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When electromagnetic radiation strikes perpendicular to a flat surface, a totally absorbing surface feels radiation pressure \(I_{0} / c\) where \(I_{0}\) is the intensity of incident electromagnetic radiation. A totally reflecting surface feels twice that pressure. More generally, a surface absorbs a proportion \(e\) of the incident radiation and reflects a complementary proportion, \(1-e,\) where \(e\) is the emissivity of the surface, as introduced in Chapter \(17 .\) Note that \(0 \leq e \leq 1 .\) (a) Determine the radiation pressure \(p_{\text {rad }}\) in terms of \(I_{0}\) and \(e .\) (b) Consider cosmic dust particles in outer space at a distance of \(1.5 \times 10^{11} \mathrm{~m}\) from the sun, where \(I_{\text {sun }}=1.4 \mathrm{~kW} / \mathrm{m}^{2}\). We can model these particles as tiny disks with \(e=0.61,\) diameter \(8.0 \mu \mathrm{m}\) and mass \(1.0 \times 10^{-10} \mathrm{grams},\) all oriented perpendicular to the sun's rays. What is the force on one of these particles that is exerted by the radiation from the sun? (c) What is the ratio of this force to the attractive force of gravity exerted by the sun on the particle?

Step by step solution

01

Derive the Radiation Pressure Formula

We know that for a fully absorbing surface, the radiation pressure \(p_{\text {rad }}\) is \(I_{0}/c\), and for a fully reflecting surface, it's \(2I_{0}/c\). If a surface absorbs a proportion \(e\), it means it reflects \(1-e\). Therefore, it feels a radiation pressure that is a linear combination of both: \n\n\[p_{\text {rad }}=e \frac{I_{0}}{c}+(1-e) \frac{2 I_{0}}{c} = \frac{I_{0}}{c}(2-e)\]

02

Calculate the Radiation Force

Using the derivation of the previous step, we can use the density of solar radiation \(I_{\text {sun }}=1.4 \mathrm{~kW/m}^{2}\) and the emissivity of a cosmic dust particle \(e=0.61\) to calculate the radiation pressure upon the particle. From that the radiation force F exerted upon the particle is obtained using: \n\n\[F = p_{\text {rad }} \times A, \]\n\nwhere \(A\) is the area of the particle, which can be computed as the area of a circle, \(A = \pi \times (\text {diameter}/2)^2\].

03

Find the Gravitational Force

The gravitational force \(F_{\text {grav }}\) between the sun and the particle is given by the universal law of gravitation: \n\n\[F_{\text {grav }} = G \frac{m_{\text {sun }} \times m_{\text {particle}}}{r^2},\]\n\nwhere \(m_{\text {particle}}\) is the mass of the dust particle, \(r\) is the distance between the dust and the sun, and \(m_{\text {sun }}\) and \(G\) are the mass of the sun and the gravitational constant respectively.

04

Compute the Ratio of Forces

Finally, calculate the ratio of the radiation force \(F\) over the gravitational force \(F_{\text {grav }}\). This will yield the relation of the two forces acting on the particle.

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Key Concepts

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

Electromagnetic Radiation

Electromagnetic radiation refers to waves of the electromagnetic field, propagating through space, carrying electromagnetic energy. These waves include visible light, radio waves, X-rays, and gamma rays.
In the context of radiation pressure, which is an intriguing concept when considering electromagnetic radiation, the pressure is exerted by the force of the photons as they strike a surface.
The interaction between electromagnetic radiation and surfaces leads to pressure due to the momentum change upon striking. This pressure can differ based on whether the surface absorbs or reflects the radiation:

• Totally absorbing surfaces feel a pressure of \( I_{0} / c \).
• Fully reflective surfaces experience twice this pressure.

The beauty of this phenomenon lies in its linearity and how it adjusts for mixed surfaces that absorb and reflect in specific proportions. For example, a surface with an emissivity of \( e \) absorbs \( e \) portion and reflects \( 1-e \) portion of the radiation, leading to a combined pressure expressed as  \[ p_{\text{rad}} = \frac{I_{0}}{c}(2-e) \]

Cosmic Dust Particles

Cosmic dust particles, scattered throughout the universe, play a vital role in various cosmic phenomena. These tiny particles, often made up of silicate and carbon materials, range from a few molecules large to several micrometers in size.
In our exercise, we're analyzing cosmic dust particles at a distance of \(1.5 \times 10^{11} \;\text{m}\) from the Sun.
These particles encounter electromagnetic radiation from the Sun, experiencing radiation pressure that exerts a force on them.
This force calculation is done by modeling the particle as a disk with defined diameter and emissivity:

• Diameter: \(8.0 \; \mu \text{m}\)
• Emissivity \( e \): 0.61

The area \( A \) of these disks can be computed using \( A = \pi \times (\text{diameter}/2)^2 \).
Given the intensity of solar radiation \( I_{\text{sun}} = 1.4 \;\text{kW/m}^2 \), we can determine the pressure and subsequently the radiation force using \( F = p_{\text{rad}} \times A \). This interaction not only introduces radiation pressure but significantly affects the dynamics of cosmic dusty regions, influencing their movements and interactions with other cosmic bodies.

Gravitational Force

Gravitational force, a fundamental force of nature, is responsible for the attraction between masses. It governs astronomical movements and interactions across the universe.
When calculating gravitational attraction, especially with tiny cosmic dust particles near massive objects like the Sun, one utilizes Newton's Law of Universal Gravitation.
The gravitational force \( F_{\text{grav}} \) between an object and the Sun is defined by:

• \( F_{\text{grav}} = G \frac{m_{\text{sun}} \times m_{\text{particle}}}{r^2} \)

Here, \( G \) is the gravitational constant, \( m_{\text{sun}} \) the solar mass, and \( m_{\text{particle}} \) the mass of the dust particle, while \( r \) denotes the distance between them.
This force is incredibly small in scale for dust particles due to their minuscule mass and vast distance from the solar body. Interestingly, when these particles are compared against the radiation force exerted by electromagnetic radiation, we often find a fascinating balance or even a dominance of one force over the other. This balance can result in various outcomes, such as causing particles to move away from the Sun or remain in stable orbits.