“And therefore the curves must cross” seems more like mathematical thinking to me. (Which has its uses in physics, obviously.) Certainly the curves must cross at some point, but it is not obvious (well, not to me) that they will do so within the range for which third-power decay and exponential decay are good approximations. (Obviously the electric field is nowhere infinite, so the x-inverse-cube relation cannot be intended as an exact description everywhere—it must break down near the zero.) To see that you’d have to put in the boundary conditions: The starting values and the constants of decay.
That said, I’m nitpicking a special case and it may well be that the professor knew the boundary conditions and could immediately see that the curves would cross somewhere in the relevant range. In general, yes, this sort of there-must-exist insight is often useful on the grand overview level; it tells you what sort of solutions you should look for. I’ve seen it used to immediately spot an error in a theory paper, by pointing out that a particular formula gave an obviously wrong answer for a limiting case. It is perhaps more useful to theorists than experimentalists.
Physicists of LW, would you care to weigh in on this? How well does Cochran’s article describe your thought processes?
“And therefore the curves must cross” seems more like mathematical thinking to me. (Which has its uses in physics, obviously.) Certainly the curves must cross at some point, but it is not obvious (well, not to me) that they will do so within the range for which third-power decay and exponential decay are good approximations. (Obviously the electric field is nowhere infinite, so the x-inverse-cube relation cannot be intended as an exact description everywhere—it must break down near the zero.) To see that you’d have to put in the boundary conditions: The starting values and the constants of decay.
That said, I’m nitpicking a special case and it may well be that the professor knew the boundary conditions and could immediately see that the curves would cross somewhere in the relevant range. In general, yes, this sort of there-must-exist insight is often useful on the grand overview level; it tells you what sort of solutions you should look for. I’ve seen it used to immediately spot an error in a theory paper, by pointing out that a particular formula gave an obviously wrong answer for a limiting case. It is perhaps more useful to theorists than experimentalists.
Not a physicist but a mathematician, but the anecdote about lightning in the first quoted paragraph sounds like everyday conversation to me.