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. `R_(1)//R_(2)` must be equal to

Options

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

Explanation

To solve the problem, we need to use the Stefan-Boltzmann law, which states that the power emitted 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 Stefan-Boltzmann Law**:
The power \( P \) emitted 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, and \( T \) is the absolute temperature.

2. **Calculate the Surface Area of Spherical Bodies**:
For a sphere, the surface area \( A \) is given by:
\[ A = 4\pi R^2 \]
where \( R \) is the radius of the sphere.

3. **Set Up the Power Equations for Both Bodies**:
Let the two spherical bodies be denoted as Body 1 and Body 2. Since both bodies radiate the same power, we can write:
\[ P_1 = P_2 \]
This gives us:
\[ \sigma (4\pi R_1^2) T_1^4 = \sigma (4\pi R_2^2) T_2^4 \]
Since \( \sigma \) and \( 4\pi \) are constants, they can be canceled out:
\[ R_1^2 T_1^4 = R_2^2 T_2^4 \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \frac{R_1^2}{R_2^2} = \frac{T_2^4}{T_1^4} \]
Taking the square root of both sides, we have:
\[ \frac{R_1}{R_2} = \frac{T_2^2}{T_1^2} \]

5. **Final Result**:
Thus, the ratio of the radii \( \frac{R_1}{R_2} \) must be equal to:
\[ \frac{R_1}{R_2} = \frac{T_2^2}{T_1^2} \]

### Conclusion:
The ratio \( \frac{R_1}{R_2} \) is equal to \( \frac{T_2^2}{T_1^2} \).