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?
Answer
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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