A laser beam is used to levitate a metal disk against the force of Earth\'s grav

Question

A laser beam is used to levitate a metal disk against the force of Earth's gravity. (a) Derive an equation giving the required intensity at lights, I, in terms of the mass m of the disk, the gravitational acceleration g, the speed of light c, and the cross-sectional area of the disk A. Assume the disk is perfectly reflecting and the beam is directed perpendicular to the disk. (b) If the disk has mass 13 g and radius 0.05 m, find the necessary light intensity. (c) Give two reasons why using light pressure as propulsion near Earth's surface it impractical.

Explanation / Answer

(a)

The intensity of laser beam is I,

Eenergy of laser falling on area A in time t is,

U = IAt

The momentum of the laser beam falling on the disk is,

P = U/c, since the light is reflected perfectly, the change in momentum is,

P = 2P = 2U/c = (2/c)IAt

So force on the disk is,

F = P/t = (2/c)IAt/t

or, F = (2/c)IA

This force balances the gravitation force on disk, so,

F = mg

or, (2/c)IA = mg

or, I = (mgc)/(2A)

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(b)

we have I = mgc/2A

or, I = [(13X10-3kg)(9.8 m/s2)(3X108 m/s)]/2[3.14X(0.05m)2]

or, I = 2.43X109 W/m2

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(c)

We saw above that to lift just a 13 g disk we need the enormous intensity of I = 2.43X109 W/m2. So the enormous energy requirement is a big impediment.

Also, the energy absorbed by the surface of a practical device would raise its temperature to high level which would be difficult to handle. This is because no material is perfect reflector.

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