Impossible Event --- An Error in a Classic Electromagnetic Induction Problem

I believe most high school science students have encountered a problem like this:

A conductor rod ab is placed on smooth horizontal rails in a uniform magnetic field with magnetic induction B. The distance between the parallel rails is L. A constant force F is applied to ab, and it starts moving from rest. The total resistance of the circuit is R. Find the maximum velocity of ab.

For a high school student, this problem is quite easy to solve. One simply lists the following equations: E=BLv, \quad I=\frac{E}{R}, \quad f_1=BIL And based on the fact that the velocity is maximized when the motion becomes uniform, from the equilibrium of forces we have f_1=F. Solving these gives: (E: induced electromotive force; I: induced current; f_1: Ampere force) v=\frac{FR}{B^2 L^2}

It seems very reasonable, but when BoJone analyzed this problem further, a very strange phenomenon was discovered: it is fundamentally impossible to reach a state of uniform motion!

Let us analyze this problem from a calculus perspective. Let the mass of ab be m. The force acting on it is F - f_1 = F - BIL = F - \frac{B^2 L^2 v}{R}. According to Newton’s second law, we have ma = F - \frac{B^2 L^2 v}{R}. Let p = \frac{F}{m} and q = \frac{B^2 L^2}{mR}. We obtain: \ddot{s} = p - q\dot{s}

This is a second-order differential equation. Integrating once gives \dot{s} = pt - qs + C_1. Since the initial displacement and velocity are both 0, we have C_1 = 0.

Solving the equation \dot{s} = pt - qs yields: s = C_2 e^{-qt} + \frac{p}{q}t - \frac{p}{q^2}

Similarly, based on the initial condition that the displacement is 0, we find C_2 = \frac{p}{q^2}. Thus, we have: v = \dot{s} = \frac{p}{q}(1 - e^{-qt})

Clearly, the maximum velocity is V_{\max} = \frac{p}{q} = \frac{FR}{B^2 L^2}. The result is the same as the classic high school method. However, we find to our surprise that reaching this value requires t \to \infty. In other words, this state can never be reached!

Reflections

Clearly, this example shows that the textbook authors did not think deeply enough and hastily presented this problem, committing a “taken-for-granted” error. BoJone cannot help but raise a question: While it might be too demanding for a Chinese high school student to perform the above analysis, the process is clearly easy for a textbook author to complete (force analysis, solving a linear differential equation). Why then does this “erroneous problem” appear again and again? Is it possible that utilitarianism is enough to overshadow scientific rigor? Or is rigor not required in high school physics? Or are they simply trying to “get away with it” by taking advantage of the fact that high school students do not understand mathematical analysis? (BoJone admits this comment was a bit too aggressive, apologies...)

On the other hand, this also tells us that we should not blindly follow authority. When we possess more scientific knowledge, we might as well look back and re-analyze the problems we once “solved”; we might find unexpected rewards!

Postscript:

Physical laws always have a matter of precision and range of applicability. Although the time in this problem is infinite, as long as a certain precision is given, there exists a finite time within which this precision can be reached, so this problem is not entirely without merit. This article (the original text) was written during my “vigorous and impulsive” high school years. I had just discovered this phenomenon and got a little excited. It was precisely because of this problem that I gained a deeper understanding of the meaning of physics. As the saying goes, “learning is everywhere in life.” (2011.02.12)