Radiation from a black body at the thermodynamic temperature `T_(1)` is measured by a small detector at distance `d_(1)` from it. When the temperature is increased to `T_(2)` and the distance to `d_(2)` , the power received by the detector is unchanged. What is the ratio `d_(2)//d_(1)`?

Options

A. `(T_(2))/(T_(1))`
B. `(T_(2))/(T_(1)^(2))`
C. `((T_(1))/(T_(2)))_(2)^(2)`
D. `((T_(2))/(T_(1)))^(4)`

Explanation

To solve the problem, we need to analyze the relationship between the power received by the detector, the temperature of the black body, and the distance from the black body to the detector.

### Step-by-Step Solution:

1. **Understanding Power and Intensity**:
The power \( P \) received by the detector is given by:
\[ P = I \cdot A \]
where \( I \) is the intensity of radiation at the detector's location and \( A \) is the area of the detector.

2. **Intensity of Radiation**:
The intensity \( I \) at a distance \( d \) from a black body can be expressed as:
\[ I = \frac{P_{\text{radiator}}}{4 \pi d^2} \]
where \( P_{\text{radiator}} \) is the total power radiated by the black body.

3. **Stefan-Boltzmann Law**:
According to the Stefan-Boltzmann law, the power radiated by a black body is given by:
\[ P_{\text{radiator}} = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the area of the radiating surface, and \( T \) is the temperature of the black body.

4. **Setting Up the Equations**:
For the first scenario (temperature \( T_1 \) and distance \( d_1 \)):
\[ I_1 = \frac{\sigma A T_1^4}{4 \pi d_1^2} \]
For the second scenario (temperature \( T_2 \) and distance \( d_2 \)):
\[ I_2 = \frac{\sigma A T_2^4}{4 \pi d_2^2} \]

5. **Equating Intensities**:
Since the power received by the detector remains unchanged, we have:
\[ I_1 = I_2 \]
This leads to:
\[ \frac{\sigma A T_1^4}{4 \pi d_1^2} = \frac{\sigma A T_2^4}{4 \pi d_2^2} \]

6. **Canceling Common Terms**:
The terms \( \sigma \), \( A \), and \( 4 \pi \) can be canceled from both sides:
\[ \frac{T_1^4}{d_1^2} = \frac{T_2^4}{d_2^2} \]

7. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{d_2^2}{d_1^2} = \frac{T_2^4}{T_1^4} \]

8. **Taking the Square Root**:
Taking the square root of both sides, we find:
\[ \frac{d_2}{d_1} = \frac{T_2^2}{T_1^2} \]

9. **Final Ratio**:
Therefore, the ratio \( \frac{d_2}{d_1} \) is:
\[ \frac{d_2}{d_1} = \left(\frac{T_2}{T_1}\right)^2 \]

### Conclusion:
The ratio \( \frac{d_2}{d_1} \) is equal to \( \left(\frac{T_2}{T_1}\right)^2 \).