Problem 45 - University Physics with Modern Physics

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.

Step-by-Step Solution

Answer

Answer: The net force on the solar sail is approximately 3.258 x 10^-6 N.

Step 1: Determine the incident power on the sail

The cosmic microwave background has a temperature of 2.725 K, and considering it as an ideal black body, we can use the Stefan-Boltzmann law to determine the incident power on the sail. The Stefan-Boltzmann law states: $$P = \sigma T^4$$ where \(P\) is the power per unit area, \(\sigma = 5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\) is the Stefan-Boltzmann constant, and \(T\) is the temperature of the black body. Plugging the values, we get: $$P = (5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}) (2.725 \mathrm{K})^4 = 3.1042 \times 10^{-6} \mathrm{W m^{-2}}$$

Step 2: Calculate the absorbed power on the sail

Next, we need to find the absorbed power on the sail. Since the incident radiation falls on both sides of the sail, we must consider both the perfectly reflecting side and the totally absorbing side. For the absorbing side, all the incident power is absorbed, so the absorbed power per unit area is the same as the incident power: $$P_\mathrm{absorbed} = P$$ For the reflecting side, no power is absorbed, so the absorbed power per unit area is zero: $$P_\mathrm{reflected} = 0$$

Step 3: Calculate the pressure difference on the sail

The pressure difference on the sail due to the absorbed and reflected power can be calculated using the following formula: $$\Delta P = \frac{1}{c} \left(P_\mathrm{absorbed} - P_\mathrm{reflected} \right)$$ where \(c\) is the speed of light, which is approximately \(3 \times 10^8 \mathrm{m s^{-1}}\). Plugging in the values, we get: $$\Delta P = \frac{1}{3 \times 10^8 \mathrm{m s^{-1}}} (3.1042 \times 10^{-6} \mathrm{W m^{-2}}) = 1.0347 \times 10^{-14} \mathrm{Pa}$$

Step 4: Calculate the net force on the sail

Finally, we can find the net force on the sail using the pressure difference and the sail area: $$F_\mathrm{net} = \Delta P \times A$$ The sail is a circle with a radius of \(10 \mathrm{km}\), so its area is given by: $$A = \pi R^2 = \pi (10 \times 10^3 \mathrm{m})^2 = 3.1416 \times 10^8 \mathrm{m^2}$$ Now, we can calculate the net force: $$F_\mathrm{net} = (1.0347 \times 10^{-14} \mathrm{Pa}) \times (3.1416 \times 10^8 \mathrm{m^2}) = 3.258 \times 10^{-6} \mathrm{N}$$ The net force on the solar sail due to its reflection and absorption of the cosmic microwave background radiation is approximately \(3.258 \times 10^{-6} \mathrm{N}\).

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is an essential concept in physics when it comes to understanding how radiation is emitted by a black body. A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle. The law gives us a way to calculate the total radiant heat energy emitted from a surface per unit area per unit time, based on its temperature. This is expressed mathematically as:

• \( P = \sigma T^4 \)

where \( P \) represents the power emitted per unit area, \( \sigma \) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\)), and \( T \) is the temperature in Kelvin.
This law helps in determining how much energy a black body at a certain temperature emits, which is particularly useful in astrophysics and climate science. In the context of solar sails, it allows us to calculate how cosmic microwave background radiation, seen as ideal black body radiation, interacts with the sail.

Cosmic Microwave Background

The Cosmic Microwave Background (CMB) radiation is a snapshot of the oldest light in our universe, a relic from the Big Bang that pervades all of space. At a temperature of about 2.725 K, it is considered a perfect black body radiation. This means it fits the predictions of the Stefan-Boltzmann law seamlessly, making it simpler to compute the energy it could impart on objects like a solar sail.
The CMB is critical in cosmology as it provides evidence about the early universe, containing tiny fluctuations that correspond to regions of differing densities, eventually leading to galaxy formation. For objects like solar sails traveling through space, the CMB can cause a measurable force due to its radiation, influencing the trajectory over long periods. While minute, this force can accumulate in a way that needs to be accounted for in spacecraft propulsion.

Radiation Pressure

Radiation pressure is the pressure exerted upon any surface due to the exchange of momentum between the surface and electromagnetic radiation. When radiation strikes a surface and is either absorbed or reflected, it transfers momentum, resulting in a force. This concept is vital for understanding the impact of light or other electromagnetic waves on surfaces, particularly in space applications like solar sails.
For perfectly absorbing surfaces, the pressure can be calculated by using the formula:

• \( F = \frac{P}{c} \)

where \( F \) is the force, \( P \) is the power, and \( c \) is the speed of light. When the radiation is perfectly reflected, an additional factor of 2 is considered due to the change in direction and momentum of the radiation. Thus, radiation pressure is crucial in designing sails that utilize light for propulsion, as it directly influences the net force experienced by the sail.

Black Body Radiation

Black body radiation refers to the electromagnetic radiation emitted by a black body at thermodynamic equilibrium. It gives a perfect thermal emission spectrum, which means it has no gaps or lines, occurring over a wide range of wavelengths.
Understanding black body radiation involves considering Planck’s law, which describes the spectral density of electromagnetic radiation emitted by a black body:

• \( B(u, T) = \frac{2hu^3}{c^2 (e^{\frac{hu}{kT}} - 1)} \)

where \( B(u, T) \) is the spectral radiance, \( u \) is the frequency, \( T \) is the temperature, \( h \) is Planck’s constant, \( c \) is the speed of light, and \( k \) is Boltzmann's constant. The concept is pivotal for studying heat and energy transfer across different systems in space. For the solar sail, black body radiation from the cosmic microwave background represents a baseline force that acts upon the sail evenly from every direction.