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Imagine a perfectly insulated box, placed inside the box is an electric heater. The heater is switched on and the box is left to reach equilibrium with its surroundings. What is the final temperature inside the box?

Clearly a real box would eventually reach an equilibrium temperature with its surroundings at the point where the energy being dissipated by the box matched the energy being added by the heater. If no energy can escape though it seems to imply that the temperature will just keep rising forever which feels wrong to me.

I suspect the answer has something to do with the maximum temperature the heating element can achieve before failing. If we had a heating element that never burnt out though what would the final temperature then be?

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I'm actually glad you asked this question. It is a perfect example of how someone's ordinary logic is just plain physically wrong, which is something that most good basic physics teachers warn about. This isn't the only problem I've encountered in my academic career where a steady-state solution was expected only to end with a reevaluation of the conceptual picture. – AlanSE Jul 24 '11 at 20:38

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You have imagined a system with constant heat input and no output. As long as those conditions pertain there is no limit to the temperature.

In practice you reach an ultimate temperature when

  • Some part of your electric circuit breaks or melts and the heater stops.
  • The insulator melts or the ever increasing interior pressure bursts the box open
  • Your approximation of perfect insulation fails and the box is leaking as much heat as is being generated by the heater.
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I agree with the answer to within its assumptions (which match the problem) but the assumptions are impossible. If the there is a mechanism for heat input, there is a mechanism for heat output. If the heating mechanism is shining a laser beam through a hole in the box, then the system can radiate back out through that hole. If the heater is a resistor connected to wires which run outside the box, the electrical conductivity of the wires provides a path for heat to leave. If the there is a mechanism for heat input, there is always a mechanism for heat output. – Anonymous Coward Jul 27 '11 at 20:36
Well, yes, but we can make it hard to identify a channel. I could for instance, couple a generator inside to a inductive rotational coupling, which demands of our unobtainium box that it lets magnetic fields pass. I'm sure there is a mechanism for energy transfer out of the box on that channel, but it is not instantly obvious to me... – dmckee♦ Jul 27 '11 at 23:20

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