The temperature of the tungsten filament of a 40 watt lamp is `1655^(@)C.` The effective surface area of the filament is `0.85 cm^(2).` Assuming that the energy radiated from the filament is 60% of that of a black body radiator at the same temperature, find the value of Stefan's constant.

Answer

To find the value of Stefan's constant based on the given parameters, we will follow these steps:

### Step 1: Convert the temperature from Celsius to Kelvin
The temperature of the tungsten filament is given as \( 1655^\circ C \). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Substituting the given temperature:
\[ T = 1655 + 273 = 1928 \, K \]

### Step 2: Convert the surface area from cm² to m²
The effective surface area of the filament is given as \( 0.85 \, cm^2 \). To convert this to square meters, we use the conversion:
\[ 1 \, cm^2 = 10^{-4} \, m^2 \]
Thus,
\[ A = 0.85 \, cm^2 = 0.85 \times 10^{-4} \, m^2 = 8.5 \times 10^{-5} \, m^2 \]

### Step 3: Determine the emissivity of the filament
It is given that the energy radiated from the filament is 60% of that of a black body radiator. Therefore, the emissivity \( \epsilon \) can be calculated as:
\[ \epsilon = \frac{60}{100} = 0.6 \]

### Step 4: Use Stefan-Boltzmann Law to find Stefan's constant
According to Stefan-Boltzmann Law, the power radiated by a body is given by:
\[ P = \sigma A \epsilon T^4 \]
Where:
- \( P \) is the power (40 W),
- \( \sigma \) is Stefan's constant,
- \( A \) is the surface area,
- \( \epsilon \) is the emissivity,
- \( T \) is the temperature in Kelvin.

Rearranging the formula to solve for \( \sigma \):
\[ \sigma = \frac{P}{A \epsilon T^4} \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:
- \( P = 40 \, W \)
- \( A = 8.5 \times 10^{-5} \, m^2 \)
- \( \epsilon = 0.6 \)
- \( T = 1928 \, K \)

Calculating \( T^4 \):
\[ T^4 = (1928)^4 = 1.928 \times 10^3 \, K^4 \approx 1.44 \times 10^{12} \, K^4 \]

Now substituting these values:
\[ \sigma = \frac{40}{(8.5 \times 10^{-5}) \times 0.6 \times (1.44 \times 10^{12})} \]

### Step 6: Calculate Stefan's constant
Calculating the denominator:
\[ (8.5 \times 10^{-5}) \times 0.6 \times (1.44 \times 10^{12}) \approx 7.344 \times 10^{7} \]
Now substituting this back into the equation for \( \sigma \):
\[ \sigma \approx \frac{40}{7.344 \times 10^{7}} \approx 5.45 \times 10^{-12} \, W/m^2K^4 \]

### Final Result
Thus, the value of Stefan's constant \( \sigma \) is approximately:
\[ \sigma \approx 5.45 \times 10^{-12} \, W/m^2K^4 \]