Question

The temperature of a black body increases from T to 2T. The factor by which the rate of emission will increase
A. 2
B. 4
C. 8
D. 16

Answer 1

Hint: Emissive power or rate of emission is defined as the energy radiated by a black body per unit area in one unit of time. It is given by $E=\sigma {{T}^{4}}$. By using this equation, find the rate of emission of the body at temperatures T and 2T. Then find the ratio of the two.

Formula used:
$E=\sigma {{T}^{4}}$

Complete step by step answer:
Every object in the universe emits thermal radiations. A body that emits radiations of all possible wavelengths is called a black body.
The energy radiated by a black body per unit area in one unit of time is called emissive power of the black body. It is also called the rate of emission of the black body. It is denoted by E.
According to Stefan’s law, the rate of emission or emissive power of a black body is directly proportional to the fourth power of its absolute temperature.
i.e. E $\propto {{T}^{4}}$
On introducing a proportionality constant it is given as
$E=\sigma {{T}^{4}}$ ….. (i).
Here, $\sigma $ is the proportionality constant called Stefan’s constant.
It is given that the temperature of a black body is increased from T to 2T.
Let the divide into two cases.
In the first case, the temperature of the black body is T.
According to equation (i), the rate of emission of the black body is $E=\sigma {{T}^{4}}$ ……. (ii).
In the second case, the temperature of the black body is 2T.
Let the rate of emission of the black body at temperature 2T be E’.
This means that the rate of emission of the black body is $E'=\sigma {{\left( 2T \right)}^{4}}=16\sigma {{T}^{4}}$ …… (iii).
Now divide equation (iii) by equation (ii).
This gives us that
$\dfrac{E'}{E}=\dfrac{16\sigma {{T}^{4}}}{\sigma {{T}^{4}}}=16$
$\Rightarrow E'=16E$
This means that the rate of emission of the given black body increases by a factor of 16 when its temperature increases from T to 2T.
Hence, the correct option is D.

Note: The given formula for rate of emission i.e. $E=\sigma {{T}^{4}}$ is applicable for a perfectly black body. However, in reality a perfectly black body does not exist.
Therefore, we defined emissivity of the black body. Emissivity (e) of a black body is the ratio of the total emissive power of the given black body to the total emissive power of a perfectly black body.
The emissive power of a given black body is directly proportional to its emissivity. Thus, equation (i) changes to $E=e\sigma {{T}^{4}}$.