Is a Thomson's lamp physically realistic? [closed]

Question

The Thomson's lamp is a philosophical puzzle that can be describe as follows:

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes.

Is it possible to have such a lamp?, and if it is possible is the lamp switch on or off after exactly two minutes?

Would the final state be different if the lamp had started out being on, instead of off?

Answer 1

If you count the number of times the switch is flicked, then when the number is even, the lamp is off, and when it's odd, the lamp is on.

So we can rephrasing your question: is infinity even or odd? That's one for mathematicians... they will probably say "both".

So the short answer is - there is no "real" answer to your question. But most likely the lamp will be off because there is no switch that will survive infinitely many actuations.

AFTERTHOUGHT

On the other hand, from a pure physics perspective, the average time that the lamp is powered during the last fraction of a second is 2/3 of the time: looking at the last two seconds, and assuming we start with the light ON, the total time it's ON will be given by the infinite series:

$$1 + \frac14 + \left(\frac14\right)^2 + ...=\\ \frac{1}{1-\frac14}= \frac43 s$$

So the lamp is ON for $\frac43$ of 2 seconds, or $\frac23$ of the time. But if we start at t=1 seconds (i.e. with the lamp OFF) we find that the lamp is OFF for the last $\frac23$ of the time. This is a strange result! We conclude that the lamp is in fact "half on, half off" at the end of the two minutes - in fact if you think of the power supply as a switched mode supply, the average power will in fact have to converge to 1/2.

So with that reasoning, the lamp is literally "half on, half off". That is actually the conclusion that mathematicians reach as well; but physically, the lamp's intensity will be exactly what it would have if you applied the voltage for half the time. And the lamp won't appear to flicker at all - any circuit has a finite inductance, and in the limit of high frequency, the current will absolutely be steady to any measurable accuracy.

Note that weird things happen to currents in lamps when you given them "voltage for half the time": the resistance of tungsten filaments is a pretty strong function of temperature, and so the average current will be more than half the full current; this also means that the power dissipation will be more than half of the "fully on" amount.

This means that you would need more information about the circuit to determine the actual intensity of the bulb at the end of the period - even assuming the switch is infinitely reliable, and this super-task can actually be performed (which Thomson claimed it could not).

Interesting stuff.

Answer 2

I think you might consider first special relativity.

We can model the problem as being in Minkowski spacetime with cartessian coordinates and put the switch at $(0,0)$ and the lamp in the coordinate $(0,L)$. Where $L$ is the distance from the switch to the lamp. Then the question is what is the state of the lamp at $(120,L)$?

Using a spacetime diagram you can notice that the point $(120,0)$ which is the state of the switch at $t=120s$ is not in causal contact with the state of the lamp which is the point $(120,L)$. In fact the last causal interaction between the switch and the lamp happened at $120-\frac{L}{c}$. So the state of the lamp will be given by the state of the switch at $(120-\frac{L}{c},0)$.

Also you might consider that eventually the scale of times where the process of changing the switch takes place as you are approaching $t=120$ is comparable to the Planck time $ t_P \equiv \sqrt{\frac{\hbar G}{c^5}}\approx 5.39106 (32) \times 10^{-44} \mbox{ s} $. Then for processes that occur in a time $t$ less than one Planck time, dimensional analysis suggests that the effects of both quantum mechanics and gravity will be important under these circumstances. So basically if you can change the switch that fast you might be probing quantum gravity effects.