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 consant, `b = 2.88 xx 10^(6) nmK`. Which of the following is correct?

Options

A. `U_(3)` = 0
B. `U_(1)gtU_(2)`
C. `U_(2)gtU_(1)`
D. `U_(1) = 0`

Explanation

To solve the problem, we will use Wien's law and the properties of black body radiation.

### Step-by-Step Solution:

1. **Understand Wien's Law**:
Wien's law states that the wavelength at which the emission of a black body spectrum is maximized (λm) is inversely proportional to the temperature (T) of the body. The relationship is given by:
\[ \lambda_m \cdot T = b \]
where \( b \) is Wien's constant.

2. **Given Values**:
- Temperature \( T = 5760 \, K \)
- Wien's constant \( b = 2.88 \times 10^6 \, nm \cdot K \)

3. **Calculate λm**:
To find the wavelength at which the maximum energy is emitted, we rearrange the formula:
\[ \lambda_m = \frac{b}{T} \]
Substituting the given values:
\[ \lambda_m = \frac{2.88 \times 10^6 \, nm \cdot K}{5760 \, K} \]
Performing the calculation:
\[ \lambda_m \approx 500 \, nm \]

4. **Identify Energy Emission**:
The problem states that:
- \( U_1 \) is the energy emitted at \( 250 \, nm \)
- \( U_2 \) is the energy emitted at \( 500 \, nm \)
- \( U_3 \) is the energy emitted at \( 1000 \, nm \)

According to the properties of black body radiation:
- The energy emitted at the peak wavelength (which is \( U_2 \) at \( 500 \, nm \)) will be the maximum.
- Energy decreases as we move away from the peak wavelength.

5. **Comparison of Energies**:
Since \( 500 \, nm \) is the wavelength of maximum emission:
\[ U_2 > U_1 \quad \text{and} \quad U_2 > U_3 \]
Therefore, we can conclude:
\[ U_2 > U_1 > U_3 \]

6. **Conclusion**:
Based on the analysis, the correct relationship among the energies is:
- \( U_2 \) is the maximum energy,
- \( U_1 \) is greater than \( U_3 \),
- \( U_3 \) is the least.

### Final Answer:
The correct option is that \( U_2 > U_1 > U_3 \).

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