A solid sphere and a hollow sphere of same material and size are heated to same temperature and allowed to cool in the same surroundings. If the temperature difference between each sphere and its surrounding is T, then

Options

A. the hollow sphere will cool at a faster rate for all, values of T
B. the solid sphere will cool at a faster rate for all value of T
C. both spheres will cool at the same rate for all values of T
D. both spheres will cool at the same rate only for small value of T.

Answer

To solve the problem, we need to analyze the cooling rates of a solid sphere and a hollow sphere made of the same material and size, using Newton's law of cooling.

### Step-by-Step Solution:

1. **Understanding Newton's Law of Cooling**:
According to Newton's law of cooling, the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. Mathematically, it can be expressed as:
\[ \frac{dT}{dt} = -k(T - T_s) \]
where \( T \) is the temperature of the object, \( T_s \) is the temperature of the surroundings, and \( k \) is a constant that depends on the properties of the object and the surroundings.

2. **Mass of the Spheres**:
The mass of the solid sphere is greater than that of the hollow sphere because the solid sphere is filled with material, while the hollow sphere has a cavity inside. The mass \( m \) of an object affects its rate of cooling; specifically, a larger mass will cool more slowly.

3. **Cooling Rate Relation**:
From the cooling law, we can deduce that the rate of temperature change is inversely proportional to the mass of the object:
\[ \frac{dT}{dt} \propto \frac{1}{m} \]
This means that if the mass \( m \) is larger, the rate of temperature change \( \frac{dT}{dt} \) will be smaller.

4. **Comparison of the Two Spheres**:
Since the solid sphere has a greater mass than the hollow sphere, it will have a smaller rate of cooling. Therefore, the hollow sphere will cool faster than the solid sphere.

5. **Conclusion**:
Based on the analysis above, we conclude that the hollow sphere will cool at a faster rate than the solid sphere for all values of temperature difference \( T \).

### Final Answer:
The hollow sphere will cool faster than the solid sphere for all values of \( T \).