Question

A black body (heat capacity = $C$ ) at absolute temperature $T$ is kept in a surrounding of absolute temperature $\dfrac{T}{2}$. Its rate of cooling is:
( $\sigma$ is Stefan’s constant)
A. $\dfrac{{\sigma A}}{C}{\left( {\dfrac{T}{2}} \right)^4}$
B. $\dfrac{{\sigma A{T^4}}}{{2C}}$
C. $\dfrac{{15\sigma A{T^4}}}{{2C}}$
D. $\dfrac{{15\sigma A}}{C}{\left( {\dfrac{T}{2}} \right)^4}$

Answer 1

This question is from the topic of Black Body Radiation and we can approach the solution to this question simply by recalling Stefan’s Law of Black Body Radiation and putting in the values that are given to us by the question.

Complete step-by-step solution:
We will try to solve the question exactly as told in the hint section of the solution and will use the Stefan’s Law of Black Body Radiation, which is as follows:
${P_{net}}\, = \,\sigma \varepsilon A\left( {{T^4}\, - \,{T^4}_{env}} \right)$
It can also be written as:
${P_{net}}\, = \,\dfrac{{\sigma A\left( {{T^4}\, - \,{T^4}_{env}} \right)}}{C}$
The question has already given us more than enough information so that we can reach the correct answer to the question.
Let us have a look at the information given to us by the question:
${T_{self}}\, = \,T$
${T_{env}}\, = \,\dfrac{T}{2}$
Putting the following values in the Stefan’s Law of Black Body Radiation, we get:
${P_{net}}\, = \,\dfrac{{\sigma A\left( {{T^4}\, - \,{{\left( {\dfrac{T}{2}} \right)}^4}} \right)}}{C}$
Upon further solving this equation, we get:
${P_{net}}\, = \,\dfrac{{15\sigma A}}{C}{\left( {\dfrac{T}{2}} \right)^4}$
We can see that this answer matches the option (D) of the question.
Hence, the correct option is option (D).

Additional Information:
A body which is a good radiator (or emitter) is also a good absorber. Imagine that radiation of all possible wavelengths is incident on a body, and it absorbs all of them. Such a body would also be capable of emitting all the wavelengths under suitable conditions. Such a perfect absorber is called a black body. The notion of black body is an idealization; in reality no object behaves like a perfect body.
A practical approximation, to a perfect black body, is having a hollow enclosure, maintained at a uniform temperature, opening that is very small compared to its size. It is designed in such a way that any radiation falling on the aperture is internally reflected and absorbed; and has negligible chance of coming out of the enclosure.

Note:- Many students won’t remember the Stefan’s Law of Black Body Radiation, but this and a few other laws and formulae are easy to remember and can gain you easy and fast marks if you can recall them quickly.