Consider a 150-W incandescent lamp. The filament of the lamp is \(5-\mathrm{cm}\) long and has a diameter of \(0.5 \mathrm{~mm}\). The diameter of the glass bulb of the lamp is \(8 \mathrm{~cm}\). Determine the heat flux, in W/ \(\mathrm{m}^{2},(a)\) on the surface of the filament and \((b)\) on the surface of the glass bulb, and (c) calculate how much it will cost per year to keep that lamp on for eight hours a day every day if the unit cost of electricity is \(\$ 0.08 / \mathrm{kWh}\).

The filament is a cylinder with length \(l=5\mathrm{~cm}=0.05\mathrm{~m}\) and diameter \(d=0.5\mathrm{~mm}=0.0005\mathrm{~m}\). The radius of the cylinder is \(r=\frac{d}{2}=0.00025\mathrm{~m}\). The surface area of a cylinder (not including the bases) is given by the formula \(A_\mathrm{filament}=2\pi rh\), where \(h\) is the height (length) of the cylinder. So, we have: \(A_\mathrm{filament}=2\pi(0.00025\mathrm{~m})(0.05\mathrm{~m}) \approx 7.854 \times 10^{-5}\mathrm{~m}^2\)

The heat flux is the power divided by the surface area. Since we are given the power of the lamp as \(150\mathrm{~W}\), the heat flux on the surface of the filament is: \(q_\mathrm{filament}=\frac{150\mathrm{~W}}{7.854 \times 10^{-5}\mathrm{~m}^2} \approx 1,909,842\mathrm{~W/m}^2\)

The glass bulb is a sphere with diameter \(D=8\mathrm{~cm}=0.08\mathrm{~m}\). The radius of the sphere is \(R=\frac{D}{2}=0.04\mathrm{~m}\). The surface area of a sphere is given by the formula \(A_\mathrm{sphere}=4\pi R^2\). So, we have: \(A_\mathrm{glass_bulb}=4\pi(0.04\mathrm{~m})^2 \approx 2.0106 \times 10^{-2}\mathrm{~m}^2\)

Using the same formula for heat flux, the heat flux on the surface of the glass bulb is: \(q_\mathrm{glass_bulb}=\frac{150\mathrm{~W}}{2.0106 \times 10^{-2}\mathrm{~m}^2} \approx 7,463\mathrm{~W/m}^2\)

The lamp is on for 8 hours a day, so the energy consumption per day is: \(E_\mathrm{daily}=150\mathrm{~W} \times 8\mathrm{~h} = 1200\mathrm{~Wh} = 1.2\mathrm{~kWh}\)

The unit cost of electricity is \(0.08\mathrm{~\$/kWh}\). First, calculate the annual energy consumption by multiplying the daily consumption by the number of days in a year: \(E_\mathrm{annual}=1.2\mathrm{~kWh} \times 365 = 438\mathrm{~kWh}\) Then, calculate the cost per year by multiplying the annual energy consumption by the unit cost: \(C_\mathrm{annual}=438\mathrm{~kWh} \times 0.08\mathrm{~\$/kWh} = \$34.84\) So, the answers are: (a) the heat flux on the surface of the filament is approximately \(1,909,842\mathrm{~W/m}^2\), (b) the heat flux on the surface of the glass bulb is approximately \(7,463\mathrm{~W/m}^2\), and (c) the cost per year to keep the lamp on for eight hours a day every day is $34.84.