A black body is at a temperature of $5760 ext{ K}$. The energy of radiation emitted by the body at wavelengt

Question

A black body is at a temperature of $5760 	ext{ K}$. The energy of radiation emitted by the body at wavelength $250 	ext{ nm}$ is $U_1$, at wavelength $500 	ext{ nm}$ is $U_2$ and that at $1000 	ext{ nm}$ is $U_3$. Wien's constant, $b = 2.88 	imes 10^6 	ext{ nm K}$. Which of the following is correct?

Accepted Answer

According to Wien's displacement law, $\lambda_m T = b$, where $\lambda_m$ is the wavelength corresponding to maximum spectral emissive power, $T$ is the absolute temperature, and $b$ is Wien's constant.
Given: $T = 5760 	ext{ K}$ and $b = 2.88 	imes 10^6 	ext{ nm K}$.
$\lambda_m = \frac{b}{T} = \frac{2.88 	imes 10^6}{5760} = 500 	ext{ nm}$.
Therefore, the maximum energy is radiated at a wavelength of $500 	ext{ nm}$.
Since $U_2$ corresponds to the energy radiated at $500 	ext{ nm}$, it is the maximum among the given energies.
Thus, $U_2 > U_1$ and $U_2 > U_3$.
Also, a black body emits continuous radiation over all wavelengths, so $U_1 
eq 0$ and $U_3 
eq 0$.
The only correct statement among the options is $U_2 > U_1$.

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