Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problem 52a

Chapter 31, Problem 52a

A laser beam shines straight up onto a flat, black foil of mass m. Find an expression for the laser power P needed to levitate the foil.

Verified step by step guidance

Step 1: Understand the physical situation. The laser beam exerts radiation pressure on the foil, which generates a force. To levitate the foil, this force must counteract the gravitational force acting on the foil. The gravitational force is given by \( F_{gravity} = m g \), where \( m \) is the mass of the foil and \( g \) is the acceleration due to gravity.

Step 2: Recall the relationship between radiation pressure and power. Radiation pressure \( P_{rad} \) is related to the intensity \( I \) of the laser beam and the speed of light \( c \). For a perfectly absorbing surface (like the black foil), \( P_{rad} = \frac{I}{c} \). The intensity \( I \) is defined as the power per unit area, \( I = \frac{P}{A} \), where \( P \) is the laser power and \( A \) is the area of the foil.

Step 3: Determine the force exerted by the radiation pressure. The force due to radiation pressure is \( F_{rad} = P_{rad} \cdot A = \frac{P}{c} \). This force must equal the gravitational force to levitate the foil, so \( F_{rad} = F_{gravity} \). Substituting \( F_{gravity} = m g \), we get \( \frac{P}{c} = m g \).

Step 4: Solve for the laser power \( P \). Rearrange the equation \( \frac{P}{c} = m g \) to isolate \( P \): \( P = m g c \). This is the expression for the laser power needed to levitate the foil.

Step 5: Summarize the result. The laser power \( P \) required to levitate the foil depends on the mass of the foil \( m \), the acceleration due to gravity \( g \), and the speed of light \( c \). The final expression is \( P = m g c \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Laser Power and Radiation Pressure

Laser power refers to the rate at which energy is emitted by the laser beam. When a laser beam strikes a surface, it exerts a force due to radiation pressure, which is the momentum transfer from the photons to the surface. The power of the laser must be sufficient to generate enough radiation pressure to counteract the gravitational force acting on the mass of the foil.

Gravitational Force

The gravitational force acting on an object is given by the equation F = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). This force pulls the foil downward, and to achieve levitation, the upward force from the laser's radiation pressure must equal this gravitational force.

Momentum of Photons

Photons, the particles of light, carry momentum despite having no mass. The momentum of a single photon is given by p = E/c, where E is the energy of the photon and c is the speed of light. When photons reflect off a surface, they impart momentum to that surface, creating an upward force that can be calculated based on the power of the laser and the area over which it is applied.

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When the Voyager 2 spacecraft passed Neptune in 1989, it was 4.5×10⁹ km from the earth. Its radio transmitter, with which it sent back data and s, broadcast with a mere 21 W of power. Assuming that the transmitter broadcast equally in all directions, What signal intensity was received on the earth?

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Textbook Question

The intensity of sunlight reaching the earth is 1360 W/m². Assuming all the sunlight is absorbed, what is the radiation-pressure force on the earth? Give your answer in newtons.

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Textbook Question

Unpolarized light of intensity I₀ is incident on a stack of 7 polarizing filters, each with its axis rotated 15° cw with respect to the previous filter. What light intensity emerges from the last filter?

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Textbook Question

For a science project, you would like to horizontally suspend an 8.5 by 11 inch sheet of black paper in a vertical beam of light whose dimensions exactly match the paper. If the mass of the sheet is 1.0 g, what light intensity will you need?

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Textbook Question

What is the total energy density in an electromagnetic wave of intensity 1000 W/m²?

Textbook Question

It has been proposed that small spacecraft could reach other planets in a fairly short time—days instead of many months—if they unfurl a reflective sail and are accelerated by a powerful laser beam generated by an earth-orbiting laser. What speed would a spacecraft need to reach Mars in 8.0 days when Mars is closest to earth?