Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problema 52a

Capitolo 31, Problema 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.

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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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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.

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Percorso guidato

07:51

Radiation Pressure

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.

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Percorso guidato

05:41

Gravitational Forces in 2D

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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Percorso guidato

05:17

Intro to Momentum

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