Question

The temperature of a furnace is $2324^\circ C$ and the intensity is maximum in its spectrum nearly at \[12000\mathop A\limits^o \]. If the intensity in the spectrum of a star is maximum nearly at $4800\mathop A\limits^o $, then the surface temperature of the star is:
(A) $8400^\circ C$
(B) $6219^\circ C$
(C) $7200^\circ C$
(D) $5900^\circ C$

Answer 1

Hint Wilhelm wein in 1893, found out that the product of temperature of a body and the maximum wavelength or the minimum frequency of the light emerging from that body is a constant. This constant has the same value universally. So, in this question, we need to equate the product of temperature and wavelength of the furnace to the temperature and wavelength of the star.

Complete step by step solution
According to Wein’s displacement law, \[{\lambda _{\max }}T = {\text{constant}} = b\]
where, \[{\lambda _{\max }}\] is the peak wavelength or the wavelength at which the intensity of the spectrum is maximum, at temperature T and b is a constant.
When the temperature of the furnace is $2324^\circ C$, the intensity of the spectrum is maximum at \[12000\mathop A\limits^o \] .
 $ \Rightarrow Peak{\text{ Wavelength}} = {\lambda _{\max }} = 12000\mathop A\limits^o $
Putting this in the equation for wien's displacement law gives us,
 $ \Rightarrow b = 2324^\circ C \times 12000\mathop A\limits^o $
When the temperature is (let’s say), T $^\circ C$, the intensity of the spectrum is maximum at $4800\mathop A\limits^o $ .
 $ \Rightarrow Peak{\text{ Wavelength}} = {\lambda _{\max }} = 4800\mathop A\limits^o $
Putting this in equation for wien's displacement law and using the previously calculated value of b,
 \[
   \Rightarrow 4800\mathop A\limits^o \times T = b = 2324^\circ C \times 12000\mathop A\limits^o \\
   \Rightarrow 4800\mathop A\limits^o \times T = b = \left( {2324 + 273K} \right) \times 12000\mathop A\limits^o \\
   \Rightarrow 4800\mathop A\limits^o \times T = b = 2597K \times 12000\mathop A\limits^o \\
   \Rightarrow T = \dfrac{{2597 \times 12000}}{{4800}} \\
   \Rightarrow T = 6492.5K \\
   \Rightarrow T = (6492.5 - 273)^\circ C \\
   \Rightarrow T = 6219.5^\circ C \\
 \]

Note According to wein’s displacement law, \[{\lambda _{\max }}T = {\text{constant}} = b\]
 \[ \Rightarrow {\lambda _{\max }}_{_1}{T_1} = {\lambda _{\max }}_{_1}{T_2}\]
Alternatively, this equation can directly be used to get a quick result.
Also, while substituting the values of temperature, we must not forget to convert \[^\circ C\] into Kelvin otherwise we will get a misleading/faulty answer.