Question

Lasers have been used to suspend spherical glass beads in the Earth's gravitational field.

(a) A black bead has a radius of 0.466 mm and a density of 0.212 g/cm³. Determine the radiation intensity needed to support the bead.

(b) What is the minimum power required for this laser?

Answer 1

Given

radius of black bead is r = 0.466 mm = 0.466*10^-3 m, density rho = 0.212 g/cm3 = 212 kg/m3

  
a)
Assuming that the bead is perfectly reflecting.

mass of bead is m = rho*Volume = rho*4/3*pir^3 = 212*(4/3)(pi*(0.466*10^-3)^3) kg = 8.98633*10^-8 kg

   the pressure P is given as P = 2*I/c; I is radiation intensity, c speed of light

here the force exerted by radiation must equal to weight of the bead mg.

so P*A = mg, (A = pi*r^2, area of cross section of the bead)

now radiation intensity is I = P*c/2
           I = 0.5(mg/(pi*r^2)(c)

           I = 0.5(8.98633*10^-8 *9.8 /(pi*(0.466*10^-3 )^2)(3*10^8)

           I = 193632383.58

the power is P = I*area = 193632383.580*pi*(0.466*10^-3)^2

       P = 132.1 w