###### By Joshua Winn, Princeton University

## As we zoom in the fundamental particles of a matter, we see the individual atoms that make up molecule. In a single water molecule, oxygen has a nucleus, with a positive electrical charge, and is surrounded by orbiting electrons, with negative electric charge. Because of those opposite charges, the nucleus and the electrons are attracted to each other. Which brings us to the second of the 4 fundamental forces of nature: electromagnetism.

### Coulomb’s Law

The Coulomb’s law says that the electric force goes as the product of the charges divided by *r*-squared. So, it’s very similar in form to Newton’s law of gravity.

For the proportionality constant, we’ll use the Greek letter *eta*, because the word “electric” comes from the Greek *uhlectron*, which starts with an *eta*.

Numerically, *eta* is 9 times 10 to the 9 Newton meter-squared or Coulomb-squared, where the Coulomb is the standard unit charge. In those units, the electron and the proton both have a charge of magnitude 1.6 times 10 to the minus-19, which we’ll represent with the letter *e*.

### Attraction and Repulsion

Also, notice the force law has a plus sign this time, not a minus sign. When the product of charges is positive—that is, when they’re both the same sign—then the force is repulsive, pushing the charges apart.

When the charges have opposite signs, like an electron and a proton, they attract. We can also write down the potential energy associated with the electric attraction or repulsion, that’s the so-called Coulomb energy, which, as in the case of gravity, varies as one over *r*.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

### Role of Velocity

The Coulomb force explains why electrons are attracted to the nucleus. But there must be something else going on, because otherwise, why don’t the electrons fall all the way down onto the nucleus, and neutralize it, so that it comes to rest?

Well, we might ask the same question about the Earth. If the Earth is attracted to the Sun, why doesn’t it fall in and burn up? The answer in that case is that the Earth has a nonzero angular momentum, a sideways velocity, in ordinary language. And the gravitational acceleration just keeps turning its velocity vector around in a circle.

So, when we look closely at an atom, we might expect to see the electrons whirling around the nucleus, like a miniature solar system. But we don’t. Instead, the electrons look indistinct, there’s an electron cloud surrounding the nucleus.

That’s because electrons, like all fundamental particles, obey the rules of quantum theory, the counter-intuitive laws of motion and interaction, that are more exact and fundamental than Newton’s laws of motion.

### Quantum Theory

The quantum theory says: when we measure the location of an electron or any fundamental particle, we get a specific answer. But when we’re not measuring it, when we’re not forcing the question of where it is, the electron spreads out into a cloud.

And there’s no way to predict exactly where we will find it, when we do measure it. All we can say is we’re likely to find it somewhere in the cloud, or, to use the technical term, the wave function.

### Wave Function

That cloud is called a wave function because the equation that governs the size and shape of the cloud, how it moves and interacts with other clouds, resembles the equation for ordinary waves.

And like regular waves, the wave function can take the form of a pattern moving through space with a certain speed. It can even interfere with other wave functions, producing fringes, like when water waves overlap.

Now, in the case of an electron near the nucleus of an atom, the wave function isn’t moving. It’s trapped by the electrical attraction to the nucleus. So, it’s more like a sound wave reverberating inside an organ pipe, or the vibrating surface of a drum.

And, the wave function obeys Heisenberg’s uncertainty principle: If you try to pin down a particle’s location, by trapping the wave function in a tiny volume, then the particle’s momentum—mass times velocity—becomes more uncertain.

Now, in the case of an electron near the nucleus of an atom, the wave function isn’t moving. It’s trapped by the electrical attraction to the nucleus. So, it’s more like a sound wave reverberating inside an organ pipe, or the vibrating surface of a drum. And, the wave function obeys Heisenberg’s uncertainty principle: If you try to pin down a particle’s location, by trapping the wave function in a tiny volume, then the particle’s momentum—mass times velocity—becomes more uncertain.

### Heisenberg’s Uncertainty Principle

Mathematically, we say *Delta-a *times *Delta-p* is greater than *h-*bar over 2, where *Delta-x* is the spatial extent of the cloud, and *Delta-p* is the extent of the momentum cloud. It’s the range in the possible values of momentum that the particle might have, if you measure it. You can’t make both *Delta-x* and* Delta-p* as small as you might want. Their product is always at least *h-*bar over 2, a fundamental constant of nature. The *h* is Planck’s constant, 6.6 times 10 to the minus-34 joule-seconds. And the little bar through the middle is shorthand for *h *over 2*pi*, because that combination comes up so often.

The uncertainty principle explains why atoms are stable. Even if you drop an electron directly onto a proton, with zero angular momentum, it doesn’t fall down and come to rest, because that would imply that *Delta-x* and *Delta-p* are both zero, which is a no-no. Instead, the wave function strikes a balance between *Delta-x *and *Delta-p*.

The proton exists as a wave function, too, but there’s a big difference. Even though the proton and the electron have charges of equal magnitude, the proton is far more massive, by a factor of 1800. This ends up causing the proton’s wave function to be much smaller in extent than the electron’s.

### Common Questions about Electromagnetism

**Q: What causes the force to be repulsive?**

When the product of charges is positive—that is, when they’re both the same sign—then the force is repulsive, pushing the charges apart.

**Q: What does the quantum theory state?**

The quantum theory says: when we measure the location of an electron or any fundamental particle, we get a specific answer. But when we’re not measuring it, the electron spreads out into a cloud.

**Q: What is a wave function?**

When the electron is not measured, it spreads out and is likely to be found somewhere in the cloud, or, the wave function. That cloud is called a wave function because the equation that governs the size and shape of the cloud, how it moves and interacts with other clouds, resembles the equation for ordinary waves.