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

a) We assume the glass bead is perfectly reflecting. The pressure P is given by P = 2I/c, where I is the radiation intensity. The force exerted by the radiation must counteract mg, the weight of the bead. Hence P A = mg, where A = πr^2 is the cross-section of the bead, and r is the radius. We can find r from m = ρ*(4/3)*a)We assume the glass bead is perfectly reflecting. The pressure P is given by P = 2I/c, where I is the radiation intensity. The force exerted by the radiation must counteract mg, the weight of the bead. Hence P A = mg, where A = πr2 is the cross-section of the bead, and r is the radius. We can find r from m = ρ*(4/3)πr^3 = 8.986*10^-8, so r = (3m/4πρ)^1/3 . Putting this all together,

I = Pc/2 = cmg/2πr^2 = (cmg/2π)*(4πρ/3m)^2/3 = (2/9π)^1/3*cgm^1/3*ρ^2/3 = 204939091.98

b) If the beam has a radius R (this is not necessarily the same as r in part a), then the power needed to produce an intensity I is

Power = I · area = I*πR² = (2π²/9)^1/3*R^2*cgm^1/3*ρ^2/3 = 139.32 W