A heated body emits radiation which has maximum intensity near the frequency `v_(0)` The emissivity of the material is `0.5` . If the absolute temperature of the body is doubled,

Options

A. the maximum intensity of radiation will be near the frequency `2v_(0)`
B. the maximum intensity of radiation will be near the frequency `v_(0)/2`
C. the total energy emitted will increases by a factor of `16`
D. the total energy emitted will increases by a factor of `8`.

Explanation

To solve the problem step by step, we will analyze the effects of doubling the absolute temperature of a heated body on the frequency of maximum intensity of radiation emitted and the total energy emitted.

### Step 1: Understanding Wien's Displacement Law
Wien's Displacement Law states that the product of the wavelength of maximum intensity (λ_max) and the absolute temperature (T) of a black body is a constant. Mathematically, it is expressed as:
\[ \lambda_{max} \cdot T = b \]
where \( b \) is Wien's displacement constant.

### Step 2: Relating Wavelength and Frequency
The relationship between wavelength (λ) and frequency (ν) is given by:
\[ \lambda = \frac{c}{\nu} \]
where \( c \) is the speed of light. Therefore, we can express the maximum wavelength in terms of frequency:
\[ \lambda_{max} = \frac{c}{\nu_{max}} \]
This means we can relate the maximum frequency and temperature using the displacement law.

### Step 3: Setting Up the Initial and Final Conditions
Let:
- \( T_1 \) be the initial temperature
- \( T_2 = 2T_1 \) be the final temperature after doubling
- \( \nu_1 \) be the initial frequency of maximum intensity
- \( \nu_2 \) be the final frequency of maximum intensity

From Wien's law, we have:
\[ \lambda_{1} \cdot T_1 = \lambda_{2} \cdot T_2 \]

### Step 4: Expressing Frequencies in Terms of Temperatures
Using the relationship between wavelength and frequency, we can write:
\[ \frac{c}{\nu_1} \cdot T_1 = \frac{c}{\nu_2} \cdot T_2 \]
Cancelling \( c \) from both sides gives:
\[ \frac{T_1}{\nu_1} = \frac{T_2}{\nu_2} \]
Rearranging this gives:
\[ \frac{\nu_2}{\nu_1} = \frac{T_2}{T_1} \]

### Step 5: Substituting the Temperature Values
Substituting \( T_2 = 2T_1 \) into the equation:
\[ \frac{\nu_2}{\nu_1} = \frac{2T_1}{T_1} = 2 \]
This implies:
\[ \nu_2 = 2\nu_1 \]

### Step 6: Conclusion on Maximum Frequency
Thus, when the absolute temperature is doubled, the frequency of maximum intensity of radiation emitted by the body becomes:
\[ \nu_2 = 2\nu_0 \]
where \( \nu_0 \) is the initial frequency. Therefore, the maximum intensity of radiation will be near the frequency \( 2\nu_0 \).

### Step 7: Analyzing Total Energy Emitted
According to the Stefan-Boltzmann Law, the total energy emitted by a black body is proportional to the fourth power of its absolute temperature:
\[ E \propto T^4 \]
If we denote the initial energy as \( E_1 \) at temperature \( T_1 \) and the final energy as \( E_2 \) at temperature \( T_2 \):
\[ \frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4 = \left(\frac{2T_1}{T_1}\right)^4 = 2^4 = 16 \]
Thus, the total energy emitted will increase by a factor of 16.

### Final Answers
1. The maximum intensity of radiation will be near the frequency \( 2\nu_0 \).
2. The total energy emitted will increase by a factor of 16.