A black body emits radiations of maximum intensity at a wavelength of 5000 A, when the temperature of the body is 1227° C. If the temperature of the body is increased by 2227 °C, the maximum intensity of emitted radiation would be observed at

Options

A. 2754.8 A
B. 3000 A
C. 3500 A
D. 4000 A

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength of maximum intensity of radiation emitted by a black body is inversely proportional to its absolute temperature. The relationship can be expressed mathematically as:

\[ \lambda \cdot T = b \]

where:
- \(\lambda\) is the wavelength of maximum intensity,
- \(T\) is the absolute temperature in Kelvin,
- \(b\) is Wien's displacement constant.

### Step-by-Step Solution:

1. **Convert the initial temperature to Kelvin:**
The initial temperature is given as \(1227^\circ C\).
To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Thus,
\[ T_1 = 1227 + 273 = 1500 \, K \]

2. **Identify the initial wavelength:**
The initial wavelength (\(\lambda_1\)) is given as \(5000 \, \text{Å}\).

3. **Use Wien's Law to find the constant \(b\):**
Using the relationship:
\[ \lambda_1 \cdot T_1 = b \]
We can substitute the values:
\[ b = 5000 \, \text{Å} \cdot 1500 \, K \]
\[ b = 7500000 \, \text{Å} \cdot K \]

4. **Calculate the new temperature:**
The temperature is increased by \(2227^\circ C\). Therefore, the new temperature (\(T_2\)) is:
\[ T_2 = 1227 + 2227 = 3454^\circ C \]
Converting this to Kelvin:
\[ T_2 = 3454 + 273 = 3727 \, K \]

5. **Use Wien's Law to find the new wavelength (\(\lambda_2\)):**
Now we can use the constant \(b\) to find the new wavelength:
\[ \lambda_2 = \frac{b}{T_2} \]
Substituting the values:
\[ \lambda_2 = \frac{7500000 \, \text{Å} \cdot K}{3727 \, K} \]
\[ \lambda_2 \approx 2018.5 \, \text{Å} \]

6. **Final Result:**
Rounding off, the maximum intensity of emitted radiation would be observed at approximately:
\[ \lambda_2 \approx 2019 \, \text{Å} \]

### Summary:
The maximum intensity of emitted radiation would be observed at a wavelength of approximately **2019 Å**.