Questions From Old Final

1. Consider a point particle of mass m in one dimension in the presence of a ‘binding’ potential of the form e2 λ V (x) = − + 2 λ>0 x x a) Sketch the potential V (x) and find the equilibrium position (take x > 0). (5 points) b) What is the frequency of small oscillations about the equilibrium position? (5 points) c) Assume we displace the particle a small distance A in the positive x-direction from the equilibrium position, then release it from rest. Assume also that there is a weak damping force of the form F = −γv where v is the speed. What is x(t)? (15 points)

2. Consider an infinitely long string stretched along the positive x-axis with an end point at x = 0, with mass density ρ and tension T . Denote the transversal string displacement by ψ. Assume ψ(x, t) = 0 for t < 0. Beginning at t = 0 and lasting till t = 1 we displace the end point at x = 0 by ψ(0, t) = t(t − 1). For t > 1 we hold the end point still with no displacement, i.e., ψ(0, t) = 0 for t > 1. a) What is ψ(x, t) for all x > 0 and t > 0? (10 points) b) Suppose instead of an infinitely long string as above we tie it to a rigid wall at x = L. How long does it take for the disturbance described in a) to reach the wall? What happens to the disturbance after that? Describe it both quantitatively and by drawing a picture of the disturbance. (15 points)

3. We have a one dimensional medium with dispersion relation ω 2 = b(k 2 + a) a) What is the speed of transmission of signal as a function of k? (10 points) b) Suppose we have a wave travelling to the right which at t = 0 is described by ψ(x, t) = Acos(kx) + Bsin(kx). Find ψ(x, t) for all t. (15 points)

4. Two blocks each of mass M are connected to each other and to two walls by springs with spring constants from left to right K1 , K2 and K3 . Find the most general solution to the above system and check your answer in the limit K2 = 0. (25 points)

5. Consider propagation of a wave in (x, y) plane consisting of two regions. Suppose for x < 0 at frequency ω the phase velocity is v1 and for x > 0 it is v2 . 1

a) suppose we have a plane wave traveling from left to right, with frequency ω. What are the boundary conditions on the wave vector for the incident (kx , ky ) reflected (kx0 , ky0 ) and refracted (kx ”, ky ”)? Whare are the frequencies for each wave? (5 points) b) Suppose v1 > v2 . From which direction should we send a wave and what condition do we need to satisfy for the angle of the direction of the wave relative to the normal to the x = 0 line, so that we have no transmitted wave? (10 points) 6. Consider a one dimensional space consisting of two regions. For x < 0 we have a medium with dispersion relation ω = kc + b for some constants b and c. For x > 0 we have a medium with dispersion relation p ω = v k 2 + a2 a) What is the group velocity in each region, and what is its significance? (5 points) b) In what regime of frequencies will part of the signal coming from x < 0 propagate to x > 0? Assuming that the wave ψ and its slope dψ/dx are continuous at x = 0, find the amplitudes of reflected and transmitted waves for the regime of frequencies where transmission is allowed. (10 points) 7. We have a plane wave of wavelength λ which passes through a screen with a slit with adjustable width d. We place another screen a distance R away in the direction of the beam. There will be a main spot formed on this screen. a) What width d will minimize the size of the spot on the screen? (10 points) b) With this width, what is the size of the spot? (5 points) 8. A Plane electromagnetic wave is propagating in vacuum in the z direction. The amplitude of the E-field is given by E = (f1 x ˆ + f2 yˆ)exp(i(kz − ωt)) where x ˆ and yˆ denote unit vectors in the x and y directions respectively, and f1 and f2 are complex numbers. a) What condition should f1 and f2 satisfy for this to be a linearly polarized wave in some direction? (5 points) b) If we pass this wave through a right handed circular polarizer we observe the intensity I1 for the transmitted wave and if instead we pass it through a left handed circular polarizer we observe intensity I2 . What is the ratio I1 /I2 (you can take any convention you wish for what is a left-handed as opposed to right-handed circular polarization)? Check your results by comparing it with the case were the light is linearly polarized or purely circularly polarized. (10 points) 9. Consider a membrane with surface tension T and mass per unit area ρ stretched along a rectangular region in the x − y plane given by 0 ≤ x ≤ L1 and 0 ≤ y ≤ L2 . 2

Moreover the end lines of the membrane at x = 0 and x = L1 and at y = 0 and y = L2 are fixed not to move. We consider transversal vibrations ψ(x, y, t) of the membrane (which p as you recall have phase velocity v = T /ρ) a) What are the boundary conditions satisfied by ψ? (5 points) b) What are the normal modes and what are the corresponding normal frequencies? (10 points) c) Write the most general solution to ψ(x, y, t) and indicate how you would go about fixing the undetermined coefficients in the general solution. (5 points) 10. Suppose we have a medium in which the index of refraction in the x and the y direction are different and given by nx and ny . Suppose that in this medium we have a light beam of plane wave with frequency ω. At some point we split the beam to two equal amplitude beams, one going in the x direction and the other in the y direction. Beams are relfected back after traversing distance L1 in the x-direction and L2 in the y direction and again recombined. What is the intensity of the recombined wave as a function of L1 , L2 , nx , ny and ω? (10 points)

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