In these blog posts, I aim to present techniques that will help you generally excel in your physics courses and exams, regardless whether you are taking your first high school physics class or your second semester of graduate-level quantum mechanics. With that goal in mind, we’ve explored ways you can breeze through the math on your homework assignments and test, such as holding off on the calculator and using dimensional analysis.
Now, let’s pull back a bit and look at how you can use physical intuition in a broader sense. If you think deeply about what is going on in the physical system and approach the problem from a different angle, can you do away with the math altogether?
A memorable homework problem
To begin our discussion of physical intuition, let me begin with the story of one of the more memorable homework problems of my undergraduate career. In my first E+M course, the professor assigned this problem:
You have a thin, uniform square of charge, with sides of length D. How far above the center of the square does the electric field become significantly different from that of an infinite sheet of charge?
That was it, the question in its entirety. In a manner very typical of a first-year undergraduate physics major – and, as I’ve come to learn, somewhat atypical of a graduate student who is actually putting that undergraduate knowledge to use in a lab – I began to pull equations out of my bag of tricks and search earnestly for the path from the statement of the problem to its solution.
Infinite sheet of charge? Easy! The field is just independent of the height.
Finite square of charge? Never fear, I know just the formula and the integral’s not so bad. You have a field pointing directly away from the square, like with an infinite sheet, and the magnitude looks like an infinite sheet when you get down close to the surface. That’s interesting, though: it looks less and less like an infinite sheet as you move farther away and kinda like … (algebra algebra algebra). Yup, it looks like a point charge when you move very far away.
That all made sense: I had applied the relevant formulas, done the math correctly, and came to answers that behaved as they should. But … “significantly?” Whence this “significantly?” On which day did he cover “significantly” in class? What page of the textbook is it on?
The skills that I had been taught, the indispensable skills that are generally the focus of an introductory physics class, had brought me through the mechanics of the problem but had left me hanging at the crux of the question. I could take the equations and physical principals that I had been taught and apply them to a novel system – I had never seen a finite square of charge, but it was no great stretch – but how should I answer the question?
I finally decided to define “significantly” for myself. I rephrased the question to read something like “How far above the center of the square does the electric field become 90% of the field that would be created if the square were an infinite sheet?” I found the corresponding height, which was unsurprisingly D multiplied by some constant number, and handed in the set.
A cheap answer to the puzzle?
I was uneasy about my answer and could not wait to see how the professor would answer the question in his solution set. What ingenious mathematical manipulations would he employ that I had overlooked? When he handed back the graded sets and solutions a few days later, I flipped to the problem and saw:
The answer is D, because that is the only relevant length scale in the problem.
I was surprised by his answer, and I felt a tiny bit cheated. Here was this brilliant theoretical physicist, and he had resorted to what I felt was a cheap answer to an unusually interesting problem.
In retrospect, I realize that the point of the problem was not to prod us to calculate the field precisely and then to make an ill-defined and somewhat haphazard leap to an answer, as I had done. The point was to encourage us to use our physical intuition, to use precisely the judgment that I found could not be looked up in our textbook. We could all do the calculus – plenty of other problems in the course made sure of that – but the point was that we did not need to do the calculus.
An intuitive approach to the problem
I should have reasoned that close to the surface, where the square looks infinite, we can treat it as an infinite sheet of charge. Imagine how a football field looks to an ant crawling on the 50-yard line: the field might as well be a continent. By the same argument, the square becomes indistinguishable from a point charge when observed from very far away. That same football field might as well have been a single blade of grass from the perspective of the Apollo astronauts as they walked on the moon.
The key is that there are different regimes in which the square of charge behaves in a very simple manner, either like an infinite sheet or a single point, and some range in the middle in which the square does not act like either. The line between the two idealized regimes is blurry: there is no point at which the square stops acting like a plane and begins acting like a point charge.
Rather, in the absence of a more quantitative criterion, we say that square acts like a square, not an infinite plane and not a point, on the scale of the square itself, i.e. at a distance D above the center. How high above the football field do you need to be before it begins to look unmistakably like a football field and not like some indefinitely large field of grass? Well, you need to be roughly 100 yards above it.
Aha! Embrace your intuition
Far from being a cheap solution to the problem, the professor’s answer turned out to be unexpectedly profound. Sometimes, we don’t need to slog through the math to find our answer; if we take a step back from the nitty-gritty of the problem and think about what’s physically going on, the answer can become obvious.
The intuitive approach is particularly useful on standardized tests, such as the physics SAT II or GRE, for two reasons. First, this approach is extremely time-efficient: every nasty equation you sidestep is another question or two that you have time to work on. Second, standardized test questions tend to focus less on brute-force calculation and more on physical intuition and scaling. For example, if you double the mass of the earth, how would a satellite need to change its orbital velocity in order to maintain its altitude?
In the next few posts, I will point out a few nice examples of physical intuition at work. Hopefully, these examples will give us a better understanding of how the equations we memorize actually do reflect the world around us. One of the most satisfying moments in a physicist’s day comes when he or she can look at a set of equations, make that mental connection to the physical world, and say, “You know what? That kinda makes sense!” Stay tuned!