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 use Stefan's Law. Here’s a step-by-step solution:

### Step 1: Understand Stefan's Law
According to Stefan's Law, the rate of heat energy (H) radiated by a black body is directly proportional to the fourth power of its absolute temperature (T):
\[ H \propto T^4 \]
This can be expressed mathematically as:
\[ H = A \sigma T^4 \]
where:
- \( H \) = rate of heat energy radiated,
- \( A \) = area of the emitting surface,
- \( \sigma \) = Stefan-Boltzmann constant,
- \( T \) = absolute temperature.

### Step 2: Calculate the New Temperature
If the temperature is increased by 5%, the new temperature \( T' \) can be calculated as:
\[ T' = T + 0.05T = 1.05T \]

### Step 3: Calculate the New Rate of Heat Energy
Using Stefan's Law, the new rate of heat energy \( H' \) at the new temperature \( T' \) is:
\[ H' = A \sigma (T')^4 \]
Substituting \( T' \):
\[ H' = A \sigma (1.05T)^4 \]

### Step 4: Expand the Expression
Now, we can expand the expression:
\[ H' = A \sigma (1.05^4) T^4 \]
Thus, we can express the new heat energy in terms of the original heat energy \( H \):
\[ H' = H \cdot (1.05^4) \]

### Step 5: Calculate the Ratio of New to Old Heat Energy
Now, we can find the ratio of the new rate of heat energy to the original rate of heat energy:
\[ \frac{H'}{H} = 1.05^4 \]

### Step 6: Calculate the Percentage Increase
To find the percentage increase in the rate of heat energy, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{H' - H}{H} \right) \times 100 \]
Substituting the ratio we found:
\[ \text{Percentage Increase} = \left( 1.05^4 - 1 \right) \times 100 \]

### Step 7: Calculate \( 1.05^4 \)
Calculating \( 1.05^4 \):
\[ 1.05^4 \approx 1.21550625 \]
Thus,
\[ \text{Percentage Increase} = (1.21550625 - 1) \times 100 \approx 21.55\% \]

### Step 8: Round the Result
Rounding to the nearest whole number gives us approximately:
\[ \text{Percentage Increase} \approx 22\% \]

### Final Answer
The rate of heat energy radiated would increase by approximately **22%**.

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