A small, conical sapphire bead of radius r and height h = r is suspended above a

Question

A small, conical sapphire bead of radius r and height h = r is suspended above a uniform laser beam which has the same radius r as the bead. When the laser light passes anywhere through this cone, it is deflected (refracted) at an angle of theta = 30.2 degree with respect to the vertical. If the laser beam has a continuous power of P = 1 W, for what value of the radius r, in microns, will the sapphire bead be suspended against gravity? [The density of sapphire is 3.97g/cm-1. The volume of a cone of radius r and height h is V = 1/3 pi r2h.]

Explanation / Answer

The upward force provided by the laser is equal to the upward momentum lost by the photons, which is 1/(3 x 10^8)*(1- cos39.2) = 7.50 x 10^-10 N. For the crystal to levitate, F = mg = Vg = (r3/3)**g, so r3 = 3F/g, and r = 3(3F/g) = (3*(7.5 x 10^-10)/(*3970*9.8)) = 26.4 m.

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