NAME: Final Exam - PHYS 611 Electromagnetic Theory Mendes, Spring 2013, April 24 2013 During the exam you can consult your textbooks (Melia, Jackson, Panofsky/Phillips, Griffiths), the print-outs of classnotes, homeworks, and mid-term exam posted on the instructor s website, your classnotes, and your own graded homework/exam. No additional material is allowed to be used for consultation during the exam. No electronic devices allowed, except for a calculator. Duration of the Exam is 150 min. 1
(37% = 1 + 1 + 25 + 5 + 3 + 2) 1) Consider a harmonic plane wave propagating along +z-axis in free space with speed c. Assume that this plane wave is linearly polarized so that the electric field is oriented along the y-axis. Also consider that this wave has an angular frequency ω and an amplitude. a) Write down the harmonic wave function for each Cartesian component of the electric field as a function of. Make sure to express the magnitude of the k-vector in terms of. b) Write down the harmonic wave function for each Cartesian component of the magnetic field. Make sure to express the amplitude of the magnetic field in terms of the amplitude of the electric field. c) Now consider an observer in an inertial frame of reference K that is moving along the +zaxis with a constant speed v with respect to the original frame of reference K considered above. By using a Lorentz transformation, find the electric and magnetic field components for the K observer of the same plane wave described above. Make sure to express your final answer in terms of the K space-time coordinates (x,y,z,t ). d) Find the angular frequency ω for the K observer in terms of ω, v, and c? This result should give you an expression for the Doppler shift in relativistic terms. e) What is the magnitude k of the k-vector for the K observer? 2
f) From the previous results, determine the speed of propagation of the plane wave for the K observer? Does your result depend on v or ω. Make sure your result is consistent with the principles of special relativity. 3
(21% = 19 + 1 + 1) 2) In class we derived that the harmonic vector potential for an electric dipole oscillating at an angular frequency can be described by (equation 4.218 in Melia): where and r is the distance from the dipole to the point of observation when the wavelength = is much larger than the dipole size D (i.e.,. a) Starting from the result above, show that the radiation pattern at frequency of an oscillating electric dipole is given by: where is the angle between the electric dipole and the distance from the dipole to the point of observation. b) Based on this result, explain why the sky is blue? c) Why the result above cannot be applied to clouds? 4
(22%) 3) A small particle of charge q moves with a constant angular velocity ω in a circle of radius R centered in the x-y plane. Find the retarded electric field at the center of the circle, i.e. at the origin. 5
(20% = 4 each question) 4) For a plane wave reflecting on a flat interface as shown in the figure below, we derived in class that boundary conditions require that: which is essentially Snell s law written in terms of the k-vector components. Now consider a plane wave that is incident from a higher refractive-index medium into a lower refractive-index medium at an angle larger than the critical angle, which is defined by. In other words, consider the case: a) Show that in this case the z-component of the k-vector in the transmission medium is a pure imaginary number,, given by: where is the wavelength of the plane wave in vacuum, and is a real number. b) As an application of the previous result, consider that the plane wave is polarized perpendicularly to the plane of incidence (x-z plane of the Figure). Therefore, the electric field has only a y-component. Show that for the y-component of the electric field in the transmission medium is given by: where is the amplitude of the electric field at the interface and is angular frequency of the plane wave. As you can see, this result represents a wave that propagates along x-axis and exponentially decays along z-axis. 6
c) Show that, in the scenario described above, the non-vanishing components of the magnetic field are given by: d) Calculate the z-component of the Poynting vector and show that its time average is equal to zero. Therefore, no power flows in the z-direction. e) Calculate the x-component of the Poynting vector and show that its time average is different than zero in general. Therefore, there is power flowing in the x-direction. 7