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I understand that a water heater element is basically just adding energy to the mass of water in the container. So does that mean that the heater element only has to keep adding energy while the water around it becomes hotter than the element itself? Or does the element warm up together with the water until both reach 100C?

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It is the same answer as to the question: "does a room heater element warm up to 20C together with the air in the room?" A simpler and painless experiment to carry out. –  anna v Nov 18 '11 at 15:13
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up vote 3 down vote accepted

The water will not warm up, if the heater has the same temperature as the water. It needs to be hotter than the current water temperature. You can not "add" thermal energy to the water by connecting an object that is in thermal equilibrium with the water.

In practice the element will get very hot without any water cooling it and it will eventually switch off to prevent any damage. If you add water this process is slowed down a lot, as the water has a high heat capacity but until the heater switches off it will always be warmer than the water.

The amount of heat flowing per time depends on the temperature difference as stated by Fourier's law: $$\partial Q \propto \Delta T$$

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this: ""The amount of heat flowing per time depends on the temperature difference as stated by Fourier's law:"" is not wrong entiely, but rather. For such a heater the heat flow is determined by the electric power fed into the heater. The temperature difference adjusts to the electical power. –  Georg Nov 18 '11 at 15:48
    
@Georg: So I personally believe that Fourier's law is correct. Entirely. That the temperature difference is a function of the thermal energy put into the heater does not change that. –  Alexander Nov 18 '11 at 17:38
    
This is not about Fouriers law, it is a question of causality in this case. Its like load independent current circuits. –  Georg Nov 18 '11 at 19:11
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