Homework Set 9, Physics 3320, Spring 2012 Due Wednesday, April 4, 2012 (start of class) 1. Consider a 3D electromagnetic plane wave in vacuum, described in complex form by i( t) , in which is a constant vector given by E E 0 xˆ ,with E 0 E0ei /2 (E0 e real) , the wave vector is k k yˆ , and is the angular frequency. As usual, the actual physical field is ( , t) E Re[ ] A) In which direction the wave is moving? What is the speed, wavelength, and period of the wave? Sketch the real field E(x=0, y, z=0, t=0) (a 2D plot with y as the horizontal axis) and E(x=0, y=0, z=0, t) (a 2D plot with t as the horizontal axis). I am using bold font to indicate a vector. Clearly indicate the direction of the field and the scale of both your axes. B) Why is this called a plane wave [where is (are) the plane(s)]? Sketch or represent this in 3D. Describe how the direction of the electric field changes in time. If E always points along the same axis, the wave is said to be linearly polarized. Is this wave linearly polarized? C) Find the associated magnetic field B(r,t) for this plane electric wave. Sketch the magnetic fields, B(x=0, y, z=0, t=0) and B(x=0, y=0, z=0, t) indicating field direction. (As above, be clear about your axes) Describe in words how B compares/contrasts with E. D) Calculate the energy density uEM (compute the energy densities of the electric and magnetic fields separately and then add them to get the total energy density; simplify your answer as much as possible) Calculate the Poynting vector S, and momentum density for these fields, as a function of position and time. Interpret the answers physically by commenting on the units, the directions, etc. Page 2 of 3 E) Suppose now that we add two plane waves, E1 and E2, (superposition works because the wave equation is a linear differential equation) to find the total electric field. Let the physical fields be ( , t) cos( t 1 ) and ( , t) cos( t 2 ) so in this simple case the waves propagate in the same direction. Let’s say the amplitudes are E1 = E1 ẑ and E2 = E2 ẑ. Use complex notation (taking the real part only at the very end) to find ET(r,t) = E1(r,t)+E2(r,t) in the form ( , t) cos( T T t T ) , giving expressions for the total amplitude , total phase shift, total k, total in terms of quantities from E1(r,t) and E2(r,t). Explicitly check your answer in the special case 1 = 2. 2. Spring constant of an atom. In Chapter 9 of Griffith, we model an atom as an electron with mass m attached to the heavy nucleus with a spring. This atomic spring has zero equilibrium length and spring constant k = 2 m0 . Here we show that this model, though crude, is not crazy. Quantum mechanics tells us that in the R d d + + ground state of hydrogen, the electron wavefunction is a spherically-symmetric cloud of charge centered on the atom. So, without too much violence to reality, we can model the electron as a uniform sphere of charge of radius R (R = bohr radius = 0.53 angstroms = 0.53 1010 m). When an external E-field is applied, the nucleus is pushed one way and the electron cloud is pushed the other way, so there is an induced dipole moment. We assume that the electron’s sphere of negative charge is not distorted in shape when it is displaced. We also assume that the nucleus is so heavy that it remains stationary, while the electron cloud can oscillate about the stationary nucleus. A) Show that when the nucleus is displaced from the center of the electron cloud by a distance d (d < R), there is a restoring force that is proportional to d. Derive an expression the spring constant k in this situation. B) Compute the value of the spring constant k, for the case of a hydrogen atom. Give the answer in both SI units ( N/m = J/m2 ) and in slightly more natural units of electron-volts per angstrom2 (eV/Å2). What is the natural frequency f0 = (in Hz) of this mass-spring system? Page 3 of 3 What is the wavelength of light that has this frequency (answer in nm ) ? What part of the spectrum does this correspond to (visible?, IR?, UV?, radio? , etc) 3. Driven, damped, harmonic oscillator. Our microscopic model for index of refraction involves a classical mass-spring model: a mass m, attached to a fixed point m 2 with a spring of spring constant k = m0 . The equilibrium length of the spring is zero. There a friction force given by Ffric = m dx/dt. The mass is driven by a x (complex) external sinusoidal force Fdrive F0 exp(it) . A) Write down the differential equation which describes the motion of this system. B) Assume a complex sinusoidal solution x x 0 exp(it) x 0 exp(i ) exp(it) and solve for x 0 , x 0 , and . (Don’t skip any steps. You can find the answer in the book, so we are grading based on the quality of your derivation, not on the correctness of your final answer.) C) Assume , and use Mathematica to make a plot of the amplitude x0 vs. and a plot of the phase shift vs.