Question

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

• A. $U_3 = 0$
• B. $U_1 > U_2$
• C. $U_2 > U_1$
• D. $U_1 = 0$

Answer 1

The Correct Option is C

Radiation Emission and Wavelength 

We are dealing with the wavelength corresponding to the maximum amount of emitted radiation, which can be derived using **Wien's Displacement Law**:

\(\lambda_m = \frac{b}{T}\)

Step 1: Calculate the Wavelength

Where: - \( \lambda_m \) is the wavelength corresponding to the maximum emitted radiation, - \( b = 2.88 \times 10^6 \, \text{nm K} \) (Wien's constant), - \( T = 5760 \, \text{K} \) (temperature).

Substituting the values into the formula:

\(\lambda_m = \frac{2.88 \times 10^6 \, \text{nm K}}{5760 \, \text{K}} = 500 \, \text{nm}\)

Step 2: Graph Interpretation

From the graph, we can see that the values of energy (represented as \( U_1 \), \( U_2 \), and \( U_3 \)) follow the relationship:

\(U_1 < U_2 > U_3\)

Conclusion:

The maximum wavelength \( \lambda_m \) corresponding to the maximum emitted radiation is \( 500 \, \text{nm} \), and the energy values follow the relationship \( U_1 < U_2 > U_3 \).