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Chapter 33: Problem 99

A beam of intensity \(I\) reflects from a long. totally reflecting cylinder of radius \(R ;\) the beam is perpendicular to the central axis of the cylinder and has a diamcter larger than \(2 R .\) What is the beam's force per unit length on the cylinder?

Short Answer

Expert verified
The beam's force per unit length on the cylinder is \( \frac{4IR}{c} \).

Step by step solution

01

Understand the Problem

We have a cylindrical object with radius \( R \), and a light beam, with intensity \( I \), that is wider than the diameter of the cylinder. The beam strikes the cylinder perpendicularly and reflects off it. We need to calculate the force per unit length the beam exerts on the cylinder.
02

Consider the Concept of Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation on a surface. When light reflects off a surface, it exerts pressure due to the momentum transfer from the light to the surface. For a perfectly reflecting surface, the pressure, \( P \), is given by: \[ P = \frac{2I}{c} \] where \( I \) is the intensity of the light and \( c \) is the speed of light in vacuum.
03

Calculate Force Per Unit Length

The force exerted by the beam is a result of the radiation pressure applied over the area it covers. Since we need the force per unit length and the width of the cylinder affected is \( 2R \), the force per unit length \( f \) is given by combining the pressure formula: \[ f = P \cdot 2R = \frac{2I}{c} \cdot 2R = \frac{4IR}{c} \] This gives us the force per unit length along the length of the cylinder as a result of the incident and reflected light.

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Key Concepts

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

Momentum Transfer
When light hits a surface, it carries momentum. This is because light has particle-like properties, which means it can exert a force upon interaction with materials.
As the light beam reflects off a surface, like our cylinder in the problem, the direction of the light changes, leading to a momentum change. This change in momentum is transferred to the surface, which can cause a mechanical effect such as pressure.
The amount of momentum transferred depends on how much light, or energy, hits the surface. This is characterized by the light's intensity, and how that light behaves upon reflection. For a perfectly reflective surface, the momentum is transferred more effectively compared to partial or non-reflective surfaces.
Light Reflection
Light reflection is the process in which light bounces off a surface rather than being absorbed or passing through.
In our cylinder scenario, the light beam reflects entirely off the cylinder's surface, akin to how a mirror reflects light. When light reflects, the angle at which it meets the surface (the angle of incidence) equals the angle at which it leaves (the angle of reflection). This principle ensures that the light beam does not penetrate the cylinder but instead exerts force via reflection.
For the case of a perfectly reflecting surface, the momentum change of light is twice as much compared to partial reflection since the change in direction also reverses the momentum vector. This total change results in the exertion of radiation pressure on the surface, proportional to the intensity of the light.
Intensity of Light
The intensity of light is a measure of the energy carried by the light beam per unit area per unit time. Higher intensity means more energy is conveyed by the light, which in turn affects how much force or pressure the light can exert upon reflection.
In our context, the intensity of light defines how much energy hits the cylinder's surface per second. The relationship between intensity and pressure is showcased in the formula \[ P = \frac{2I}{c} \]where this pressure partly arises due to the reflection characteristics of the light. Higher intensity leads directly to greater radiation pressure when the light is fully reflected.
Understanding intensity helps in visualizing how light can apply forces to surfaces, particularly when dealing with reflective materials like the cylinder in the problem.

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Most popular questions from this chapter

In about A.D. \(150,\) Claudius Ptolemy gave the following measured values for the angle of incidence \(\theta_{1}\) and the angle of refraction \(\theta_{2}\) for a light beam passing from air to water: $$ \begin{array}{lcll} \hline \theta_{1} & \theta_{2} & \theta_{1} & \theta_{2} \\ \hline 10^{\circ} & 8^{\circ} & 50^{\circ} & 35^{\circ} \\\ 20^{\circ} & 15^{\circ} 30^{\prime} & 60^{\circ} & 40^{\circ} 30^{\prime} \\\ 30^{\circ} & 22^{\circ} 30^{\prime} & 70^{\circ} & 45^{\circ} 30^{\prime} \\\ 40^{\circ} & 29^{\circ} & 80^{\circ} & 50^{\circ} \\ \hline\end{array}$$ Assuming these data are consistent with the law of refraction, use them to find the index of refraction of water. These data are interesting as perhaps the oldest recorded physical measurements.

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?

Some neodymium鈥揼lass lasers can provide 100 TW of power in 1.0 ns pulses at a wavelength of \(0.26 \mu \mathrm{m}\). How much energy is contained in a single pulse?

Project Seafarer was an ambitious program to construct an enormous antenna, buried underground on a site about \(10000 \mathrm{~km}^{2}\) in area. Its purpose was to transmit signals to submarines while they were decply submerged. If the effective wavelength were \(1.0 \times 10^{4}\) Earth radii, what would be the (a) frequency and (b) period of the radiations emitted? Ordinarily, electromagnetic radiations do not penetrate very far into conductors such as seawater, and so normal signals cannot reach the submarines.

In Fig. \(33-41,\) a beam of unpolarized light, with intensity \(43 \mathrm{~W} / \mathrm{m}^{2}\), is sent into a system of two polarizing sheets with polarizing dircctions at angles \(\theta_{1}=70^{\circ}\) and \(\theta_{2}=90^{\circ}\) to the \(y\) axis. What is the intensity of the light transmitted by the system?
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