A 100W incandescent lamp of filament area 45 mm^2, emits all its energy by radiation into the surrounding space of room at 22C. If the filament emissivity is 0.32, find the temperature of the filament?

To find the temperature of the filament, we can use the Stefan-Boltzmann law, which relates the power radiated by an object to its temperature. The law is given by:

P = εσA(T^4 - T0^4)

where:
P is the power radiated by the object (in watts)
ε is the emissivity of the object
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A is the area of the object's surface (in square meters)
T is the temperature of the object (in Kelvin)
T0 is the temperature of the surroundings (in Kelvin)

In this case, we are given:
P = 100W (power of the lamp)
ε = 0.32 (emissivity of the lamp filament)
A = 45 mm^2 = 45 x 10^-6 m^2 (area of the filament)
T0 = 22°C = 22 + 273 = 295K (temperature of the surroundings)

Plugging these values into the formula, we can solve for T:

100 = 0.32 * 5.67 x 10^-8 * 45 x 10^-6 * (T^4 - 295^4)

Dividing both sides of the equation by (0.32 * 5.67 x 10^-8 * 45 x 10^-6), we get:

T^4 - 295^4 = (100 / (0.32 * 5.67 x 10^-8 * 45 x 10^-6))

Simplifying the right side of the equation, we have:

T^4 - 295^4 = 470,585,714,285,714

Taking the fourth root of both sides of the equation, we get:

T = (470,585,714,285,714)^(1/4)

Using a calculator, we find that T ≈ 1446 K.

Therefore, the temperature of the filament is approximately 1446 Kelvin.