Fear of Physics (7 page)

This is absurd because, among other things, the distance between Aspen, Colorado (at noon, Mountain Standard Time), and the sun, and Cleveland, Ohio (at noon, Eastern Standard Time), and the sun differ by 8,000 feet, or about 250,000 centimeters—the difference between the heights of Aspen and Cleveland above
sea level. Thus, we would have to specify where on Earth we made such a measurement to make it meaningful. Next, even if we specify that this distance is from the center of the Earth to the center of the sun (a reasonable choice), this implies that we can measure the size of the Earth and sun to the nearest centimeter, not to mention the distance between the two, to this precise accuracy. (If you think about any practical physical way in which you might go about measuring the distance from the Earth to the sun, you can convince yourself that a measurement with this kind of accuracy is unlikely, if not impossible.)

No: It is clear that when we write 14,960,000,000,000 cm, we are rounding off the distance to a tidy number. But with what accuracy do we really know the number? There is no such ambiguity when we write 1.4960 × 10
¹³
cm, however. It tells us exactly how well we know this distance. Specifically, it tells us that the actual value lies somewhere between 1.49595 × 10
¹³
cm and 1.49605 × 10
¹³
cm. If we knew the distance with 10 times greater accuracy, we would write instead 1.49600 × 10
¹³
cm.

Thus, there is a world of difference between 1.4960 × 10
¹³
cm and 14,960,000,000,000 cm. More than a world, in fact, because if you think about the uncertainty implicit in the first number, it is about 0.0001 × 10
¹³
, or 1 billion, centimeters—greater than the radius of the Earth!

This leads to an interesting question. Is this number accurate? While an uncertainty of 1 billion centimeters seems like an awful lot, compared to the Earth-to-sun distance it is small—less than 1 ten-thousandth of this distance, to be exact. This means that we know the Earth-to-sun distance to better than 1 part in 10,000. On a relative scale, this is highly accurate. It would be like measuring your own height to an accuracy of a tenth of a millimeter.

The beauty of writing down a number like 1.4960 × 10
¹³
is that because the 10
¹³
scale sets the “scale” of the number, you can immediately see how accurate it is. The more decimal places that are filled in, the higher the accuracy. In fact, when you think about it this way, numbers written in scientific notation tell you above all what you can
ignore!
The moment you see 10
¹³
cm, you know that physical effects that might alter the result by centimeters, or even millions or billions of centimeters, are probably irrelevant. And as I stressed in the last chapter, knowing what to ignore is usually the most important thing of all.

 

 

I have so far ignored perhaps the most crucial fact that makes 1.49600 × 10
¹³
cm a physical quantity and not a mathematical one. It is the “cm” tacked on the end. Without this attribute, we have no idea what kind of quantity it refers to. The “cm” tells us that this is a measurement of length. This specification is called the
dimension
of a quantity, and it is what connects numbers in physics with the real world of phenomena. Centimeters, inches, miles, light-years—all are measurements of distance and so carry the dimensions of length.

The thing probably most responsible for simplifying physics is a fascinating property of the world. There are only three kinds of fundamental dimensional quantities in nature: length, time, and mass.
^b
Everything, all physical quantities,
can be expressed in terms of some combination of these units. It doesn’t matter whether you express velocity in miles/hour, meters/sec, furlongs/fortnight, they are all just different ways of writing length/time.

This has a remarkable implication. Because there are just three kinds of dimensional quantities, there are a limited number of independent combinations of these quantities you can devise. That means that every physical quantity is related to every other physical quantity in some simple way, and it strongly limits the number of different mathematical relations that are possible in physics. There is probably no more important tool used by physcists than the use of dimensions to characterize physical observables. It not only largely does away with the need to memorize equations but it underlies the way we picture the physical world. As I will argue, using dimensional analysis gives you a fundamental perspective of the world, which gives a sensible basis for interpreting the information obtained by your senses or by other measurements. It provides the ultimate approximation: When we picture things, we picture their dimensions.

When we earlier analyzed the scaling laws of spherical cows, we really worked with the interrelationship between their dimensions of length and mass. For example, what was important there was the relationship between length and volume and, more explicitly, the ratio of volumes of objects that were scaled up in size. Thinking about dimensions, we can go further and figure out how to estimate the volume itself of any object. Think of any system of units to describe the volume of something: cubic inches, cubic centimeters, cubic feet. The key word is
cubic.
These measurements all describe the same dimensions: lengh × length × length = length
³
. Thus, it is a good bet that the volume of an object can be estimated by picking some characteristic length, call it
d,
and then cubing it, or taking d
³
. This is usually good to within an order of magnitude. For example, the volume of a sphere I gave earlier can be rewritten as π/6 d
³
≈ [1/2] d
³
, where
d
is its diameter.

Here’s another example: Which is correct: distance = velocity x time, or distance = velocity/time? Though the simplest kind of “dimensional analysis” can immediately give the correct answer, generation after generation of students taking physics insists on trying to memorize the formula, and they invariably get it wrong. The dimensions of velocity are length/time. The dimensions of distance are length. Therefore, if the left-hand side has dimensions of length, and velocity has dimensions of length/time, clearly you must multiply velocity by time in order for the righthand side to have the dimensions of length.

This kind of analysis can never guarantee you that you have the right answer, but it can let you know when you are wrong. And even though it doesn’t guarantee you’re right, when working with the unknown it is very handy to let dimensional arguments be your guide. They give you a
framework
for fitting the unknown into what you already know about the world.

