A heated body emits radiation which has maximum intensity at frequency `v_m` If the temperature of the body is doubled:

Options

A. the maximum intensity radiation will be at frequency `2v_(m)`
B. the maximum intensity radiation will be at frequency `(1//2)v_(m)`
C. the total emitted energy will increase by a factor16
D. the total emitted energy will increase by a factor 2

Answer

To solve the problem step by step, we will use Wien's displacement law and Stefan-Boltzmann law.

### Step 1: Understanding Wien's Displacement Law
Wien's displacement law states that the wavelength at which the intensity of radiation is maximum (λ_m) is inversely proportional to the temperature (T) of the body. Mathematically, it can be expressed as:
\[ \lambda_m \propto \frac{1}{T} \]
This means that if the temperature increases, the maximum wavelength decreases.

**Hint:** Remember that Wien's law relates wavelength and temperature inversely.

### Step 2: Relating Wavelength and Frequency
The relationship between wavelength (λ) and frequency (ν) is given by the equation:
\[ v = f \lambda \]
From this, we can deduce that:
\[ \lambda \propto \frac{1}{\nu} \]
This implies that if the wavelength decreases, the frequency increases.

**Hint:** Keep in mind the relationship between frequency and wavelength.

### Step 3: Establishing the Relationship Between Frequency and Temperature
From the two relationships established in Steps 1 and 2, we can combine them. Since λ_m is inversely proportional to T, and λ is inversely proportional to ν, we can conclude that:
\[ \nu \propto T \]
This means that frequency is directly proportional to temperature.

**Hint:** Use the proportionality to connect frequency and temperature.

### Step 4: Doubling the Temperature
If the temperature of the body is doubled (T becomes 2T), then according to our relationship from Step 3:
\[ \nu' = 2 \nu_m \]
This indicates that the new frequency (ν') at which the maximum intensity occurs will also be doubled.

**Hint:** Apply the direct proportionality to find the new frequency.

### Step 5: Energy Emission According to Stefan-Boltzmann Law
According to the Stefan-Boltzmann law, the total energy emitted per unit time (E) is given by:
\[ E \propto A \cdot T^4 \]
Where A is the surface area of the body. If the temperature is doubled (T becomes 2T), the energy emitted will be:
\[ E' \propto A \cdot (2T)^4 = A \cdot 16T^4 \]
This shows that the energy emitted increases by a factor of 16.

**Hint:** Remember that energy emission is related to the fourth power of temperature.

### Conclusion
1. The maximum intensity frequency will be doubled when the temperature is doubled.
2. The total energy emitted will increase by a factor of 16 when the temperature is doubled.

**Final Answers:**
- Maximum intensity frequency: \( 2 \nu_m \) (Option 1 is correct)
- Total energy emitted: 16 times (Option 3 is correct)