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quiz-1-01-26-10-answers

quiz-1-01-26-10-answers - Quiz 1 for Physics 176 Solutions...

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Quiz 1 for Physics 176: Solutions Professor Greenside Problems That Require Writing 1. A laser consists of a long cylindrical tube of some medium that has mirrors at both ends such that light bounces back and forth between the mirrors while its intensity is amplified. (Light not parallel to the axis of the cylinder is lost rapidly since the outer surface of the cylinder is often transparent.) Assume that the laser light is monochromatic with frequency f , which means that the laser cavity is filled with photons (quantum light particles) that each have momentum p = ( h/c ) f where h is Planck’s constant and c is the speed of light. The total energy of the light in the laser is given by E = Nhf where N is the number of photons in the cavity. (a) (6 points) Use a simple kinetic theory to derive an expression for the pressure P that the light exerts on a mirror in terms of the photon number N , the volume V of the cylinder, and the light frequency f . Answer: This problem is identical to the one discussed in lecture and that you were thinking about in a homework assignment. To obtain the answer, all you had to do was start with the expression for the total momentum transfer over a time Δ t and over an area A Δ p total = A ( v Δ t ) × N V × 1 6 × 2 mv, (1) and change the height of the prism v Δ t c Δ t , change the momentum transfer per particle 2 mv 2[( h/c ) f ], and replace the factor 1 / 6 with 1 / 2 to get the answer P = F A = p total / Δ t ) A = 1 A Δ t × A ( c Δ t ) × N V × 1 2 × 2 h c f ‚¶ = N V hf. (2) You had to be observant to make the substitution 1 / 6 1 / 2: unlike the atoms in a gas, the photons only move along two possible directions with respect to the axis of the laser, so only half the particles in the laser are moving in a given direction at any given time. In a quiz, you don’t have time for lengthy descriptions but I do ask that you justify what you are doing with short phrases or diagrams. Here you needed to say something to the effect that the pressure on the mirror could be determined by calculating the total momentum change Δ p total over some short arbitrary time Δ t over the area A of a mirror; you needed to say enough that it was clear that you knew what the symbols meant and what the strategy was for the calculation. Alternatively, you could have drawn a cylindrical laser with a small prism over one mirror and sketch in the appropriate details, with short phrases. In a bit more detail: the total momentum transfer could be calculated by counting the number of photons that strike the mirror within a short time Δ t and by using the fact that each photon transfers an amount of momentum 2[( h/c ) f ] if the photon reflects perfectly from the mirror. (You lost a point if you didn’t realize that the factor of 2 in the momentum change 2[( h/c ) f ] came from the reversal of direction of the photon by reflection.) The number of photons that strike within time Δ t are all photons that are closer than the distance c Δ t from the mirror and therefore lie in prism of volume A ( c Δ t ). But only half the photons in this prism are moving in the right direction
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