Conforme a Principios de Contabilidad Generalmente Aceptados (PCGA)

Forces: What Holds it All Together

At our present level of knowledge, there appear to exist four forces. These forces are gravity, the electromagnetic force, the strong (or nuclear) force and the radiation-causing weak force. Gravity is per-haps the most familiar. It keeps us on Earth and guides the stars and planets through the cosmos. Gravity is always an attractive force, which means gravity will always make two particles want to move closer to one another. When one thinks about forces, an important question is always “What governs the strength of the force?” For gravity, just three things are relevant: (a) the mass of each of the two objects, (b) the distance separating the centers of the two objects and (c) a constant factor which is related to how strong the gravity force is, once the other two factors are taken into account.

Gravity

Mass is a somewhat tricky concept, with which most of us have a mildly incorrect familiarity. Everyone is familiar with the concept of weight (in my case an often depressing familiarity). While weight is not mass (weight is really the force due to gravity), weight is related to mass. A person who weighs more also has a greater mass. However, while weight goes away in outer space, mass does not. Further, while your weight would change if you were to stand on a different planet, again your mass would remain unchanged. So this is a very important idea: weight can change, but mass doesn’t. If it makes you more com-fortable, you can use the two interchangeably as long as you stay on Earth (just don’t tell my physicist colleagues that I said it was OK).

Really weight is a force; a greater weight means that you experience a greater force. The reason that one’s weight can change, while one’s mass is unchanged is because of how gravity works. The force due to gravity is proportional to the mass of one object, multiplied by the mass of the second object. Since Jupiter is the biggest planet, it has a much larger mass than Earth. So, if you were standing on Jupiter, you would feel a greater force than on Earth because, while your mass is unchanged, Jupiter’s mass is much greater. Since your weight is

related to your mass multiplied by the planet’s mass, voilà, you’re heavier on Jupiter. And, when you’re deep in outer space, there is no nearby planet, so the planet’s mass is zero. Now you multiply your (unchanged) mass by the zero mass of the planet and the result is zero (recall that anything multiplied by zero is zero). So, no force in outer space.

Actually, what I just told you is a tiny lie. This is because two objects that have mass always feel an attractive force, no matter how far they’re separated. The force due to gravity extends to the edge of the universe. So even if you’re extremely far from Earth, Earth will always exert a force on you. So why don’t we feel a force due to grav-ity from Jupiter if it is so much more massive than Earth? This is because the mass of the two objects is not the only thing that affects the force of gravity that an object feels. The distance that separates the two objects also affects gravitational force. Physicists say that the force goes down as the square of the distance (physics-ese for the distance multiplied by itself). So, if you have two objects which feel a particu-lar attractive force, when you double the distance between them (2), the force goes down by a factor of four (2  2  4). Similarly if you increase the distance by a factor of ten (10), the force goes down by a factor of a hundred (10 10  100). Thus one sees that the force due to gravity drops off rather quickly; but while the force gets weaker, it never becomes exactly zero. However, as the distance increases, the force drops until it can be neglected (i.e. gets “close enough” to zero).

The final component relevant to determining the force due to gravity is a single, universal constant. While the amount of the mass involved is important, as well as the distance between the two objects, one also needs to include the strength of gravity itself. It turns out that gravity is really a very tiny force. The only reason that it appears to be so strong is that the force is always in one direction and further each proton or neutron (recall we call them collectively nucleons) in your body feels gravity from each nucleon that makes up the Earth;

and that’s a whole bunch of nucleons (you ⬃10²⁸ nucleons and

Earth
10⁵¹ nucleons). When you think about it, with so many atoms involved, it’s a good thing that gravity is so weak, otherwise we’d be squashed like an unlucky bug. The essential points in the pre-ceding few paragraphs are illustrated in Figure 4.1.

The knowledgeable reader will recognize that we’ve been dis-cussing Newton’s theory of gravity. In 1916, Albert Einstein realized that there were situations where the theory breaks down.

If one has a huge amount of mass concentrated in a small space, then gravity is better thought of as a warping of space (which is a very Figure 4.1The effect of gravity depends on three things. The mass of the two interacting bodies, the distance that separates them and the fundamen-tal strength of gravity. (Drawing courtesy of Dan Claes.)

cool idea!). So one might be inclined to say that Newton was wrong, but in fact he really wasn’t. It’s more correct to say that his theory was incomplete, that is, it applied only in limited circumstances. One can think of ample examples where a theory is correct and yet incomplete.

If you punch a brick, what happens is the brick is unaffected and your hand hurts a lot. However, if a karate expert hits the brick, the brick breaks and his (or her!) hand doesn’t hurt (well much anyway). So a hypothetical Newton’s “law of bricks” might be something like

“Hitting a brick doesn’t affect the brick and hurts your hand, with the degree of pain proportional to how fast the hand was moving.” This is a good theory, which works very well over a vast range of hand speeds.

Einstein’s theory would be more like “When you hit a brick, the brick flexes an imperceptible amount (i.e. little enough that zero flexing is a good approximation) and your hand hurts in an amount proportional to hand speed (although the flexing does reduce the pain by an equally imperceptible amount). As the speed of the hand increases, the amount that the brick flexes increases, although the flexing remains very small.

