If the temperature of the body is increased by 10%, the percentage increase in the emitted radiation will be

Options

A. `46%`
B. `40%`
C. `30%`
D. `80%`

Answer

To solve the problem of finding the percentage increase in emitted radiation when the temperature of a body is increased by 10%, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Stefan's Law**:
According to Stefan's Law, the emissive power (P) of a body is proportional to the fourth power of its absolute temperature (T). This can be expressed mathematically as:
\[ P = \sigma T^4 \]
where \(\sigma\) is the Stefan-Boltzmann constant.

2. **Initial Temperature**:
Let the initial temperature of the body be \(T\).

3. **Increased Temperature**:
If the temperature is increased by 10%, the new temperature \(T'\) can be calculated as:
\[ T' = T + 0.1T = 1.1T \]

4. **Calculating New Emissive Power**:
The new emissive power \(P'\) at the increased temperature \(T'\) is given by:
\[ P' = \sigma (T')^4 = \sigma (1.1T)^4 \]
Expanding this, we have:
\[ P' = \sigma (1.1^4) T^4 \]

5. **Finding the Ratio of New to Old Emissive Power**:
We can now find the ratio of the new emissive power to the old emissive power:
\[ \frac{P'}{P} = \frac{\sigma (1.1^4) T^4}{\sigma T^4} = 1.1^4 \]

6. **Calculating \(1.1^4\)**:
Now we need to calculate \(1.1^4\):
\[ 1.1^4 = 1.4641 \quad (\text{approximately } 1.464) \]

7. **Calculating the Percentage Increase**:
The percentage increase in emitted power can be calculated as:
\[ \text{Percentage Increase} = \frac{P' - P}{P} \times 100 = \left(\frac{1.464P - P}{P}\right) \times 100 \]
Simplifying this gives:
\[ \text{Percentage Increase} = (1.464 - 1) \times 100 = 0.464 \times 100 = 46.4\% \]

8. **Final Answer**:
Therefore, the percentage increase in the emitted radiation is approximately **46.4%**.