If the temperature of hot black body is raised by 5%, rate of heat energy radiated would be increased by how much percentage ?

Options

A. `12%`
B. `22%`
C. `32%`
D. `42%`

Explanation

To solve the problem of how much the rate of heat energy radiated by a hot black body increases when its temperature is raised by 5%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Stefan-Boltzmann Law**:
The rate of heat energy radiated by a black body is given by the formula:
\[ E = \sigma T^4 \]
where \( E \) is the energy radiated, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the absolute temperature in Kelvin.

2. **Initial Temperature**:
Let the initial temperature of the black body be \( T \). Thus, the initial rate of heat energy radiated is:
\[ E_{\text{initial}} = \sigma T^4 \]

3. **Final Temperature**:
If the temperature is raised by 5%, the new temperature \( T' \) becomes:
\[ T' = T + 0.05T = 1.05T \]

4. **Calculate the Final Rate of Heat Energy**:
The rate of heat energy radiated at the new temperature is:
\[ E_{\text{final}} = \sigma (T')^4 = \sigma (1.05T)^4 \]
Expanding this gives:
\[ E_{\text{final}} = \sigma (1.05^4) T^4 \]

5. **Calculate \( 1.05^4 \)**:
We need to calculate \( 1.05^4 \):
\[ 1.05^4 \approx 1.21550625 \]
Thus, we can write:
\[ E_{\text{final}} \approx \sigma (1.2155) T^4 \]

6. **Determine the Increase in Energy**:
The increase in energy \( \Delta E \) is given by:
\[ \Delta E = E_{\text{final}} - E_{\text{initial}} = \sigma (1.2155 T^4) - \sigma T^4 \]
Simplifying this, we get:
\[ \Delta E = \sigma T^4 (1.2155 - 1) = \sigma T^4 (0.2155) \]

7. **Calculate the Percentage Increase**:
The percentage increase in the rate of heat energy radiated is:
\[ \text{Percentage Increase} = \left( \frac{\Delta E}{E_{\text{initial}}} \right) \times 100 = \left( \frac{0.2155 \sigma T^4}{\sigma T^4} \right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} \approx 21.55\% \]

8. **Final Answer**:
Rounding to the nearest whole number, the increase in the rate of heat energy radiated is approximately **22%**.