At a particular hand speed, the flexing of the brick becomes large enough that the brick breaks and the hand no longer hurts.” We see that as long as one’s hand is moving slowly enough and one doesn’t measure the flexing of the brick too precisely, that Newton’s and Einstein’s law of bricks are nearly identical. However, at high enough hand speed and for good enough brick flexing measurements, Newton’s law is no longer sufficiently accurate. Newton’s law of bricks should rightfully be called “Newton’s law of brick hitting at low hand speeds.”

Similarly, with gravitation, it should be “Newton’s Law of Universal Gravitation, as long as speeds aren’t huge, masses aren’t enormous and distances aren’t galactic.” Einstein’s law of gravitation should be

“Einstein’s Law of General Relativity (i.e. gravity) unless the sizes involved are tiny.” I’m sad to report that Einstein’s theory, cool as it may be, also fails under particular circumstances. It’s also true that nobody knows how to write a new theory that supersedes Einstein’s theory as Einstein’s theory superceded Newton’s. We’ll come back to this when we discuss the modern mechanism for forces and again in Chapter 8.

Electromagnetism

The force of electromagnetism, the reader will no doubt recall, is one that explains both the phenomena of electricity and magnetism. So let’s start out with the electric force. The electric force is in many respects similar to the gravitational force. Instead of mass, the equiv-alent quantity for electrical force is electric charge. However, unlike the gravitational force, the electrical force can be either attractive or repulsive. This stems from the fact that there are two “flavors” of elec-trical charge, which have been named, positive () and negative (
).

While the reasons for this naming convention are historical (and arbi-trary, as any two names would do), it turns out that these names are handy when one is doing the math that one needs to do to calculate things. This is because if you put an equal amount of positive and negative charge in the same place, they cancel each other out, just like positive and negative numbers in math class, and the result is zero net charge.

So what governs how strong the electric force is between two electric charges and what direction the force points (i.e. attractive or repulsive)? Well the strength is governed by three things (which should sound familiar): (a) the amount of electric charge carried by each of the two objects, (b) the distance between their centers and (c) a constant which turns out to be vastly larger than the similar grav-itational constant (about 10²⁰ or 100 quintillion times greater, in fact). The direction depends on the flavor of not one, but both charges. If both charges are of the positive type, or if both are of the negative type, then the two charges will be repelled. If the two charges are of opposite flavor (that is, one is positive while the other is negative … it doesn’t matter which), the two charges will be attracted. This is where the phrase “opposites attract” comes from (and not that old girlfriend or boyfriend about whom all of your friends asked “What were you thinking?” after the fact).

As illustrated in Figure 4.2, just like gravity, the electric force felt by each particle is dependent on the properties (in this case the elec-trical charge) of both. Increase either particle’s elecelec-trical charge and

the force on both increases. It’s also true that how the electrical force varies with particle separation is identical to that of the gravitational force (e.g. double the distance, reduce the force by a factor of four;

increase the distance by ten, reduce the force by a factor of 100). So except for the fact that the electrical force can repel as well as exhibit-ing gravity’s attractive behavior, the two forces appear very similar.

Figure 4.2Like gravity, the electromagnetic force depends on three things.

The electric charge of the two interacting bodies, the distance that separates them and the fundamental strength of electromagnetism.

The two forces also differ enormously in their strength. The electrical force is vastly stronger than the gravitational force. I cannot tell you in general how much they differ (remember that the forces also depend on mass and charge) but if one uses “obvious” units (one kilogram and one coulomb for the technically minded), the electrical force overpowers the gravitational force by that mind-boggling factor of 10²⁰(100 billion billion). Wow!

If you’re still awake at this point, I hope your first reaction will be to join me in that “Wow!” You’re second reaction should be “Wait a cotton-picking minute. That can’t be right. If the electrical force is that much bigger, why doesn’t it dominate the universe rather than gravity?” To this, I reply “Good question. I’m glad you’re awake!”

The answer stems from the fact that most objects have a very small total electrical charge. Since each atom has the same amount of positive charge (in the nucleus) as negative charge (in the electrons surrounding the nucleus), the net charge is zero (remember that pos-itive and negative charge cancel). So it doesn’t matter how strong the electrical force could be, if one (or especially if both) of the charges were essentially zero, they would feel no electrical force.

So why talk about the electrical force? Because there are situations where it matters and where it matters a whole bunch. Recall that the electrical force gets much larger as the particles get closer together.

Since the size of an atom is about 10
¹⁰(one ten billionth) of a meter, it stands to reason that in this situation, the electromagnetic force must be very strong. This is because of the fact that the electrical charge of the atomic nucleus and the charge of the electrons “see”

one another. Because the electrical force dominates the gravitational, it is electricity that holds the atoms together. If you put in the correct charges and masses of the electrons and atomic nucleus of a hydrogen atom, you see that in this case the electrical force is 10³⁹times larger than the gravitational force.

