In Fig. 33-38, a laser beam of power 4.60 W and diameter D=2.60 mm is directed upward at one c

step 1 In this question, we have a cylinder of diameter small D. Okay, it has a high H, okay, and then

In Fig. 33-38, a laser beam of power 4.60 W and diameter...

In Fig. 33-38, a laser beam of power $4.60 \mathrm{~W}$ and diameter $D=2.60 \mathrm{~mm}$ is directed upward at one circular face (of diameter $d<2.60 \mathrm{~mm}$ ) of a perfectly reflecting cylinder. The cylinder is levitated because the upward radiation force matches the downward gravitational force. If the cylinder's density is $1.20 \mathrm{~g} / \mathrm{cm}^{3},$ what is its height $H ?$

Key Concepts

Radiation Pressure

Radiation pressure is the pressure exerted upon any surface due to the momentum transfer of light. When light reflects off or is absorbed by a surface, it imparts momentum, exerting a force over the area of incidence. In the context of a perfectly reflecting surface, the momentum change is doubled, making the radiation pressure effective at counteracting gravitational forces.

Light Intensity

Light intensity is defined as the power per unit area carried by an electromagnetic wave. For a laser with a given power and beam diameter, the intensity can be determined by dividing the total power by the cross-sectional area of the beam. This concept is crucial when calculating the radiation pressure because it directly influences the force exerted by the laser beam on the surface.

Momentum Transfer from Light

The transfer of momentum from light to a surface is a fundamental concept in electromagnetism where the momentum of photons is related to their energy. In cases of reflection, each photon reverses its momentum, resulting in a force that is twice as effective as if the photon were merely absorbed. This phenomenon underlies the ability of light to exert a force capable of lifting objects, such as in optical levitation.

Force Equilibrium

Force equilibrium in this context refers to the balance between the upward force due to radiation pressure and the downward gravitational force acting on the cylinder. When these forces are equal, the net force is zero, allowing the cylinder to levitate. Setting these forces equal provides the necessary condition to determine unknown parameters such as the height of the cylinder.

Density and Mass of an Object

An object's mass can be determined from its density and volume. For a cylinder, the volume is calculated as the product of its cross-sectional area and its height. Given the density, the mass—and therefore the gravitational force—can be expressed in terms of its geometric dimensions. This relationship is essential when equating the gravitational force to the radiation force in order to solve for the cylinder's height.

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