Question

You are working in a laser laboratory. Your current project involves suspending spherical glass beads in the Earth's gravitational field using a vertically directed laser beam. Today's experiment involves a black bead of radius r and density rho. Assuming that the radius of the laser beam is the same as that of the bead and that the beam is centered on the bead, determine the minimum laser power required to suspend this bead in equilibrium. (Use any variable or symbol stated above along with the following as necessary: g and c.)

Answer 1

The minimum laser power required to suspend the bead in equilibrium is (3c/8πr³ρg).

Step-by-step explanation:

In order to suspend the bead in equilibrium, the radiation pressure from the laser beam must be equal to the gravitational force on the bead.

The radiation pressure is given by P = 2I/c, where P is the pressure, I is the laser intensity, and c is the speed of light. The gravitational force is given by F = (4/3)πr³ρg, where F is the force, r is the radius of the bead, ρ is the density, and g is the acceleration due to gravity.

Equating the radiation pressure and gravitational force, we have 2I/c = (4/3)πr³ρg. Rearranging this equation, we get I = (3c/8πr³ρg).

Therefore, the minimum laser power required to suspend the bead in equilibrium is (3c/8πr³ρg).

Answer 2

P = m g c

Step-by-step explanation:

For this exercise we use the transactional equilibrium equation

∑F = 0

F-W = 0

F = W

to determine the force exerted by the laser beam use the concept of pressure

P = F / A

F = P A

The pressure is the radiation pressure, as it indicates that the beads are black, they absorb all the radiation, therefore the root pressure is

P = S / c

where S is the pointing vector that is equal to the intensity of the incident beam

P = I / c

The area of a circle is

A = π r²

we substitute in force

F = I /c π r²

of the equilibrium equation

I / c π r² = mg

intensity is power per unit area, so

P = I π r²

P = m g c

detone P is the laser power