The temperature of a black body is increased by 50% , then the percentage of increases of radiation is approximetaly

Options

A. `100%`
B. `25%`
C. `400 %`
D. `500 %`

Explanation

To solve the problem, we need to determine the percentage increase in radiation when the temperature of a black body is increased by 50%. We will use Stefan's Law, which states that the energy radiated by a black body is proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Define Initial Temperature**:
Let the initial temperature \( T_1 \) be represented as 100% (or simply \( T_1 \)).

2. **Calculate New Temperature**:
If the temperature is increased by 50%, the new temperature \( T_2 \) can be calculated as:
\[ T_2 = T_1 + 0.5 \times T_1 = 1.5 \times T_1 \]
Thus, \( T_2 = \frac{3}{2} T_1 \).

3. **Apply Stefan's Law**:
According to Stefan's Law, the energy radiated \( E \) is proportional to \( T^4 \):
\[ E \propto T^4 \]
Therefore, the ratio of the energies at the two temperatures can be expressed as:
\[ \frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4 \]
Substituting \( T_2 = \frac{3}{2} T_1 \):
\[ \frac{E_2}{E_1} = \left(\frac{3/2}{1}\right)^4 = \left(\frac{3}{2}\right)^4 = \frac{81}{16} \]

4. **Calculate the Increase in Radiation**:
To find the percentage increase in radiation, we use the formula:
\[ \text{Percentage Increase} = \frac{E_2 - E_1}{E_1} \times 100 \]
Substituting \( E_2 = \frac{81}{16} E_1 \):
\[ \text{Percentage Increase} = \frac{\frac{81}{16} E_1 - E_1}{E_1} \times 100 = \left(\frac{81}{16} - 1\right) \times 100 \]
Simplifying:
\[ = \left(\frac{81 - 16}{16}\right) \times 100 = \frac{65}{16} \times 100 \]

5. **Final Calculation**:
Now, calculate \( \frac{65}{16} \):
\[ \frac{65}{16} = 4.0625 \]
Therefore, the percentage increase in radiation is approximately:
\[ 4.0625 \times 100 = 406.25\% \]
Rounding off, we find that the percentage increase in radiation is approximately **400%**.

### Conclusion:
The percentage increase in radiation when the temperature of a black body is increased by 50% is approximately **400%**.