The radiation emitted by a black body at a temperature of 2618 K with the wavelength between 599 nm and 600 nm is `U_a` , between 1099 nm and 1100 nm is` _b`U and between 1599 nm and 1600 nm is `U_c` . If the Wien's constant is `b = 2.88 xx 10^6 nm K` then,

Options

A. `U_a ne 0 `
B. `U_c = 0`
C. `U_b gt U_a`
D. `U_b gt U_c`

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength at which the radiation emitted by a black body is maximum (λ_max) is inversely proportional to its temperature (T). The formula is given by:

\[ \lambda_{max} \cdot T = b \]

Where:
- \( \lambda_{max} \) is the wavelength at maximum emission,
- \( T \) is the absolute temperature in Kelvin,
- \( b \) is Wien's constant.

Given:
- Temperature \( T = 2618 \, K \)
- Wien's constant \( b = 2.88 \times 10^6 \, nm \cdot K \)

### Step 1: Calculate the wavelength at maximum emission (λ_max)

Using the formula:

\[ \lambda_{max} = \frac{b}{T} \]

Substituting the values:

\[ \lambda_{max} = \frac{2.88 \times 10^6 \, nm \cdot K}{2618 \, K} \]

Calculating this gives:

\[ \lambda_{max} \approx 1100 \, nm \]

### Step 2: Analyze the given ranges of wavelengths

The problem states the following ranges:
- \( U_a \) for wavelengths between 599 nm and 600 nm,
- \( U_b \) for wavelengths between 1099 nm and 1100 nm,
- \( U_c \) for wavelengths between 1599 nm and 1600 nm.

### Step 3: Compare the values of \( U_a \), \( U_b \), and \( U_c \)

Since \( λ_{max} \) is approximately 1100 nm, we can conclude:
- The range for \( U_b \) (1099 nm to 1100 nm) is where the maximum radiation occurs.
- The range for \( U_a \) (599 nm to 600 nm) is much lower than \( λ_{max} \).
- The range for \( U_c \) (1599 nm to 1600 nm) is higher than \( λ_{max} \).

### Step 4: Determine the relationships between \( U_a \), \( U_b \), and \( U_c \)

1. Since \( U_b \) is at the maximum wavelength, it will emit more radiation than both \( U_a \) and \( U_c \).
2. \( U_a \) will have some radiation since it is not zero, but it will be less than \( U_b \).
3. \( U_c \) will also have some radiation, but it will be less than \( U_b \) because it is further from the maximum wavelength.

### Conclusion

From our analysis, we can conclude:
- \( U_a \) is not equal to 0 (there will be some radiation).
- \( U_c \) is also not equal to 0 (there will be some radiation).
- \( U_b \) is greater than both \( U_a \) and \( U_c \).

### Final Relationships:
- \( U_b > U_a \)
- \( U_b > U_c \)