A solid sphere and a hollow sphere of the same material and of equal radii are heated to the same temperature

Options

A. both will emit equal amount of radiation per unit time in the beginning
B. both will absorb equal amount of radiation per second from the surroundings in the beginning
C. the initial rate of cooling will be the same for both the spheres
D. the two spheres will have equal temperatures at any instant

Explanation

To solve the problem regarding the solid sphere and the hollow sphere of the same material and equal radii heated to the same temperature, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two spheres: one solid and one hollow, made of the same material and having the same radius.
- Both spheres are heated to the same initial temperature.

2. **Applying Stefan-Boltzmann Law**:
- According to the Stefan-Boltzmann law, the power radiated by a body is proportional to the fourth power of its absolute temperature and is given by:
\[ P = \sigma A T^4 \]
where \( P \) is the power radiated, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the absolute temperature.

3. **Surface Area Calculation**:
- The surface area \( A \) of both spheres can be calculated using the formula for the surface area of a sphere:
\[ A = 4\pi r^2 \]
- Since both spheres have the same radius \( r \), they will have the same surface area.

4. **Radiation Emission**:
- Since both spheres are made of the same material and have the same surface area and temperature, they will emit radiation at the same rate initially:
\[ P_{\text{solid}} = P_{\text{hollow}} \]

5. **Mass Consideration**:
- The solid sphere has more mass than the hollow sphere because the hollow sphere has a cavity. This means the solid sphere will have a greater thermal inertia.

6. **Rate of Cooling**:
- Due to the difference in mass, the rate at which the temperature of the two spheres changes will differ. The solid sphere, having more mass, will cool down more slowly than the hollow sphere.

7. **Conclusion**:
- Initially, both spheres will radiate and absorb heat at the same rate, but as time progresses, the hollow sphere will cool down faster than the solid sphere due to its lesser mass.

### Final Answer:
- Both spheres will emit equal amounts of radiation per unit time initially, but their rates of cooling will differ due to the difference in mass.