It is understood that waves are produced by a disturbance: dropping a stone in water, clapping your hands. So, what kind of disturbance produces electromagnetic waves? The answer is the acceleration of an electrically charged particle. When a charge accelerates, it radiates. And it goes backwards, too. Charges can absorb electromagnetic energy, causing them to accelerate. These facts come from solving Maxwell’s equations of electromagnetism.

Consider a positively charged particle at rest. Coulomb’s law says it produces an electric field, pointing away from the charge, varying in strength as 1 over *r *squared. We can represent that field with lines that emanate from the charge and diverge into the surrounding space. The electric field is everywhere parallel to these field lines. If we now shake the charge, the field lines quiver. A disturbance propagates outward in the form of sharp bends in the field lines. That’s a transverse pattern of electric fields, which is accompanied by magnetic fields.

The pattern and the energy of those waves depend on the details of the acceleration. An electron slowing down in a block of lead produces a certain pattern; an electron whirling around a magnetic field produces a different pattern; and an electron falling from one orbit to another inside an atom produces yet another kind of radiation. But, if the charges are moving randomly, if there are zillions of charges rattling around colliding with each other and they’ve been at it a long time, enough to reach a steady state, then there’s an enormous simplification. The radiation takes on a universal character, depending only on the temperature.

### Normal Particles

For now, though, let’s take leave of waves, and talk about particles. In quantum theory, we learn that if you look closely enough at electromagnetic energy, you’ll see it comes in tiny lumps, called photons. In principle, you could count the photons arriving from the Sun, like counting raindrops falling from the sky, or grains of sand in an hourglass. But photons have some weird properties that normal particles do not have.

So, first, let’s talk about normal particles. Think of a box full of tiny particles, like little billiard balls, whizzing around, knocking into each other. This is an idealized model of a gas. Each particle has a mass, *m*, and a speed, *v*, from which we can compute the particle’s kinetic energy, 1/2 *mv *squared, and its momentum, *mv*. We’ll use the symbol *epsilon* for the energy of a single particle, to distinguish it from the total energy *E* of the entire gas.

### Photons

Now, what about photons? The first thing to know about photons is they have zero mass. The second thing is they always travel at the same speed, *c*, 3 times 10 to the 8th meters per second.

Even though they have zero mass, photons do have energy and momentum. But the equations are different. The energy of an individual photon is *h *time*s nu*, or equivalently, *hc* over *lambda*. So, even though it’s a single particle, a photon has an associated frequency, *nu*, and a wavelength, *lambda*. That constant of proportionality, little *h*, that’s Planck’s constant. Whenever we see little *h* in an equation, we know we’re dealing with quantum theory.

As for the momentum, that turns out to be equal to energy divided by *c*. So, the momentum of a single photon is *h nu* over *c*.

I realize it’s hard to imagine something that’s both a particle and a wave. I try to imagine photons as wave packets, glowing little bundles of energy with dark-and-light fringes. But that’s just a mental image, and in quantum theory, none of our mental images are quite right.

### Common Questions about Waves, Particles, and Photons

**Q: How can we define wavelength and frequency of a wave?**

We can define the wavelength, *lambda*, as the distance between maximum values of whatever’s waving, and we can define the frequency, *nu*, as the rate at which the pattern oscillates.

**Q: What do we get when we multiply wavelength and frequency of a wave?**

When we multiply wavelength and frequency of a wave, we get the wave’s phase velocity; the speed with which the pattern moves.

**Q: What are two specific features of photons?**

Firstly, photons have zero mass. Secondly, they always travel at the same speed, *c*, 3 times 10 to the 8th meters per second.