A surface at temperature `T_(0)K` receives power `P` by radiation from a small sphere at temperature `T ltT_(0)` and at a distance d. If both `T` and d are doubled the power received by the surface will become .

Options

A. `8P`
B. `4P`
C. `2P`
D. `P`

Explanation

To solve the problem, we will use the principles of Stefan-Boltzmann law and the inverse square law of radiation.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- A surface at temperature \( T_0 \) receives power \( P \) from a small sphere at temperature \( T \) (where \( T < T_0 \)) at a distance \( d \).

2. **Applying Stefan-Boltzmann Law**:
- According to Stefan-Boltzmann law, the power \( P \) received by the surface is directly proportional to the fourth power of the temperature of the sphere and inversely proportional to the square of the distance from the sphere:
\[ P \propto \frac{T^4}{d^2} \]
- We can express this relationship as:
\[ P = k \frac{T^4}{d^2} \]
- Here, \( k \) is a constant of proportionality.

3. **Changing the Conditions**:
- Now, both the temperature \( T \) and the distance \( d \) are doubled:
\[ T' = 2T \quad \text{and} \quad d' = 2d \]

4. **Calculating the New Power**:
- We need to find the new power \( P' \) received by the surface under these new conditions:
\[ P' = k \frac{(T')^4}{(d')^2} \]
- Substituting the new values:
\[ P' = k \frac{(2T)^4}{(2d)^2} \]

5. **Simplifying the Expression**:
- Now, simplify the expression:
\[ P' = k \frac{16T^4}{4d^2} = k \frac{16}{4} \frac{T^4}{d^2} = 4k \frac{T^4}{d^2} \]
- Since \( k \frac{T^4}{d^2} = P \), we can substitute:
\[ P' = 4P \]

6. **Conclusion**:
- Therefore, the power received by the surface when both \( T \) and \( d \) are doubled becomes:
\[ P' = 4P \]

### Final Answer:
The power received by the surface will become \( 4P \).

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