It is said that fortune favors the prepared mind. Nothing could be more true in the history of physics. And dimensional analysis can prepare our minds for the unexpected. In this regard, the ultimate results of simple dimensional analysis are often so powerful that they can seem magical. To demonstrate these ideas graphically, I want to jump to a modern example based on research at the forefront of physics—where the known and unknown mingle together. In this case, dimensional arguments helped lead to an understanding of one of the four known forces in nature: the “strong” interactions that bind “quarks” together to form protons and neutrons, which in turn make up the nuclei of all atoms. The arguments might seem a little elusive at first reading, but don’t worry. I present them because they give you the chance to see explicitly how pervasive and powerful dimensional arguments can be in guiding our physical intuition. The flavor of the arguments
is probably more important to carry away with you than any of the results.

Physicists who study elementary-particle physics—that area of physics that deals with the ultimate constituents of matter and the nature of the forces among them—have devised a system of units that exploits dimensional analysis about as far as you can take it. In principle, all three dimensional quantities—length, time, and mass—are independent, but in practice nature gives us fundamental relations among them. For example, if there existed some universal constant that related length and time, then I could express any length in terms of a time by multiplying it by this constant. In fact, nature has been kind enough to provide us with such a constant, as Einstein first showed. The basis of his theory of relativity, which I will discuss later, is the principle that the speed of light, labeled
c,
is a universal constant, which all observers will measure to have the same value. Since velocity has the dimensions of length/time, if I multiply any time by
c,
I will arrive at something with the dimension of length—namely, the distance light would travel in this time. It is then possible to express all lengths unambiguously in terms of how long it takes light to travel from one point to another. For example, the distance from your shoulder to your elbow could be expressed as 10
^–9
seconds, since this is approximately the time it takes a light ray to travel this distance. Any observer who measures how far light travels in this time will measure the same distance.

The existence of a universal constant, the speed of light, provides a one-to-one correspondence between any length and time. This allows us to eliminate one of these dimensional quantities in favor of the other. Namely, we can choose if we wish to express all lengths as equivalent times or vice versa. If we want to do this, it is simplest to invent a system of units where the speed of light is
numerically equal to unity. Call the unit of length a “light-second” instead of a centimeter or an inch, for example. In this case, the speed of light becomes equal to 1 light-second/second. Now all lengths and their equivalent times will be numerically equal!

We can go one step further. If the numerical values of light-lengths and light-times are equal in this system of units, why consider length and time as being separate dimensional quantities? We could choose instead to equate the dimensions of length and time. In this case all velocities, which previously had the dimensions of length/time, would now be dimensionless, since the dimensions of length and time in the numerator and denominator would cancel. Physically this is equivalent to writing all velocities as a (dimensionless) fraction of the speed of light, so that if I said that something had a velocity of [1/2], this would mean that its velocity was [1/2] the speed of light. Clearly, this kind of system requires the speed of light to be a universal constant for all observers, so that we can use it as a reference value.

Now we have only two independent dimensional quantities, time and mass (or, equivalently, length and mass). One of the consequences of this unusual system is that it allows us to equate other dimensional quantities besides length and time. For example, Einstein’s famous formula E = mc
²
equates the mass of an object to an equivalent amount of energy. In our new system of units, however, c (= 1) is dimensionless, so that we find that the “dimensions” of energy and mass are now equal. This carries out in practice what Einstein’s formula does formally: It makes a one-to-one connection between mass and energy. Einstein’s formula tells us that since mass can be turned into energy, we can refer to the mass of something in either the units it had before it was transformed into energy or the units of the equivalent amount of energy that it transforms into. We need no longer speak of the
mass of an object in kilograms, or tons, or pounds, but can speak of it in the equivalent units of energy, in, say, “Volts” or “Calories.” This is exactly what elementary-particle physicists do when they refer to the mass of the electron as 0.5 million electron Volts (an electron Volt is the energy an electron in a wire gets when powered by a 1-Volt battery) instead of 10
^–31
grams. Since particle-physics experiments deal regularly with processes in which the rest mass of particles is converted into energy, it is ultimately sensible to use energy units to keep track of mass. And that is one of the guidelines: Always use the units that make the most physical sense. Similarly, particles in large accelerators travel at close to the speed of light, so that setting c = 1 is numerically practical. This would
not
be practical, however, for describing motions on a more familiar scale, where we would have to describe velocities by very small numbers. For example, the speed of a jet airplane in these units would be about 0.000001, or 10
^–6
.

Things don’t stop here. There is another universal constant in nature, labeled
h
and called Planck’s constant, after the German physicist Max Planck (one of the fathers of quantum mechanics). It relates quantities with the dimensions of mass (or energy) to those with the dimensions of length (or time). Continuing as before, we can invent a system of units where not only c = 1 but h = 1. In this case, the relation between dimensions is only slightly more complicated: One finds that the dimension of mass (or energy) becomes equivalent to 1/length, or 1/time. (Specifically, the energy quantity 1 electron Volt becomes equivalent to 1/6 × 10
^–16
seconds.) The net result of all this is that we can reduce the three
a priori
independent dimensional quantities in nature to a single quantity. We can then describe all measurements in the physical world in terms of just one-dimensional quantity, which we can choose to be mass, time, or length at our convenience. To
convert between them, we just keep track of the conversion factors that took us from our normal system of units—in which, for example, the speed of light c = 3 × 10
⁸
meters/sec—to the system in which c = 1. For example, volume, with the dimensions of length × length × length = length
³
in our normal system of units, equivalently has the dimensions of 1/mass
³
(or 1/energy
³
) in this new system. Making the appropriate conversions to these new units one finds, for example, that a volume of 1 cubic meter (1 meter
³
) is equivalent to (1/[10
^–20
electron Volts
³
]).