Problem 26 - University Physics with Modern Physics

Question

In the 25-ft Space Simulator facility at NASA's Jet Propulsion Laboratory, a bank of overhead arc lamps can produce light of intensity 2500 W/m\(^2\) at the floor of the facility. (This simulates the intensity of sunlight near the planet Venus.) Find the average radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing section of the floor and (b) a totally reflecting section of the floor. (c) Find the average momentum density (momentum per unit volume) in the light at the floor.

Step-by-Step Solution

Answer

(a) 8.33 x 10^{-6} Pa (8.22 x 10^{-11} atm), (b) 1.67 x 10^{-5} Pa (1.65 x 10^{-10} atm), (c) 2.78 x 10^{-14} kg/m^3.

Step 1: Identify Key Concepts

Understand that radiation pressure is caused by light being absorbed or reflected. When light is absorbed, the pressure is given by \( P = \frac{I}{c} \) where \( I \) is the intensity and \( c \) is the speed of light. For perfectly reflecting surfaces, \( P = \frac{2I}{c} \). Average momentum density is given by \( \frac{I}{c^2} \). The speed of light \( c \) is approximately \( 3 \times 10^8\, \text{m/s} \). Atmospheric pressure is \( 1 \text{atm} = 101325 \text{Pa} \).

Step 2: Calculate Pressure on Absorbing Surface

Use the formula \( P = \frac{I}{c} \). Substitute the given intensity, \( I = 2500 \, \text{W/m}^2 \): \[ P_a = \frac{2500}{3 \times 10^8} \approx 8.33 \times 10^{-6} \, \text{Pa} \]}.

Step 3: Convert Pressure to Atmospheres

Convert the result from Step 2 from pascals to atmospheres: \[ P_a = \frac{8.33 \times 10^{-6}}{101325} \approx 8.22 \times 10^{-11} \, \text{atm} \]}.

Step 4: Calculate Pressure on Reflecting Surface

Since the surface is perfectly reflecting, use the formula \( P = \frac{2I}{c} \). Substitute the given intensity: \[ P_r = \frac{2 \times 2500}{3 \times 10^8} \approx 1.67 \times 10^{-5} \, \text{Pa} \]}.

Step 5: Convert Reflecting Pressure to Atmospheres

Convert the pressure on the reflecting surface from pascals to atmospheres: \[ P_r = \frac{1.67 \times 10^{-5}}{101325} \approx 1.65 \times 10^{-10} \, \text{atm} \]}.

Step 6: Calculate Average Momentum Density

Use the formula for momentum density \( \frac{I}{c^2} \). Substitute the given intensity: \[ \text{Momentum Density} = \frac{2500}{(3 \times 10^8)^2} \approx 2.78 \times 10^{-14} \, \text{kg/m}^3 \]}.

Key Concepts

Momentum Density

Momentum density is a way to measure the amount of momentum in a unit volume of light. It tells us how much momentum is carried by light in a given space. To calculate the momentum density of light, you use the formula: \[\text{Momentum Density} = \frac{I}{c^2} \]where \(I\) represents light intensity, and \(c\) is the speed of light (approximately \(3 \times 10^8\) m/s). For a light intensity of 2500 W/m\(^2\), as found in the NASA Space Simulator, the momentum density results in a very small value of \(2.78 \times 10^{-14}\) kg/m\(^3\). Reflecting on this value helps us understand how even the tiny forces in light contribute to influences in a space environment.

Light Intensity

Light intensity measures how much light power passes through a unit area. In physics, it's denoted by \(I\) and is measured in watts per square meter (W/m\(^2\)). At NASA's Space Simulator, an intensity of 2500 W/m\(^2\) is used to mimic the sunlight near Venus. This large value ensures accurate simulation of the harsh conditions in space, helping researchers test the resilience and performance of various spacecraft materials. Understanding light intensity is vital because it's a key factor in calculating radiation pressure and momentum density, and influences the development of space equipment.

Reflecting and Absorbing Surfaces

Surfaces respond differently when exposed to light, particularly when it comes to reflecting or absorbing light. These responses affect how much radiation pressure a surface experiences. Radiation pressure is the force exerted by light per unit area. For absorbing surfaces, the pressure is given by:\[P = \frac{I}{c}\]Reflecting surfaces experience double the pressure:\[P = \frac{2I}{c}\]The reason behind the increased pressure on reflecting surfaces is that they send the light back, causing the momentum to change direction, effectively doubling the force. insights about these concepts are fundamental for developing solar sails, which harness radiation pressure efficiently for movement in space.

NASA Space Simulator

The NASA Space Simulator is a 25-ft facility designed to replicate the conditions of outer space. With high-intensity lights simulating sunlight, this simulator is crucial for testing spacecraft components against the rigors of environments like those found near planets such as Venus. By reproducing light intensities around 2500 W/m\(^2\), the simulator allows engineers to assess how different materials and designs handle radiation pressure and thermal exposure. This process aids in perfecting spacecraft for long missions where exposure to solar radiation is continuous and relentless.

Radiation Pressure Calculation

Radiation pressure is the pressure exerted by light upon any surface it encounters. Calculating it helps scientists and engineers understand the forces exerted by light in space. The formulas used differ based on whether the light is absorbed or reflected:- For absorbing surfaces: \[P = \frac{I}{c}\]- For reflecting surfaces: \[P = \frac{2I}{c}\]By knowing the intensity of light \(I\), engineers calculate the pressure in pascals, and then often convert it to atmospheres for practical understanding. These calculations are crucial for project development in space industries, like designing lightweight structures that can withstand varying pressures without damage.