A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
(A) U1 = 0
(B) U3 = 0
(C) U1>U2
(D) U2>U1
The Correct Option is D
Approach Solution - 1
Let λm be the wavelength corresponding to the maximum energy emitted by the black body at 2880k.
Using Wien's displacement law,
λmT = b
⇒ λm × 2880K = 2.88 × 106 nm-k
⇒ λm = (2.88 × 106 nm-k)/2880K
⇒ λm = 1000 nm
Thus, the maximum energy emitted by the star at temperature 2880K is corresponding to the wavelength of 1000 nm.
Graph of energy versus wavelength of the black body is shown below:
In the above graph, λ corresponds to the point A. This clearly shows that U is the point where energy is maximum and U > U3 or U1Hence, the correct option is (D).
Approach Solution -2
Wien’s Displacement Law
According to Wein’s displacement law, the wavelength of the maximum intensity of emission of black body radiation is inversely proportional to the absolute temperature of the black body i.e.
λm ∝ 1/T
⇒ λm = b/T
Where
- λm = wavelength of maximum intensity of radiation
- T = absolute temperature
- b = Wien’s constant or Wien’s displacement constant
Wien’s Constant
Aa physical constant that establishes a relationship between the thermodynamic temperature and wavelength of a black body is known as Wien’s constant.
- It is a combination of temperature and the wavelength of the black body, which becomes shorter as the temperature rises and the wavelength approaches its maximum.
- The value of Wien’s displacement constant is 2.898 x 10-3 meter kelvin (m K).
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