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I've constructed a 25cm x 25cm hot plate out of an aluminum plate with flat nichrome wire under it as a resistive element.

I blackened the top with a pitch black spray paint, and placed it at the base of a 23cm x 23cm x 23cm styrofoam box. I lined the inside walls and ceiling of the box with aluminum foil, but the top of the hot plate remains black.

At room temperature the plate is 24.4C or so. According to the Stefan Boltzmann Law, it's thus emitting 23.513W of thermal radiation.

However, when I run 13.997W of current through it and let it attain a steady state, the steady state temperature is 77.6C. According to the same law, it's now emitting 45.4W of radiation.

So I've increased the rate of energy input into the system by 13.997W, but now it's radiating 21.89W more energy as a result! That's about a 1.56x multiplier effect.

How does this not violate the laws of physics? It seems to violate the first law of thermodynamics in that it's generating more energy than I put in, or perhaps the second law as it's being heated more by its own heat reflected from within the box?

However, it actually happened, so I know that it must not be!

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    $\begingroup$ The black in question is probably not black at all! Especially in range for the (supposedly) maximum wave length, were the bulk of the power should come from. Maybe you should try to measure the emitted "light" (highly IR) as well to see if the assumption of black body is actually fulfilled $\endgroup$
    – Confuse-ray30
    Commented Apr 9 at 16:43
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    $\begingroup$ Aluminium foil is very far from a black body and will heat up to a far greater temperature when forced to get into radiative equilibrium. $\endgroup$
    – naturallyInconsistent
    Commented Apr 9 at 16:50
  • $\begingroup$ The emissivity of aluminum varies greatly with surface characteristics (clean in vacuum, oxidized, etc.), ranging from roughly 0.03 to 0.3 $\endgroup$
    – Jon Custer
    Commented Apr 9 at 16:53
  • $\begingroup$ @Confuse-ray30: I tested it by going outside with the plate and an IR thermometer. Without painting it, the thermometer on the plate essentially reads the sky temperature (i.e. what it's reflecting). But after blackening it, it essentially reads its own temperature instead $\endgroup$
    – Cloudyman
    Commented Apr 9 at 17:02
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    $\begingroup$ I meant you should read out the emission spectrum, which als determines the (after integration over "all frequencies") the total amount of power. I don't know what you meant with IR thermometer. The temperature is independent of the... well, whatever IR here means. $\endgroup$
    – Confuse-ray30
    Commented Apr 10 at 8:33

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I've increased the rate of energy input into the system by 13.997W, but now it's radiating 21.89W more energy as a result!

That would be surprising if all of the energy were leaving the system. But this is just a raw calculation of the amount of energy emitted from the plate. Some of that is staying inside the box and reflecting back.

How does this not violate the laws of physics

Before you turned on the wire, the plate was emitting 25W. But that was only because there was also 25W being absorbed by it from the surroundings.

When you turned on the wire, the temperature in the box rose and more radiation fell on the plate. So it's a lot more incoming radiation plus the 14W wire.

But the net heat flow from the box to the surroundings will be 14W. The more you insulate the box, the higher the temperature needs to be before that much power escapes.

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