If the temperature of the Sun were to increase from T to 2T and its radius from R to 2R. The rat io of power, radiated by it would become

Options

A. 64 times
B. 16 times
C. 32 times
D. 4 times

Explanation

To solve the problem of finding the ratio of power radiated by the Sun when its temperature increases from T to 2T and its radius increases from R to 2R, we can use the Stefan-Boltzmann law. According to this law, the power radiated by a black body is proportional to the fourth power of its temperature and the surface area of the body.

### Step-by-Step Solution:

1. **Understand the formula for power radiated**:
The power radiated \( P \) by a body is given by:
\[ P = A \cdot \sigma \cdot T^4 \]
where \( A \) is the surface area, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the temperature.

2. **Calculate the initial power radiated by the Sun**:
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi R^2 \]
Therefore, the initial power radiated by the Sun \( P \) when the temperature is \( T \) and radius is \( R \) is:
\[ P = 4\pi R^2 \cdot \sigma \cdot T^4 \]

3. **Calculate the new power radiated when temperature and radius increase**:
When the temperature increases to \( 2T \) and the radius increases to \( 2R \), the new surface area \( A' \) becomes:
\[ A' = 4\pi (2R)^2 = 4\pi \cdot 4R^2 = 16\pi R^2 \]
The new power radiated \( P' \) is then:
\[ P' = A' \cdot \sigma \cdot (2T)^4 = 16\pi R^2 \cdot \sigma \cdot 16T^4 = 256\pi R^2 \cdot \sigma \cdot T^4 \]

4. **Find the ratio of the new power to the initial power**:
Now, we can find the ratio of the new power \( P' \) to the initial power \( P \):
\[ \text{Ratio} = \frac{P'}{P} = \frac{256\pi R^2 \cdot \sigma \cdot T^4}{4\pi R^2 \cdot \sigma \cdot T^4} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{256}{4} = 64 \]

5. **Conclusion**:
The ratio of the power radiated by the Sun after the changes in temperature and radius is:
\[ \text{Ratio} = 64 \]

### Final Answer:
The ratio of the power radiated by the Sun when its temperature increases from \( T \) to \( 2T \) and its radius from \( R \) to \( 2R \) is \( 64 \).

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