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Stefan's law tells gives an expression for thermal radiation emitted per unit time by a body of surface area A and temperature $T$

$$ u = \sigma A e T^4$$

In my book, it is written that in thermal equilibrium the energy of a body radiated out by stefan's law is equal to the energy radiated out. So, if a body is initially at temperature $T_o$ then it's absorbed heat is given by :

$$ u = \sigma A e (T_o)^4$$

And now suppose the body has it's temperature raised but room temperature is constant, then the energy radiated out per unit time is $ u = e \sigma A T^4$ and the absorbed is said to be $ u_o = e \sigma (T_o)^4$

Now, this is where I'm confused, why is the heat absorbed out at thermal equilibrium the same amount which is absorbed out when the temperature of the body is raised? Is there a proof for this?


Reference: In Concepts of Physics part-2 by H.C.-Verma

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  • $\begingroup$ " why is the heat absorbed out at thermal equilibrium the same amount which is absorbed out when the temperature of the body is raised? Is there a proof for this?" This is only true if the emissivity and the area of the body do not change with the increase in temperature. Why is it so? Because the room radiation towards the object is independent of the body's temperature. $\endgroup$ Oct 19 '20 at 8:01
  • $\begingroup$ ...the room radiation? $\endgroup$
    – Buraian
    Oct 19 '20 at 8:02
  • $\begingroup$ The radiation of the environment of the body. $\endgroup$ Oct 19 '20 at 9:32
  • $\begingroup$ You gave the conditions but you haven't explained why it is true in the first case $\endgroup$
    – Buraian
    Oct 19 '20 at 9:49
  • $\begingroup$ The environment is assumed to be so much bigger than the object that the temperature of said environment does not change when the body is heated up. It is similar to the case of the temperature of reservoirs in thermodynamics, which are assumed to be constant through thermodynamics processes. $\endgroup$ Oct 19 '20 at 10:21
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If the question is why the energy absorbed per unit time is the same irrespective of the body's temperature, then the answer is: this is not strictly true, because emissivity $\epsilon$ and area of the surface of the body $A$ depend on temperature. But if the temperature changes are small, this effect may be too small to make a difference and the simple model with $\epsilon$ and $A$ independent of temperature is used.

If the temperature changes by a lot, then emissivity $\epsilon$ may change substantially enough so that the simple model may not be accurate enough.

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  • $\begingroup$ I made two terrible typos at the end. Check the edit please. $\endgroup$
    – Buraian
    Oct 19 '20 at 3:10
  • $\begingroup$ Could you give a reference for where I could read more about this? $\endgroup$
    – Buraian
    Oct 23 '20 at 10:10
  • $\begingroup$ I don't know any reference off-hand that would discuss this approximation. I would look into classic works on heat radiation (such as M. Planck: The theory of heat radiation, 1914 and references given there) and available textbooks on that topic, and papers on thermal radiation of real bodies. See e.g. irsm.cas.cz/materialy/cs_content/2016_doi/… $\endgroup$ Oct 28 '20 at 19:05

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