Calculate the temperature at which a perfect black body radiates at the rate of `1 W cm^(-2)`, value of Stefan's constant,
`sigma = 5.67 xx 10^(-8) W m^(-2)K^(-4)`

Options

A. 576 K
B. 648 K
C. 695 K
D. 766 K

Answer

To calculate the temperature at which a perfect black body radiates at the rate of \(1 \, \text{W cm}^{-2}\), we will use Stefan-Boltzmann Law. The law states that the power radiated per unit area of a black body is directly proportional to the fourth power of its absolute temperature. The formula is given by:

\[ E = \sigma T^4 \]

where:
- \(E\) is the power radiated per unit area (in \( \text{W m}^{-2} \)),
- \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}\)),
- \(T\) is the absolute temperature in Kelvin.

### Step 1: Convert the given power from \( \text{W cm}^{-2} \) to \( \text{W m}^{-2} \)

Given:
\[ E = 1 \, \text{W cm}^{-2} \]

To convert \( \text{W cm}^{-2} \) to \( \text{W m}^{-2} \):
\[ 1 \, \text{W cm}^{-2} = 1 \, \text{W} \times (100 \, \text{cm/m})^2 = 1 \times 10^4 \, \text{W m}^{-2} \]

So,
\[ E = 10^4 \, \text{W m}^{-2} \]

### Step 2: Substitute the values into the Stefan-Boltzmann equation

Now we can substitute \(E\) and \(\sigma\) into the equation:
\[ 10^4 = 5.67 \times 10^{-8} T^4 \]

### Step 3: Solve for \(T^4\)

Rearranging the equation to solve for \(T^4\):
\[ T^4 = \frac{10^4}{5.67 \times 10^{-8}} \]

### Step 4: Calculate \(T^4\)

Calculating the right-hand side:
\[ T^4 = \frac{10^4}{5.67 \times 10^{-8}} \approx 1.76 \times 10^{12} \]

### Step 5: Calculate \(T\)

Now, take the fourth root to find \(T\):
\[ T = (1.76 \times 10^{12})^{1/4} \]

Calculating \(T\):
\[ T \approx 648 \, \text{K} \]

### Final Answer

Thus, the temperature at which a perfect black body radiates at the rate of \(1 \, \text{W cm}^{-2}\) is approximately:
\[ T \approx 648 \, \text{K} \]

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