Two spherical black bodies of radii `r_1` and `r_2` and with surface temperature `T_1` and `T_2` respectively radiate the same power. Then the ratio of `r_1` and `r_2` will be

Options

A. `((T_2)/(T_1))^2`
B. `((T_2)/(T_1))^4`
C. `((T_1)/(T_2))^2`
D. `((T_1)/(T_2))^4`

Explanation

To solve the problem, we need to find the ratio of the radii \( r_1 \) and \( r_2 \) of two spherical black bodies that radiate the same power. We will use the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature and its surface area.

### Step-by-Step Solution:

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

2. **Surface Area of a Sphere**:
The surface area \( A \) of a sphere with radius \( r \) is given by:
\[ A = 4\pi r^2 \]

3. **Set Up the Power Equations for Both Bodies**:
For the first body with radius \( r_1 \) and temperature \( T_1 \):
\[ P_1 = \sigma (4\pi r_1^2) T_1^4 \]
For the second body with radius \( r_2 \) and temperature \( T_2 \):
\[ P_2 = \sigma (4\pi r_2^2) T_2^4 \]

4. **Equate the Powers**:
Since both bodies radiate the same power, we have:
\[ P_1 = P_2 \]
This leads to:
\[ \sigma (4\pi r_1^2) T_1^4 = \sigma (4\pi r_2^2) T_2^4 \]

5. **Cancel Common Terms**:
We can cancel \( \sigma \) and \( 4\pi \) from both sides:
\[ r_1^2 T_1^4 = r_2^2 T_2^4 \]

6. **Rearrange the Equation**:
Rearranging gives:
\[ \frac{r_1^2}{r_2^2} = \frac{T_2^4}{T_1^4} \]

7. **Take the Square Root**:
Taking the square root of both sides:
\[ \frac{r_1}{r_2} = \frac{T_2^2}{T_1^2} \]

8. **Final Ratio**:
Thus, the ratio of the radii \( r_1 \) and \( r_2 \) is:
\[ \frac{r_1}{r_2} = \left( \frac{T_2}{T_1} \right)^2 \]

### Conclusion:
The ratio of the radii \( r_1 \) and \( r_2 \) of the two spherical black bodies is given by:
\[ \frac{r_1}{r_2} = \left( \frac{T_2}{T_1} \right)^2 \]