Final answer:
To find the power usage of the light bulb filament at 2600°C, the Stefan-Boltzmann law is used, incorporating the area of the filament, its emissivity, temperature in Kelvin, and the Stefan-Boltzmann constant, to calculate the power radiated.
Step-by-step explanation:
To calculate the electrical power being used by the light bulb filament at a temperature of 2600 degrees Celsius, we need to apply the Stefan-Boltzmann law for black-body radiation, which states that the power radiated per unit area of a black body is directly proportional to the fourth power of the black body's thermodynamic temperature.
The formula for the Stefan-Boltzmann law is:
P = ε·A·σ·T^4
Where:
P is the power radiated,
ε is the emissivity of the material (0.8 for the filament),
A is the area of the emitting body in square meters (6.45 x 10⁻⁴ m² for the filament),
σ is the Stefan-Boltzmann constant (≈ 5.67 x 10⁻⁸ W/m²K⁴),
T is the absolute temperature in Kelvins (2600°C = 2873K).
We can rearrange this formula to solve for P (the power) and substitute in the values:
P = 0.8 · 6.45 x 10⁻⁴ m² · 5.67 x 10⁻⁸ W/m²K⁴ · (2873K)⁴
Upon calculating, we find that the filament's power usage is significant, which is typical for an incandescent bulb operating at such high temperatures.