A 100 W bulb has a tungsten filament of total length 1.0 m and radius $4 imes 10^{-5} \mathrm{~m}$

Question

A 100 W bulb has a tungsten filament of total length 1.0 m and radius $4 	imes 10^{-5} \mathrm{~m}$. The emissivity of the filament is 0.8 and $\sigma=6.0 	imes 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$. If the temperature of the filament when the bulb is operating at the correct wattage is $T \mathrm{~K}$, find the value of $T / 100$ to the nearest integer. (Take $\pi=3.14$ )

Professor PC Thomas Classes · Answer

At a steady state, the electrical power consumed by the bulb is equal to the thermal power radiated by the filament (ignoring the temperature of the surroundings). The power radiated is given by the Stefan-Boltzmann law: $$ P=e \sigma A T^4 $$ The surface area $A$ of the cylindrical filament is: $$ A=2 \pi r L $$ Substituting the given values into the equation: $$ \begin{gathered} 100=0.8 	imes\left(6.0 	imes 10^{-8}ight) 	imes\left(2 	imes 3.14 	imes 4 	imes 10^{-5} 	imes 1.0ight) 	imes T^4 \ 100=120.576 	imes 10^{-13} 	imes T^4 \ T^4=\frac{100}{120.576 	imes 10^{-13}} \approx 8.29 	imes 10^{12} \mathrm{~K}^4 \ T \approx 1698 \mathrm{~K} \end{gathered} $$ Therefore, the value of $T / 100$ is: $$ \frac{1698}{100}=16.98 $$