Question

Two stars A and B radiate maximum energy at $3600{A^ \circ }$and $4800{A^ \circ }$ respectively. Then the ratio of absolute temperature of A and B is:

A. $256:81$

B. $81:256$

C. $4:3$

D. $3:4$

Answer 1

Hint: In this problem, Wein’s displacement law will be used. It is useful for determining the temperature of hot objects such as stars. Hotter objects emit radiation that has a short wavelength. That is why they appear to be blue. This law was derived in 1893.

Complete step by step solution:

As per this law, black body radiation has different peaks of temperature at wavelengths. It is inversely proportional to the temperature. The wavelength of the peak of the blackbody helps in measuring the temperature.

Using Wein displacement law

\[{\lambda _m}T = {b_0}\]-(i)

Where ${b_0}$ is the Wein’s displacement constant and $T$ is the temperature.

The above equation (i) can be written as

${\lambda _m} = \dfrac{{{b_0}}}{T}$

Given that there are two stars A and B radiate maximum energy. The ratio of their absolute temperature is written by

$ \Rightarrow \dfrac{{{\lambda _A}}}{{{\lambda _B}}} = \dfrac{{{T_B}}}{{{T_A}}}$

Substituting the values of temperatures, in the above equation

$\dfrac{{{\lambda _A}}}{{{\lambda _B}}} = \dfrac{{4800}}{{3600}}$

$ \Rightarrow \dfrac{{{\lambda _A}}}{{{\lambda _B}}} = \dfrac{4}{3}$

The ratio of the absolute temperatures of A and B is

$\therefore {T_A}:{T_B} = 4:3$

Therefore, option (C) is the right answer.

Note: It is important to remember that the cool objects emit radiations of longer wavelengths. The objects appear to be reddish. A black body is a substance that has the ability to absorb or emit all the frequencies of light. It is the relationship between the temperature and the wavelength at which it emits the light. When the temperature increases, the energy that is radiated will also increase. This law is also used in determining the temperature of any radiant object that has a higher temperature than its surroundings.