A spherical body of 5 cm radius is maintained at a temperature of `327^(@)C`. The wavelength at which maximum energy radiated will be nearly `(b=2.898xx10^(-3)m" "K)`

Options

A. 482 Å
B. 4.82 Å
C. 482 `mu` m
D. 4.82 `mu` m

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength at which the maximum energy is radiated by a black body is inversely proportional to its absolute temperature. The formula for Wien's Displacement Law is given by:

\[ \lambda_m = \frac{b}{T} \]

where:
- \(\lambda_m\) is the wavelength at which maximum energy is radiated,
- \(b\) is Wien's displacement constant, approximately \(2.898 \times 10^{-3} \, \text{m} \cdot \text{K}\),
- \(T\) is the absolute temperature in Kelvin.

### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Celsius is given as \(327^\circ C\). To convert this to Kelvin, we use the formula:

\[ T(K) = T(°C) + 273 \]

Substituting the given value:

\[ T = 327 + 273 = 600 \, K \]

### Step 2: Apply Wien's Displacement Law
Now that we have the temperature in Kelvin, we can substitute the values into Wien's law formula:

\[ \lambda_m = \frac{b}{T} \]

Substituting the values of \(b\) and \(T\):

\[ \lambda_m = \frac{2.898 \times 10^{-3} \, \text{m} \cdot \text{K}}{600 \, K} \]

### Step 3: Calculate the wavelength
Now we perform the division:

\[ \lambda_m = \frac{2.898 \times 10^{-3}}{600} \approx 4.83 \times 10^{-6} \, m \]

### Step 4: Convert to micrometers
To express the wavelength in micrometers (\(\mu m\)), we convert meters to micrometers:

\[ \lambda_m \approx 4.83 \, \mu m \]

### Final Answer
The wavelength at which maximum energy is radiated is approximately \(4.83 \, \mu m\).

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