4 Photon Momentum

A particle of mass m moves though a viscous medium. (See figure.) In this medium, if the particle

moves with a velocity v, it experiences a drag force Fa given by Fa = -mv/T. T is known as the

relaxation time. If a particle moves through this medium without an external driving force, its

velocity will decay exponentially to zero with a characteristic time 7 according to v(t) = v(t)e-t/T.

Essentially, the particle quot;forgetsquot; what it had been doing more that a few relaxation times in the

past.

(F=mv/tau)

– – – – – – – – –

m

lambda

In this problem, the particle is illuminated from the left and absorbs photons of wavelength A at

an average rate of I photons per unit time. As a result, the particle experiences an average force

and is driven to the right at an average velocity lt; v gt;. Because the photons arrive randomly as

discrete packets, the velocity of the particle is not steady and fluctuates about the average with

fluctuations of typical size Av. Your task is to compute lt; v gt; and Av.

4.1 a)

What is the average force experienced by the particle as a result of absorbing the photons?

4.2 b)

Find the average speed of the particle after a long time lt; v gt; from the condition that the total

average force on the particle is zero.

4.3

Your answer to b) should involve a factor IT. At any instant, this factor represents the total number

of photons received by the particle during the preceding relaxation time 7. Because the photons

arrive randomly, this number fluctuates. What is the size of these fluctuations? If everything else

in your expression for lt; v gt; is constant except for these fluctuations, what is Av, the size of the

fluctuations in the particle’s velocity?Science