A black body is at a temperature of 2880 K. The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is`U_(1)`, between 999 nm and 1000 nm is `U_(2)` and between 1499 nm and 1500 nm is `U_(3)`. The Wein's constant `b = 2.88 xx 10^(6) "nm K"`. Then

Options

A. `U_1 = 0`
B. `U_2 = 0`
C. `U_1 = U_2`
D. `U_2 gt U_1`

Answer

To solve the problem, we need to analyze the radiation emitted by a black body at a temperature of 2880 K and compare the energies of radiation emitted in specific wavelength ranges. We will use Wien's Displacement Law to find the wavelength at which the intensity of radiation is maximum.

### Step-by-Step Solution:

**Step 1: Calculate the wavelength at maximum intensity (λ_max)**
According to Wien's Displacement Law, the wavelength at which the intensity is maximum is given by the formula:
\[ \lambda_{\text{max}} = \frac{b}{T} \]
where \( b = 2.88 \times 10^6 \, \text{nm K} \) and \( T = 2880 \, \text{K} \).

**Calculation:**
\[ \lambda_{\text{max}} = \frac{2.88 \times 10^6 \, \text{nm K}}{2880 \, \text{K}} = 1000 \, \text{nm} \]

**Step 2: Analyze the wavelength ranges for U1, U2, and U3**
- \( U_1 \) is the energy emitted between 499 nm and 500 nm.
- \( U_2 \) is the energy emitted between 999 nm and 1000 nm.
- \( U_3 \) is the energy emitted between 1499 nm and 1500 nm.

**Step 3: Determine the position of λ_max relative to U1, U2, and U3**
Since \( \lambda_{\text{max}} = 1000 \, \text{nm} \):
- The wavelength range for \( U_1 \) (499 nm to 500 nm) is far below \( \lambda_{\text{max}} \).
- The wavelength range for \( U_2 \) (999 nm to 1000 nm) is at the peak intensity.
- The wavelength range for \( U_3 \) (1499 nm to 1500 nm) is far above \( \lambda_{\text{max}} \).

**Step 4: Compare the energies**
- Since \( U_2 \) is at the maximum intensity, it will have the highest energy.
- \( U_1 \) is significantly lower than \( \lambda_{\text{max}} \) and thus will have less energy.
- \( U_3 \) is also far from the peak, and its energy will be lower than \( U_2 \) but could be comparable to \( U_1 \).

**Conclusion:**
From the analysis, we can conclude:
- \( U_2 > U_1 \)
- \( U_2 > U_3 \)
- \( U_1 \) and \( U_3 \) will have lower energies than \( U_2 \).

Thus, the correct answer is that \( U_2 \) is greater than \( U_1 \).