In physics, it is often necessary to make certain assumptions to simplify a complicated problem to make it tractable. We might make assumptions about symmetry, say that a certain small value is essentially zero, talk about what happens when a certain value is infinity, or any of a host of other simplifications that would make a mathematician cringe and make a layman wonder how he can take our results seriously.
We often make light of this tendency by talking about spherical cows. Obviously, a spherical cow is a pretty ridiculous picture:
However, in certain situations, estimating the cow as a sphere with a characteristic radius might not be as ridiculous as it seems. For example, if the car were flying through the air (or standing in a strong wind, if flying spherical cows are too much for you to accept), the air resistance on a sphere the size of a cow would be a pretty accurate approximation.
But of course, the spherical cow is most useful as a metaphor for the approximation techniques we do use. So what are these simplificaton techniques? Read on to find out.
- Arguments from Symmetry
This sort of corresponds to the spherical cow, in that it’s much easier to think about (and write equations about) things that have nice symmetry, like a sphere, than something that has a strange shape, like a cow. A common example of this is in the application of Gauss’ Law. In theory, the Law should allow us to calculate the electric field due to any arbitrary distribution of electric charge. In practice, however, the math is incredibly difficult to solve except for certain cases where we can invoke a spherical, cylindrical, or planar symmetry. This is all fine when you’re doing textbook problems that begin “consider a uniform sphere of charge…”, but is such a symmetry argument really useful in practice?
It does, in fact, come in handy when solving real-world problems, something that I had direct experience with during my REU research at Indiana University in the summer of 2007. I was working on a computer program that was part of a simulation of the electric fields produced by the proton beam in a linear particle accelerator. In reality, a beam of protons is a collection of discrete charges distributed randomly across a certain cylindrical region. But to develop my program, I used the “Beer Can Model” of a proton beam: approximating it as a continuous cylinder of charge. I developed my method (which was a technique of removing numerical noise from the simulation) based upon this model. When we tested the method on data generated from a more realistic beam, the technique still worked. This showed that the symmetrical approximation that we made was valid.
- “This number is really small, so it goes away”
Everybody who’s taken any sort of math class knows that a statement like N+1 = N is simply ridiculous. Everyone, that is, except for the physicist. Let’s say that N is a really huge number, like if someone dumped an entire truckload of M&M’s in your driveway. If you turned your back on me to watch the truck drive away, and I threw another M&M in the pile while you weren’t looking, would you really notice? What if I snuck one while you were looking to the sky to thank God for this miracle? No, you’d really have no idea. So in this case, for all practical purposes, N+1 = N-1 = N. We make this approximation all the time in my statistical mechanics class, where N represents some astronomically huge number, like the number of water molecules in your glass.
This argument is probably one of the most commonly used in derivation of physics concepts. Besides adding and subtracting, you can say that a small number divided by a big number equals zero. Or that a big number divided by a small number is infinity. Or that the sine of a small angle is equal to the angle measure.
Of course, using this approximation judiciously is extremely important, and knowing when it’s valid to do so is as much of an art as a science. This is all a part of a physics education, as there is not really a way to sit down and say “here’s where you can make this argument, and here’s where you can’t.” Rather, it’s something that you gradually pick up over the years of watching your professors and and textbook authors make the argument.
- Higher-Order Terms Vanish
This is similar to the previous case, but in a slightly more abstract way. The argument basically goes that if a number is small (much smaller than 1), then its square will be tiny, and any terms involving this square can be thrown out. The same argument goes for products of two different but very small numbers.
This comes into play when you are working with expressions that involve differentials (the dx and dt stuff that you may remember from calculus). If, for example, you multiply two expressions containing differentials together, you would keep the terms having just one differential, but you would throw away anything involving dx^2, dt^2, or dx*dt. This is called the first-order approximation, and second-order will be more accurate, but sometimes that level of accuracy is not necessary. Some problems will have even higher-order terms that you may or may not have to calculate.
This also comes applies when approximating a function by a series. Many common functions can be expressed as series, but these series have infinitely many terms. Since we cannot possibly deal with an infinite number of terms, the higher order terms must be discarded.
- It’s on the order of…
This is the jargon that physicists use when we are doing really rough estimates. Often, you don’t care what the actual numerical value of something is, you just want to know whether it’s about a meter, a nanometer, or a kilometer. This gives us free license to be sloppy. Pi is equal to 3, or 1 if we really feel like it. We don’t have to remember whether the equation has a factor of 2 in it, or the exact values of constants, and we can feel free to make any of the other approximations as we see fit. These calculations are sometimes called “back of the envelope calculations,” since we might work them out on any scrap of paper lying around, as the answer will not be quite worth saving. This is not to say that the answer is unimportant, however, as the calculation can give us a conceptual understanding of how the system in question behaves.
The new blog Shores of the Dirac Sea has a nice list of how such approximations are called in the jargon of technical papers:
- “Toy model”: hopefully this model has something to do with the original problem that I couldn’t do.
- “This is a very rough approximation”: We are calling a spherical cow a spherical cow. If you are lucky the order of magnitude is right.
- “An approximation”: One can probably do better, but this is all I can do now.
- “To zero-th order”: This suggests that there is a systematic way to improve the calculation: these are called first order, second order, etc.
- “An uncontrolled approximation”: Something that seems to work even if I don’t know why.
- Setting Constants Equal to Unity
A trick that theorists often use is to set all their units equal to 1 (or unity, as they seem to like to call it, even though it’s pretty much just a fancy word for one). I’m not really sure how they get away with this (I think it has to do with how they define their units), but I guess I’ll find out eventually, since I intend to become a theorist. This isn’t quite in the same vein as the other methods of approximation, but it is a trick that physicists use to make our lives easier, which is really what making assumptions is all about.
Finally, if all else fails, just wave your hands, sketch a graph, or look at limiting cases where you know the answer must tend toward a certain value. Although I don’t think all the hand-waving in the world will persuade a farmer to take that spherical cow off of your hands.