A heated body maitained at T K emits thermal radiation of total energy E with a maximum intensity at frequency v. The emissivity of the material is 0.5. If the temperature of the body be increased and maintained at temperature 3T K, then :-
(i) The maximum intensity of the emitted radiation will occur at frequency v/3
(ii) The maximum intensity of the emitted radiation will occur at frequency 3v.
(iii) The total energy of emitted radiation will become 81 E
(iv) The total energy of emitted radiation will become 27 E

Options

A. i & ii
B. ii & iii
C. i & iv
D. i, ii & iv

Answer

To solve the problem, we need to analyze the effects of increasing the temperature of a heated body on the frequency of maximum intensity and the total energy of emitted radiation.

### Step-by-Step Solution:

1. **Understanding Wien's Displacement Law**:
- Wien's Displacement Law states that the wavelength of maximum intensity (λ_max) is inversely proportional to the temperature (T).
- Mathematically, it can be expressed as:
\[ \lambda_{\text{max}} \propto \frac{1}{T} \]
- Since frequency (ν) is inversely related to wavelength (ν = c/λ), we can infer that frequency is directly proportional to temperature:
\[ \nu \propto T \]

2. **Calculating the New Frequency**:
- Let the initial temperature be T and the initial frequency be ν.
- When the temperature is increased to 3T, we can express the new frequency (ν') as:
\[ \frac{\nu}{\nu'} = \frac{T}{3T} \implies \nu' = 3\nu \]
- Therefore, the maximum intensity of the emitted radiation will occur at frequency **3ν**.

3. **Understanding the Total Energy of Emitted Radiation**:
- The total energy (E) emitted by a black body is proportional to the fourth power of its temperature:
\[ E \propto T^4 \]
- If the initial energy at temperature T is E, then at temperature 3T, the new energy (E') can be calculated as:
\[ E' = E \left(\frac{3T}{T}\right)^4 = E \cdot 3^4 = E \cdot 81 \]
- Thus, the total energy of emitted radiation will become **81E**.

### Summary of Results:
- The maximum intensity of the emitted radiation will occur at frequency **3ν** (Option ii).
- The total energy of emitted radiation will become **81E** (Option iii).

### Final Answers:
- (ii) The maximum intensity of the emitted radiation will occur at frequency **3ν**.
- (iii) The total energy of emitted radiation will become **81E**.

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