The temperature of a radiation body increases by `30 %`. Then the increase in the amount of radiation is

Options

A. `185 %`
B. `285%`
C. `325%`
D. `130 %`

Explanation

To solve the problem of how much the amount of radiation increases when the temperature of a radiating body increases by 30%, we can follow these steps:

### Step 1: Define the Initial Temperature
Let the initial temperature of the body be \( T \).

### Step 2: Calculate the Final Temperature
The temperature increases by 30%, so the final temperature \( T' \) can be calculated as:
\[ T' = T + 0.30T = 1.30T \]

### Step 3: Use Stefan-Boltzmann Law
According to the Stefan-Boltzmann law, the amount of radiation emitted by a body is proportional to the fourth power of its absolute temperature. This can be expressed as:
\[ E \propto T^4 \]
Let \( E \) be the initial amount of radiation at temperature \( T \) and \( E' \) be the amount of radiation at temperature \( T' \).

### Step 4: Write the Radiation Equations
The initial radiation can be expressed as:
\[ E = kT^4 \]
And the final radiation can be expressed as:
\[ E' = k(T')^4 = k(1.30T)^4 \]

### Step 5: Simplify the Final Radiation Expression
Now, substituting \( T' \):
\[ E' = k(1.30^4)T^4 \]
Thus, we can express the ratio of the final radiation to the initial radiation:
\[ \frac{E'}{E} = \frac{k(1.30^4)T^4}{kT^4} = 1.30^4 \]

### Step 6: Calculate \( 1.30^4 \)
Now we need to calculate \( 1.30^4 \):
\[ 1.30^4 = 2.8561 \quad (\text{approximately}) \]

### Step 7: Find the Increase in Radiation
The increase in radiation can be expressed as:
\[ \Delta E = E' - E = (2.8561E - E) = (2.8561 - 1)E = 1.8561E \]

### Step 8: Calculate the Percentage Increase
To find the percentage increase in radiation, we use:
\[ \text{Percentage Increase} = \left(\frac{\Delta E}{E}\right) \times 100 = \left(\frac{1.8561E}{E}\right) \times 100 = 185.61\% \]

### Final Answer
Thus, the increase in the amount of radiation when the temperature of the body increases by 30% is approximately **185.61%**.

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