A 100 W bulb has a tungsten filament with a total length of 1.0 m and a radius of 4 10⁻⁵ m . The emissivity of the filament is 0.8 , and = 6.0 10⁻⁸ W m ⁻² K ⁻⁴ . Calculate the temperature of the filament when the bulb is operating at its correct wattage.

Step-by-step solution

The power radiated by the filament is given by the Stefan-Boltzmann law: P = e A T^4 The surface area of the cylindrical filament is: A = 2 r L Substituting the given values: A = 2 (4 10⁻⁵) 1.0 = 8 10⁻⁵ m ^2 Substituting A into the power equation: 100 = 0.8 (6.0 10⁻⁸) (8 10⁻⁵) T^4 100 = 38.4 10⁻¹³ T^4 T^4 = 100 38.4 10⁻¹³ T^4 = 10¹⁵ 120.64 8.29 10¹² K ^4 Taking the fourth root: T (8.29)^ 1/4 10^3 K Since 1.7^4 = 8.3521 , we get: T 1.7 10^3 K = 1700 K