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A black body is at a temperature of 5760 K5760 \text{ K}. The energy of radiation emitted by the body at wavelength 250 nm250 \text{ nm} is U1U_1, at wavelength 500 nm500 \text{ nm} is U2U_2 and that at 1000 nm1000 \text{ nm} is U3U_3. Wien's constant, b=2.88×106 nm Kb = 2.88 \times 10^6 \text{ nm K}. Which of the following is correct?

A

U3=0U_3 = 0

B

U1>U2U_1 > U_2

C

U2>U1U_2 > U_1

D

U1=0U_1 = 0

Step-by-Step Solution

According to Wien's displacement law, λmT=b\lambda_m T = b, where λm\lambda_m is the wavelength corresponding to maximum spectral emissive power, TT is the absolute temperature, and bb is Wien's constant. Given: T=5760 KT = 5760 \text{ K} and b=2.88×106 nm Kb = 2.88 \times 10^6 \text{ nm K}. λm=bT=2.88×1065760=500 nm\lambda_m = \frac{b}{T} = \frac{2.88 \times 10^6}{5760} = 500 \text{ nm}. Therefore, the maximum energy is radiated at a wavelength of 500 nm500 \text{ nm}. Since U2U_2 corresponds to the energy radiated at 500 nm500 \text{ nm}, it is the maximum among the given energies. Thus, U2>U1U_2 > U_1 and U2>U3U_2 > U_3. Also, a black body emits continuous radiation over all wavelengths, so U10U_1 \neq 0 and U30U_3 \neq 0. The only correct statement among the options is U2>U1U_2 > U_1.

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