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Unformatted text preview: Cornell University Department of Physics Classical Electrodynamics P561 Second Exam November 29, 2007 • You have 3 hours to work on the exam. • This is an open notes open Jackson exam (but no other book or printed material may be used): you can use any of your handwritten notes, but no book other than Jackson or anything printed is permitted. You may use calculators but no computers. Also NO collaboration is allowed and you may not be in real or virtual contact with a living person. • Do not open the exam until you are told to do so. • If something is not clear about the exam, ask Johannes. • Each of the four problems carry the same weight (25 points). However not each subpart will be graded equally, the value of the subparts is noted everywhere. • Graded exams will be available Monday afternoon, Dec. 3 from Johannes. NAME: SIGNATURE: 1. Multipole moments a) A sphere of radius R has a charge density in its volume given by ρ = ρ0 + ρ1 cos θ. Calculate the - total charge (1 point) - electric dipole moment vector (2 points) - electric quadrupole moment tensor (3 points) of this setup. Calulate the electric field far away from this setup up to the quadrupole term (2 points). b) (8 points) We now start rotating this sphere with angular frequency ω around the z-axis. What will be the magnetic dipole moment vector? What will be the leading approximation for the magnetic field in this setup? c) (9 points) Assume now that there is no rotation, but the radius of the sphere is changing as R(t) = R + r0 cos ωt. Use the following general formulae for radiation of a time dependent diple p to find the radiation electric field E and the angular distribution of the radiation: 1 ˆ r¨ E = 2 r × (ˆ × p ) cr ¨ |p |2 dP = sin2 θ, dΩ 4πc3 where r = r/r and the double dots represent the second time derivative. At what frequencies ˆ (in terms of ω ) will the system radiate? Note that in this case ρ(x, t) = ρ(x)eiωt , so we cannot apply the formulae (9.19) and (9.23) in chapter 9 of Jackson. 2a) n=1 ki kr 11111111111 00000000000 11111111111 00000000000 ǫ, σ 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 kc 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 2b) n=1 ki 11111 00000 11111 00000 ǫ, σ 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 n=1 kt 2. Reflection of plane waves from a conductor (Note: Depending on our approach, this problem may turn out to be the calculationally most intensive part of the exam. You may consider leaving this problem to the end, after you have worked on the other problems.) A linearly polarized plane wave in free empty space is orthogonally incident on a conductor with dielectric constant ǫ and conductivity σ (for the magnetic permeability you can assume that µ = 1 everywhere). a) (13 points) Calculate the amplitude of the reflected and refracted waves. Assuming that the conductivity is very small ( 4πσ ≪ ǫ), calculate the leading correction to the usual ω Fresnel equations. What fraction would be reflected for a very good conductor ( 4πσ ≫ ǫ)? ω b) (12 points) Assume that the material is a very good conductor and has a thickness d. What fraction of the incoming wave will get transmitted in the limit 4πσ ≫ ǫ ∼ O(1)? ω Hint: You may find it useful to work with a complex index of refraction n = though you can solve the probem without that too. ǫ + i 4πσ , ω y 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 z x 3. TM modes in a triangular wave guide A wave guide is made of an√ ideal conductor, which has an inner cross section of an isosceles right triangle with sides a, a, 2a. The goal in this problem is to find the T Mmn modes in this waveguides, identify the cutoff frequencies and the amount of energy transmitted. a) (4 points) What are the boundary conditions that Ez (x, y ) has to satisfy? b) (5 points) To find the relevant modes for the triangular wave guide, first consider the modes in the corresponding square wave guide with sides of length a. What is the form of the TMnm modes in this square wave guide? c) (6 points) Because of the discrete symmetry x ↔ y for both the square and triangular wave guides, the solutions TMnm and T Mmn from b) are degenerate. Write a superposition of the TMnm and TMmn fields obtained in b) which also satisfy the BC at the third side of the triangle at y = a − x. d) (4 points) What is the cutoff frequency for the TMnm mode obtained in part c? What is the lowest cutoff frequency? e) (6 points) Find the transverse components of the E and B -fields for the triangular wave guide. 4. Radiation of the classical electron Assume that an electron with charge e is in a circular orbit around the proton (which you can substitute with the Coulomb potential). a) (8 points) How much energy is radiated out by the electron per unit time in terms of the constants me , c, e and the initial velocity v of the electron (assume the electron is in a non-relativistic circular orbit). b) (8 points) Re-expressing your answer from part a) in terms of the total energy E (kinetic + potential) of the electron, find a differential equation relating dE/dt to E using only me , e and c as constants. c) (9 points) How long would it take for the electron to fall into the center? Give a formula for this time and a numerical answer as well. Assume that the initial radius that the electron starts at is at a0 = 5 · 10−9 cm. Relevant constants for the numerical part will be me = 9.110 · 10−28 g, e = 4.803 · 10−10 esu and c = 3 · 1010 cm/s. ...
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