A grey body (ε = 0.8) emits the same amount of heat as the black body at 1075 K. The required temperature of the grey body will be _______.
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A.
113.672oC
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B.
1136.72 K
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C.
672oC 2. 1136.72 K 3. 113.672 K
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D.
1136.72oC
Correct Answer:
B. 1136.72 K
Explanation:
To determine the required temperature of the grey body, we use the Stefan-Boltzmann Law, which states that the total energy radiated per unit surface area is proportional to the fourth power of the absolute temperature. For a black body, the emissive power is given by E = σT⁴, while for a grey body, it is E = εσT⁴, where ε is the emissivity.
Given that the grey body (ε = 0.8) emits the same amount of heat as a black body at 1075 K, we set their emissive powers equal:
ε × σ × (T\_grey)⁴ = σ × (T\_black)⁴
By canceling the Stefan-Boltzmann constant (σ) from both sides, the equation simplifies to:
0.8 × (T\_grey)⁴ = (1075)⁴
Solving for T\_grey:
(T\_grey)⁴ = (1075)⁴ / 0.8
T\_grey = 1075 / (0.8)^(1/4)
T\_grey = 1075 / 0.9457
T\_grey ≈ 1136.72 K
Therefore, the required temperature for the grey body to match the heat emission of the black body is 1136.72 K.
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