The wavelength of maximum emitted energy of a body at 700 K is `4.08 mu m`. If the temperature of the body is raised to 1400 K , the wavelength of maximum emitted energy will be

Options

A. `1.02 mum`
B. `16.32 mu m`
C. `8.16 mu m`
D. `2.04 mu m`

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength of maximum emitted energy (λ_max) is inversely proportional to the temperature (T) of the body. The relationship can be expressed as:

\[ \lambda_{max} \cdot T = K \]

where \( K \) is a constant known as Wien's constant.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial temperature (T1) = 700 K
- Wavelength at T1 (λ1) = 4.08 μm
- New temperature (T2) = 1400 K

2. **Calculate Wien's Constant (K):**
Using the initial conditions:
\[ K = \lambda_1 \cdot T_1 = 4.08 \, \mu m \cdot 700 \, K \]
\[ K = 2856 \, \mu m \cdot K \]

3. **Apply Wien's Law for the New Temperature:**
Now, we need to find the new wavelength (λ2) at the new temperature (T2):
\[ \lambda_{max} \cdot T = K \]
Rearranging for λ2:
\[ \lambda_2 = \frac{K}{T_2} \]
Substituting the values:
\[ \lambda_2 = \frac{2856 \, \mu m \cdot K}{1400 \, K} \]

4. **Calculate λ2:**
\[ \lambda_2 = \frac{2856}{1400} \, \mu m \]
\[ \lambda_2 = 2.04 \, \mu m \]

5. **Final Answer:**
The wavelength of maximum emitted energy at 1400 K is **2.04 μm**.