The thermal radiation emitted by a body is proportional to `T^(n)` where T is its absolute temperature. The value of n is exactly 4 for

Options

A. a blackbody
B. all bodies
C. bodies painted black only
D. polished bodies only.

Explanation

To solve the question, we need to analyze the relationship between thermal radiation and temperature as described by Stefan-Boltzmann's law.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The thermal radiation emitted by a body is proportional to \( T^n \), where \( T \) is the absolute temperature. We need to determine the value of \( n \) for different types of bodies.

2. **Stefan-Boltzmann Law**:
According to Stefan-Boltzmann's law, the total energy radiated per unit surface area of a black body is given by the equation:
\[ U = \epsilon \sigma A T^4 \]
where:
- \( U \) is the total energy radiated,
- \( \epsilon \) is the emissivity of the body (which varies for different materials),
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the absolute temperature.

3. **Identifying the Black Body**:
A black body is defined as an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. For a perfect black body, the emissivity \( \epsilon = 1 \).

4. **Value of n**:
From the equation, we can see that the thermal radiation emitted by a perfect black body is proportional to \( T^4 \) (since \( \epsilon = 1 \)). Therefore, for a black body, the value of \( n \) is exactly 4.

5. **Analyzing the Options**:
- **a) Black body**: Correct, as we established that \( n = 4 \) for a black body.
- **b) All bodies**: This is misleading; while the Stefan-Boltzmann law applies to all bodies, the value of \( n \) is specifically 4 for black bodies.
- **c) Bodies painted black only**: This does not guarantee \( n = 4 \) as the emissivity may vary.
- **d) Polished bodies only**: Polished bodies typically have low emissivity, hence \( n \) would not be 4.

6. **Conclusion**:
The value of \( n \) is exactly 4 for a black body.

### Final Answer:
The value of \( n \) is exactly 4 for a **black body**.

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