The surface temperature of a hot body is `1227^(@)`C. Find the wavelength at which it radiates maximum energy. Given Wien's constant = 0.2892 cm.K.

Explanation

To find the wavelength at which a hot body radiates maximum energy, we can use Wien's Displacement Law. Here are the steps to solve the problem:

### Step 1: Convert the temperature from Celsius to Kelvin
The given surface temperature of the hot body is \( 1227^\circ C \). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Substituting the given temperature:
\[ T = 1227 + 273 = 1500 \, K \]

### Step 2: Write down Wien's Displacement Law
Wien's Displacement Law states that the wavelength \( \lambda_m \) at which the radiation is maximum is inversely proportional to the temperature \( T \):
\[ \lambda_m \cdot T = b \]
where \( b \) is Wien's constant.

### Step 3: Substitute the value of Wien's constant
The value of Wien's constant given in the problem is \( b = 0.2892 \, \text{cm} \cdot K \). To use it in our calculations, we convert it to meters:
\[ b = 0.2892 \, \text{cm} \cdot K = 0.2892 \times 10^{-2} \, \text{m} \cdot K \]

### Step 4: Rearrange the equation to solve for \( \lambda_m \)
From Wien's Displacement Law, we can express \( \lambda_m \) as:
\[ \lambda_m = \frac{b}{T} \]

### Step 5: Substitute the values into the equation
Now we can substitute the values of \( b \) and \( T \) into the equation:
\[ \lambda_m = \frac{0.2892 \times 10^{-2} \, \text{m} \cdot K}{1500 \, K} \]

### Step 6: Calculate \( \lambda_m \)
Performing the calculation:
\[ \lambda_m = \frac{0.2892 \times 10^{-2}}{1500} = 1.93 \times 10^{-6} \, \text{m} \]

### Conclusion
The wavelength at which the hot body radiates maximum energy is:
\[ \lambda_m \approx 1.93 \, \mu m \, (or \, 1930 \, nm) \]

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