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The spectral distribution of radiation from a black body at T1 = 3000 K has a maximum at wavelength λmax. The body cools down to a temperature T2. If the wavelength corresponding to the maximum of the spectral distribution at T2 is 1.2 times of the original wavelength λmax, then the temperature T2 is ________ K (round off to the nearest integer).
Correct answer is '2500'. Can you explain this answer?
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The spectral distribution of radiation from a black body at T1 = 3000 ...
From Wien’s Displacement law,
λmT = constant ⇒ λm1T1 = λm2T2
λm1 × 3000 = 1.2 λm1 × T2
λmT = constant ⇒ λm1T1 = λm2T2
λm1 × 3000 = 1.2 λm1 × T2
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The spectral distribution of radiation from a black body at T1 = 3000 ...
Explanation:
Step 1: Finding the relationship between temperature and peak wavelength
According to Wien's displacement law, the peak wavelength (λmax) of radiation emitted by a black body is inversely proportional to its temperature. Mathematically, this relationship can be expressed as:
λmax ∝ 1/T
Step 2: Given information
Given that the peak wavelength at T2 is 1.2 times the original peak wavelength (λmax), we can write:
λmax(T2) = 1.2 × λmax(T1)
Step 3: Substituting the relationship between temperature and peak wavelength
From Step 1, we know that λmax ∝ 1/T. Therefore, we can rewrite the equation from Step 2 in terms of temperature:
1/T2 = 1.2/T1
Step 4: Finding the temperature T2
Given T1 = 3000 K, we can substitute this value into the equation from Step 3 to find T2:
1/T2 = 1.2/3000
T2 = 1/(1.2/3000)
T2 ≈ 2500 K
Therefore, the temperature T2 is approximately 2500 K.
Step 1: Finding the relationship between temperature and peak wavelength
According to Wien's displacement law, the peak wavelength (λmax) of radiation emitted by a black body is inversely proportional to its temperature. Mathematically, this relationship can be expressed as:
λmax ∝ 1/T
Step 2: Given information
Given that the peak wavelength at T2 is 1.2 times the original peak wavelength (λmax), we can write:
λmax(T2) = 1.2 × λmax(T1)
Step 3: Substituting the relationship between temperature and peak wavelength
From Step 1, we know that λmax ∝ 1/T. Therefore, we can rewrite the equation from Step 2 in terms of temperature:
1/T2 = 1.2/T1
Step 4: Finding the temperature T2
Given T1 = 3000 K, we can substitute this value into the equation from Step 3 to find T2:
1/T2 = 1.2/3000
T2 = 1/(1.2/3000)
T2 ≈ 2500 K
Therefore, the temperature T2 is approximately 2500 K.
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The spectral distribution of radiation from a black body at T1 = 3000 K has a maximum at wavelength λmax. The body cools down to a temperature T2. If the wavelength corresponding to the maximum of the spectral distribution at T2 is 1.2 times of the original wavelength λmax, then the temperature T2 is ________ K (round off to the nearest integer).Correct answer is '2500'. Can you explain this answer?
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