Question

A solar sail is a giant circle (with a radius \(R=10.0 \mathrm{~km}\) ) made of a material that is perfectly reflecting on one side and totally absorbing on the other side. In deep space, away from other sources of light, the cosmic microwave background will provide the primary source of radiation incident on the sail. Assuming that this radiation is that of an ideal black body at \(T=2.725 \mathrm{~K}\), calculate the net force on the sail due to its reflection and absorption. Also assume that any heat transferred to the sail will be conducted away, and that the photons are incident perpendicular to the surface of the sail.

Short answer

Answer: The net force acting on the solar sail due to its reflection and absorption of cosmic microwave background radiation is approximately \(4.92 \times 10^{-14} \mathrm{N}\).

Step-by-step solution

Calculate the power of black body radiation

To calculate the power of the black body radiation, we use the Stefan-Boltzmann law. The equation for the power radiated per unit area by a black body at temperature T is given by: $$P = \sigma T^4$$ where \(\sigma = 5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\) is the Stefan-Boltzmann constant and T is the temperature. First, we calculate the power per unit area of the cosmic microwave background radiation: $$P = \sigma T^4 = (5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}})(2.725 \mathrm{K})^4 = 3.13 \times 10^{-14} \mathrm{W m^{-2}}$$

Calculate the total power incident on the solar sail

Now, we calculate the total area of the solar sail, which is a circle with radius \(R = 10.0 \mathrm{km}\). The area of the solar sail is given by: $$A = \pi R^2 = \pi (10.0 \times 10^3 \mathrm{m})^2 = 3.1416 \times 10^8 \mathrm{m^2}$$ Next, we calculate the total power incident on the solar sail, by multiplying the power per unit area by the total area of the sail: $$P_{total} = P \times A = (3.13 \times 10^{-14} \mathrm{W m^{-2}})(3.1416 \times 10^8 \mathrm{m^2}) = 9.83 \times 10^{-6} \mathrm{W}$$

Calculate the net force on the solar sail

To calculate the net force on the solar sail, we need to find the change in momentum of the absorbed and reflected photons. The momentum of a photon is given by: $$p = \frac{E}{c}$$ where \(E\) is the energy of the photon and \(c \approx 3 \times 10^8 \mathrm{m s^{-1}}\) is the speed of light. Now, since the solar sail is perfectly reflecting on one side and totally absorbing on the other side, for each photon absorbed, the change in momentum is \(p\), and for each photon reflected, the change in momentum is \(2p\). Therefore, the average change in momentum per photon is \(\frac{3}{2}p\). For the total change in momentum per unit time (which is force F), we have: $$F = \frac{3}{2}\frac{P_{total}}{c} = \frac{3}{2}\frac{9.83 \times 10^{-6} \mathrm{W}}{3 \times 10^8 \mathrm{m s^{-1}}} = 4.92 \times 10^{-14} \mathrm{N}$$ The net force acting on the solar sail due to its reflection and absorption of the cosmic microwave background radiation is approximately \(4.92 \times 10^{-14} \mathrm{N}\).