A solid at temperature `T_(1)` is kept in an evacuated chamber at temperature `T_(2)gtT_(1)` . The rate of increase of temperature of the body is propertional to

Options

A. `T_(2)-T_(1)`
B. `T_(2)^(2)-T_(1)^(2)`
C. `T_(2)^(3)-T_(1)^(3)`
D. `T_(2)^(4)-T_(1)^(4)` .

Explanation

To solve the problem, we need to analyze the situation where a solid at a lower temperature \( T_1 \) is placed in an evacuated chamber at a higher temperature \( T_2 \). We want to find out what the rate of increase of temperature of the solid is proportional to.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a solid with an initial temperature \( T_1 \).
- This solid is placed in an evacuated chamber at a higher temperature \( T_2 \) (where \( T_2 > T_1 \)).
- Since the chamber is evacuated, heat transfer occurs only through radiation.

**Hint**: Identify the temperatures involved and their relationship.

2. **Applying the Stefan-Boltzmann Law**:
- According to the Stefan-Boltzmann law, the power (rate of energy transfer) emitted by a body due to thermal radiation is given by:
\[ P = \sigma A T^4 \]
where \( P \) is the power, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the absolute temperature in Kelvin.

**Hint**: Recall the formula for power in terms of temperature and understand that it applies to both the solid and the surrounding.

3. **Rate of Heat Transfer**:
- The solid will absorb radiation from the surrounding at temperature \( T_2 \) and emit radiation at its own temperature \( T_1 \).
- The rate of heat absorbed by the solid from the surroundings is proportional to \( T_2^4 \).
- The rate of heat emitted by the solid is proportional to \( T_1^4 \).

**Hint**: Write down the expressions for the power absorbed and emitted.

4. **Net Rate of Heat Transfer**:
- The net rate of heat transfer (which affects the rate of increase of temperature of the solid) can be expressed as:
\[ \text{Net Rate} \propto T_2^4 - T_1^4 \]
- This indicates that the increase in temperature of the solid is influenced by the difference in the fourth powers of the temperatures.

**Hint**: Think about how the difference in temperatures affects the net heat transfer.

5. **Conclusion**:
- Therefore, the rate of increase of temperature of the body is proportional to \( T_2^4 - T_1^4 \).
- This matches with option 4 from the provided choices.

**Hint**: Compare your final expression with the options given in the question to confirm the answer.

### Final Answer:
The rate of increase of temperature of the body is proportional to \( T_2^4 - T_1^4 \).