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Unformatted text preview: EAD 234B: E&M Homework #7 Due Thursday, May 27, 2010 (1) Consider a transformation of the longitudinal position x and time t from one inertial frame to another moving with speed v, subject to the following requirements. • Linearity: x′ = a(v)(x−vt) t′ = b(v)x+e(v)t x = a(v)(x′+vt) t = b(v)x′+ f (v)t′ • The composition of two such transformations is also a transformation of the same form. Show that the Galilean transform and the Lorentz transform are the only possibilities consistent with these requirements. (2) You are given that a rocket moves with velocity v along the x axis, with the back end of the rocket, at x’ = 0, at position x = vt in the lab frame. Someone in the rocket bounces a ball off of the wall (at x’ = 0) at the back end of the rocket, with initial and final 3‐velocities and . (Be careful: v is the velocity of the rocket relative to the lab frame, and V’ is the velocity of the ball, according to an observer in the rocket frame.) Find the initial and final 3‐velocities of the ball as observed by someone in the lab frame. (3) The plane z = 0 has surface charge density σ. The plane z = h has the opposite surface charge density. There are no current densities in this lab reference frame. a) Find the surface charge density σ’ and surface current on those surfaces as seen by an observer in a rocket, moving along the +x axis with velocity v. b) Find the electric and magnetic fields frame. Using your answer to part (a), compute everywhere in space according to an observer in the lab and as measured by an observer in the rocket frame. Now verify that the relation you found between the fields in the two frames of reference agree with the Lorentz transformation formula . (4) A plane, electromagnetic wave has its wave vector k making an angle θ with respect to the positive x‐axis.: it has components kx=kcosθ, ky=ksinθ, and kz=0. The wave is linearly polarized with the electric field in the x‐y plane. The flux (energy received per unit area and time) in the wave is F. Write down an expression for the electric field and magnetic field of the wave (assume an arbitrary initial phase at r=0 and t=0). Transform these fields to a frame of reference K’ moving at velocity v along the x‐axis. Show that the fields in this frame of reference imply that the wave is propagating at an angle θ’ with respect to the x’ axis given by and that the flux measured in K’ is (5) Each of two very thin, long, parallel beams of electrons of the same velocity u carries electric charge of density λ per unit length (as observed in the coordinate frame moving with electrons). (i) Calculate the distribution of the electric and magnetic fields in the system (outside the beams), as measured in the lab frame. (ii) Calculate the interaction force between the beams (per particle) and the resulting acceleration, both in the lab frame, and in the system moving with the electrons. Compare the results and give a brief discussion of the comparison. (6) Consider the motion of a point charge in crossed (i.e., perpendicular) uniform fields E and B. (a) Show that the fields in the frame drifting with velocity v = cE × B / B 2 are E ′ = 0 , and ⎛ E2 ⎞ B′ = B ⎜ 1 − 2 ⎟ . ⎝ B⎠
What is the significance of this frame? (b) Show that in this drifting frame, the particle undergoes circular motion with angular frequency 1/ 2 ′ u⊥ ′ where R′ is the radius of the circle and u⊥ is the component perpendicular to the magnetic R′ field of the particle’s velocity relative to the drifting frame. Find R′, ω ′ . ω′ = (c) Discuss the relativistic distortions to the non‐relativistic cycloidal motion, for the special case of a particle starting from rest in the laboratory, with particular attention to the circulation radius (and therefore the cycle length). ...
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- Spring '10