Two solid spheres of radii `R_(1)` and `R_(2)` are made of same material and have similar surface. The spheres are raised to the same temperature and then allowed to cool under identical conditions. Assuming spheres to be perfect conductors of heat, their initial ratio of rates of loss of heat is:

Options

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

Explanation

To solve the problem, we need to find the initial ratio of rates of loss of heat for two solid spheres with radii \( R_1 \) and \( R_2 \).

### Step-by-Step Solution:

1. **Understanding Heat Loss**:
The rate of heat loss from a body can be described by the Stefan-Boltzmann law, which states that the power (rate of heat loss) is proportional to the surface area and the fourth power of the temperature difference. For perfect black body radiators, the formula is given by:
\[ P = \sigma A (T^4) \]
where \( P \) is the power (rate of heat loss), \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the absolute temperature.

2. **Surface Area of the Spheres**:
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi R^2 \]
Therefore, for the two spheres, the surface areas will be:
\[ A_1 = 4\pi R_1^2 \quad \text{and} \quad A_2 = 4\pi R_2^2 \]

3. **Rate of Heat Loss for Each Sphere**:
The rate of heat loss for each sphere can be expressed as:
\[ P_1 = \sigma A_1 (T^4) = \sigma (4\pi R_1^2) (T^4) \]
\[ P_2 = \sigma A_2 (T^4) = \sigma (4\pi R_2^2) (T^4) \]

4. **Finding the Ratio of Heat Loss**:
To find the ratio of the rates of heat loss, we take the ratio \( \frac{P_1}{P_2} \):
\[ \frac{P_1}{P_2} = \frac{\sigma (4\pi R_1^2) (T^4)}{\sigma (4\pi R_2^2) (T^4)} \]
The \( \sigma \), \( 4\pi \), and \( T^4 \) terms cancel out, leading to:
\[ \frac{P_1}{P_2} = \frac{R_1^2}{R_2^2} \]

5. **Conclusion**:
Therefore, the initial ratio of rates of loss of heat for the two spheres is:
\[ \frac{P_1}{P_2} = \frac{R_1^2}{R_2^2} \]