A hollow copper sphere & a hollow copper cube of same surface area & negligible thickness , are filled with warm water of same temperature and placed in an enclosure of constant temperature a few degrees below that of the bodies. Then in the beginning:-

Options

A. The rate of energy lost by the sphere is greater than that by the cube
B. The rate of energy lost by the two are equal
C. The rate of energy lost by the sphere is less than that by the cube
D. The rate of fall of temperature for sphere is less than that for the cube

Explanation

To solve the problem, we need to analyze the rate of energy loss and the rate of fall of temperature for a hollow copper sphere and a hollow copper cube that have the same surface area and are filled with warm water at the same temperature.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a hollow copper sphere and a hollow copper cube.
- Both have the same surface area and negligible thickness.
- They are filled with warm water at the same temperature and placed in an enclosure with a constant temperature slightly lower than that of the bodies.

2. **Rate of Energy Loss**:
- According to Stefan-Boltzmann law, the rate of heat loss (energy loss) from a body is given by:
\[ \frac{dQ}{dt} = \epsilon \sigma A (T^4 - T_s^4) \]
where:
- \( \epsilon \) is the emissivity (which is the same for both since they are made of copper),
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area (same for both),
- \( T \) is the temperature of the body,
- \( T_s \) is the surrounding temperature.

- Since both the sphere and cube have the same surface area and are at the same initial temperature, the rate of energy loss will be the same for both.

3. **Rate of Fall of Temperature**:
- The rate of fall of temperature can be analyzed using the formula:
\[ \frac{dT}{dt} = -\frac{1}{mc} \frac{dQ}{dt} \]
where:
- \( m \) is the mass of the body,
- \( c \) is the specific heat capacity of water (same for both).

- Since the rate of energy loss is the same, the rate of fall of temperature will depend on the mass of the water inside each body.

4. **Comparing Masses**:
- The mass of water inside each body can be calculated based on their volumes.
- The volume of the sphere \( V_s \) is given by:
\[ V_s = \frac{4}{3} \pi R^3 \]
- The volume of the cube \( V_c \) is given by:
\[ V_c = A^3 \]
- Given that both have the same surface area, we can derive the relationship between the radius \( R \) of the sphere and the side length \( A \) of the cube.

5. **Calculating Volume Ratio**:
- Since both shapes have the same surface area, we can express the surface area of the sphere and cube:
- For the sphere: \( A_s = 4\pi R^2 \)
- For the cube: \( A_c = 6A^2 \)
- Setting these equal gives us a relationship between \( R \) and \( A \).

6. **Conclusion on Temperature Fall**:
- The sphere, having a larger volume due to its shape, will contain more water than the cube.
- Therefore, since the mass of water in the sphere is greater, the rate of fall of temperature in the sphere will be less than that in the cube.

### Final Result:
- The rate of energy loss is the same for both the sphere and the cube.
- The rate of fall of temperature is less for the sphere compared to the cube due to the larger mass of water it contains.