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Chapter 33: Problem 28

The average intensity of the solar radiation that strikes normally on a surface just outside Earth's atmosphere is \(1.4 \mathrm{~kW} / \mathrm{m}^{2} .\) (a) What radiation pressure \(p_{r}\) is exerted on this surface, assuming complete absorption? (b) For comparison, find the ratio of \(p_{r}\) to Earth's sea-level atmospheric pressure, which is \(1.0 \times 10^{5} \mathrm{~Pa}\).

Short Answer

Expert verified
The radiation pressure is approximately \(4.67 \times 10^{-6} \text{ Pa}\) and the ratio of radiation to atmospheric pressure is \(4.67 \times 10^{-11}.\)

Step by step solution

01

Understanding Radiation Pressure Formula

The radiation pressure \( p_r \) for complete absorption is given by the formula \( p_r = \frac{I}{c} \), where \( I \) is the intensity of the radiation and \( c \) is the speed of light in a vacuum \( \approx 3 \times 10^8 \text{ m/s} \).
02

Calculation of Radiation Pressure

Using the intensity \( I = 1.4 \times 10^3 \text{ W/m}^2 \) and the speed of light \( c = 3 \times 10^8 \text{ m/s} \), substitute into the formula: \[ p_r = \frac{1.4 \times 10^3}{3 \times 10^8} \approx 4.67 \times 10^{-6} \text{ Pa}. \]
03

Comparison with Earth鈥檚 Atmospheric Pressure

Find the ratio of the radiation pressure to Earth's atmospheric pressure. Given that the atmospheric pressure is \( 1.0 \times 10^5 \text{ Pa} \), the ratio is: \[ \text{Ratio} = \frac{p_r}{p_{atm}} = \frac{4.67 \times 10^{-6}}{1.0 \times 10^5} \approx 4.67 \times 10^{-11}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Intensity of Solar Radiation
When talking about the intensity of solar radiation, we refer to the amount of solar power received per unit area. This value is crucial for calculating radiation pressure because it directly relates to how much energy reaches a surface, such as the Earth. The average intensity of solar radiation just at the edge of Earth's atmosphere is approximately 1.4 kW/m虏.

Here are some key aspects about the intensity of solar radiation:
  • It is measured in watts per square meter (W/m虏), reflecting how much energy hits each square meter every second.
  • This intensity can vary slightly based on Earth's position related to the sun and atmospheric conditions.
  • The value given (1.4 kW/m虏) is often referred to as the solar constant, providing a baseline for calculations involving solar energy.
Knowing this intensity helps us understand the potential for solar energy conversion and its impact on Earth's climate systems.
Grasping the Speed of Light
The speed of light, denoted as 'c', is a fundamental constant in physics, essential in calculations involving electromagnetic radiation, such as light itself. In a vacuum, the speed of light is approximately \(3 \times 10^8\ \text{m/s}\). This impressive speed tells us how fast light travels from the sun to us, and it plays a critical role in calculating radiation pressure.

Important points about the speed of light include:
  • It is the ultimate speed limit of the universe, meaning nothing can travel faster than light in a vacuum.
  • Understanding this speed helps us appreciate the time it takes for sunlight, or any electromagnetic radiation, to reach Earth's surface.
  • The speed of light is a crucial factor in Einstein's famous equation \(E=mc^2\), linking energy and mass in physics.
In the context of radiation pressure, the speed of light is used to determine how much force solar radiation exerts upon absorbing surfaces, as seen in the pressure formula \( p_r = \frac{I}{c} \).
Exploring Atmospheric Pressure Comparison
Atmospheric pressure is the force exerted by the weight of the air in the atmosphere pushing down on Earth's surface. At sea level, this pressure averages around \(1.0 \times 10^5\ \text{Pa}\) (Pascals), a measure that provides context when comparing it with other types of pressures, like radiation pressure.

Some vital elements of atmospheric pressure include:
  • It varies with altitude, decreasing as one moves higher above sea level.
  • Atmospheric pressure affects weather patterns and the boiling point of water.
  • When comparing radiation pressure to atmospheric pressure, it becomes apparent how comparatively tiny the force from solar radiation is when fully absorbed.
The comparison between the two pressures highlights that while solar radiation exerts a measurable force, it is minuscule compared to the air pressure we experience daily, underscoring how Earth's atmosphere plays a significant role in maintaining our planet鈥檚 surface conditions.

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Most popular questions from this chapter

A plane electromagnetic wave, with wavelength \(3.0 \mathrm{~m}\), travels in vacuum in the positive direction of an \(x\) axis. The electric field, of amplitude \(300 \mathrm{~V} / \mathrm{m}\), oscillates parallel to the \(y\) axis. What are the (a) frequency, (b) angular frequency, and (c) angular wave number of the wave? (d) What is the amplitude of the magnetic field component? (e) Parallel to which axis does the magnetic field oscillate? (f) What is the timeaveraged rate of energy flow in watts per square meter associated with this wave? The wave uniformly illuminates a surface of area \(2.0 \mathrm{~m}^{2}\). If the surface totally absorbs the wave, what are \((\mathrm{g})\) the rate at which momentum is transferred to the surface and (h) the radiation pressure on the surface?

(a) How long does it take a radio signal to travel \(150 \mathrm{~km}\) from a transmitter to a receiving antenna? (b) We see a full Moon by reflected sunlight. How much earlier did the light that enters our eye leave the Sun? The Earth-Moon and Earth-Sun distances are \(3.8 \times 10^{5} \mathrm{~km}\) and \(1.5 \times 10^{8} \mathrm{~km}\), respectively. (c) What is the round-trip travel time for light between Earth and a spaceship orbiting Saturn, \(1.3 \times 10^{9} \mathrm{~km}\) distant? (d) The Crab nebula, which is about 6500 light-years (ly) distant, is thought to be the result of a supernova explosion recorded by Chinese astronomers in A.D. 1054 . In approximately what year did the explosion actually occur? (When we look into the night sky, we are effectively looking back in time.)

Someone plans to float a small, totally absorbing sphere \(0.500\) \(\mathrm{m}\) above an isotropic point source of light, so that the upward radiation force from the light matches the downward gravitational force on the sphere. The sphere's density is \(19.0 \mathrm{~g} / \mathrm{cm}^{3}\), and its radius is \(2.00 \mathrm{~mm}\). (a) What power would be required of the light source? (b) Even if such a source were made, why would the support of the sphere be unstable?

An electromagnetic wave with frequency \(4.00 \times 10^{14} \mathrm{~Hz}\) travels through vacuum in the positive direction of an \(x\) axis. The wave has its electric field directed parallel to the \(y\) axis, with amplitude \(E_{m}\). At time \(t=0\), the electric field at point \(P\) on the \(x\) axis has a value of \(+E_{m} / 4\) and is decreasing with time. What is the distance along the \(x\) axis from point \(P\) to the first point with \(E=0\) if we search in (a) the negative direction and (b) the positive direction of the \(x\) axis?

The electric component of a beam of polarized light is $$E_{y}=(5.00 \mathrm{~V} / \mathrm{m}) \sin \left[\left(1.00 \times 10^{6} \mathrm{~m}^{-1}\right) z+\omega t\right]$$ (a) Write an expression for the magnetic field component of the wave, including a value for \(\omega\). What are the (b) wavelength, (c) period, and (d) intensity of this light? (e) Parallel to which axis does the magnetic field oscillate? (f) In which region of the electromagnetic spectrum is this wave?
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