A hollow and a solid sphere of same material and identical outer surface under identical condition are heated to the same temperature at the same time (both have same e, a):

Options

A. ) In the beginning both will emit equal amount of radiation per unit time
B. In the beginning both will absorb equal amount of radiation per unit time
C. Both spheres will have same rate of fall of temperature `(dT)//dt)`
D. Both spheres will have equal temperatures at any moment

Explanation

To solve the problem, we need to analyze the behavior of a hollow sphere and a solid sphere made of the same material, having identical outer surfaces and heated to the same temperature under identical conditions. We will look at the power emitted, absorption, and the rate of change of temperature.

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Both spheres are made of the same material.
- Both have identical outer surfaces (same radius).
- Both are heated to the same initial temperature.

2. **Power Emitted**:
- The power emitted by a body due to radiation can be expressed using the Stefan-Boltzmann law:
\[ P = \sigma E A T^4 \]
- Here, \(\sigma\) is the Stefan-Boltzmann constant, \(E\) is the emissivity, \(A\) is the surface area, and \(T\) is the absolute temperature.
- Since both spheres have the same radius, their surface areas \(A\) are equal. Therefore, the power emitted by both spheres will be the same, given that \(E\), \(\sigma\), and \(T\) are also the same.

3. **Power Absorbed**:
- Similarly, the power absorbed by both spheres from their surroundings (assuming they are in thermal equilibrium with the surroundings) will also be the same since they are under identical conditions.
- Thus, both spheres will absorb equal amounts of radiation per unit time.

4. **Rate of Change of Temperature**:
- The rate of change of temperature (\( \frac{dT}{dt} \)) can be related to the power emitted and absorbed. The relationship can be expressed as:
\[ P = m c \frac{dT}{dt} \]
- Here, \(m\) is the mass of the sphere and \(c\) is the specific heat capacity of the material.
- Since the mass of the hollow sphere is less than that of the solid sphere (given they have the same outer dimensions), the rate of change of temperature will be different for both spheres. The solid sphere will have a greater mass and thus will change its temperature at a different rate compared to the hollow sphere.

5. **Conclusion**:
- **Option 1**: Both spheres will emit the same power - **Correct**.
- **Option 2**: Both spheres will absorb the same power - **Correct**.
- **Option 3**: Both spheres will have the same rate of change of temperature - **Incorrect**.
- **Option 4**: Both spheres will have equal temperature at any moment - **Incorrect**.

### Final Answer:
Options 1 and 2 are correct.