Since magnetism is just caused by electrical charges in motion, we do not go over this force in detail. Things are a little different, because velocity now matters too. However, as Maxwell showed, electricity and

magnetism are two faces of the same phenomenon; as the magnetic force increases, the electric force compensates. How it does this is really interesting, but a little technical. So I do not discuss the details, but simply state that much of what was said about the electrical force also applies to the magnetic force.

We now know enough to be perplexed. If positive charges attract negative charges, why don’t the negative electrons get sucked into the positively charged atomic nucleus, instead of swirling around the atom in a little “planetary system”? Similarly, why don’t the planets crash into the Sun? Remember in Chapter 1, when it was revealed that Newton said that things moving would go in a straight line unless a force acts on them? Well, if the electric force was somehow magically

“turned off,” the electrons would instantly (ignoring for the moment a few of Einstein’s ideas) start traveling in a straight line, with the directions determined by where they were going when the electric force was turned off. However, the electric force does exist and the electrons are always attracted to the atomic nucleus. As we see in Figure 4.3, the electrons do get pulled towards the nucleus, but since they’re moving, they miss. A little later, they’re still moving (but in a different direction), but still getting pulled towards the center. The net effect is that the electrons keep moving in a circle, always being pulled towards the nucleus, but always missing it.

So why is this interesting? The reason is it shouldn’t be possible.

In the 1860s, Maxwell (remember him?) showed that electricity and magnetism were the same. According to his theory, the electrons should lose energy (physics-ese for slow down) as they felt the elec-trical force and they should have spiraled down into the nucleus of the atom in a brief fraction of a second. So either Maxwell was wrong (heresy!) or something else was going on. Maxwell’s equations have been heavily tested under lots of circumstances. They predicted radio and most of the electrical phenomena that makes our modern tech-nology possible. So his theories obviously applied. Except. Just as Newton’s laws made wrong predictions under some circumstances, Maxwell’s theory only worked when the sizes involved were large.

(Note: large in this context means large when compared to the size of an atom. Maxwell’s laws work rather well even for charges separated by distances that are so small that the eye can’t see them.)

The fact that Maxwell’s equations didn’t work for atoms caused no end of consternation. How this quandary was resolved is a very inter-esting story, about which many books have been written. The birth and growth of quantum mechanics is a fascinating tale, involving some of the most brilliant and storied physicists of the 20th century. Bohr, Heisenberg, Schroedinger and Pauli, legends among physicists, are but a few of the people involved. This book is not about the story of quan-tum mechanics, but some of quanquan-tum mechanics’ ideas are needed to further our tale. Bohr postulated that electrons could only orbit the nucleus only at fixed distances, although why this should be so he was quite uncertain. His postulate worked though and broadly explained Figure 4.3Even if two particles experience an attractive force, they will not necessarily come together and collide. If the particles have a velocity, they will orbit each other, much in the same way as the planets orbit the Sun.

why atoms gave off the light that they do (particular atoms, hydrogen for example, are observed to only give off certain discrete colors of light and no others). Clearly his idea had merit. It was the work of Schroedinger and Heisenberg that generated all of the fuzzy and counter-intuitive aspects of quantum mechanics.

There’s a great story about how Schroedinger made his great con-tribution. He is said to have gone on a holiday (European for vaca-tion) in the Austrian Alps, having brought some paper, a pen, two pearls and a mistress. He placed a pearl in each ear to screen out dis-tractions, put the mistress in bed for inspiration and tried to work out the mystery of the atom. Somehow he had to keep his woman happy while simultaneously creating a new physics theory that explained many difficult mysteries. When I tell this story, I often add that, as a physicist, of course he was up to the challenge and succeeded at every-thing that he set out to do.

As I’m writing this, I’m in an airplane, returning from a physics conference that was held in the Italian Alps. I am returning without any new and brilliant theories. Of course, I was in Italy, not Austria, I brought no pearls and I was unaccompanied. As a scientist who really would like to make a great discovery, I really feel that I need to do the experiment to determine which factor was critical to Schroedinger’s success. Luckily, my wife is a caring and understanding woman, so I’m sure she would agree to both Austria and the pearls.

The upshot of the theory of quantum mechanics is that electrons sort of orbit atomic nuclei. While it is in principle impossible to know where any particular electron is at any particular time, you can know where it is on average. Quantum mechanics also explained the partic-ular colors of light emitted by each kind of atom. Perhaps most inter-estingly, physicists were finally able to explain Mendeleev’s Periodic Table of the elements (introduced in Chapter 1). This fact gave sub-stantial credence to the theory. Prior to the full understanding of quantum mechanics, scientists knew of about 100 atomic elements and knew of electrons and atomic nuclei. Now they knew of the rules governing electrons and the known elements were just a consequence

of electrons, nuclei and quantum mechanics. The world was thereby greatly simplified.

While Schroedinger had extended physics understanding to the ultra-small, the theory had an obvious flaw. It had not included Einstein’s special theory of relativity and thus was not guaranteed to work at speeds nearing the speed of light. Clearly an extension of

While Schroedinger had extended physics understanding to the ultra-small, the theory had an obvious flaw. It had not included Einstein’s special theory of relativity and thus was not guaranteed to work at speeds nearing the speed of light. Clearly an extension of