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Download Midterm Exam No. 03 (Spring 2015) PHYS 520B: Electromagnetic Theory
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Midterm Exam No. 03 (Spring 2015) PHYS 520B: Electromagnetic Theory Date: 2014 Apr 15 1. (40 points.) A particle of mass m and charge q moving in a uniform magnetic field B and uniform electric field E experiences a velocity dependent force F given by the expression m dv = q [E + v × B] , dt (1) where v(t) = dx/dt is the velocity of the particle in terms of its position x(t). Choose the magnetic field to be along the positive z direction, B = Bẑ, and the electric field to be along the positive y direction, E = E ŷ. (a) For the case when the particle starts at rest at the origin at time t = 0, use the initial conditions v(0) = 0 x̂ + 0 ŷ + 0 ẑ, x(0) = 0 x̂ + 0 ŷ + 0 ẑ, (2) (3) to solve the differential equation in Eq. (1) to find the position x(t) and velocity v(t) as a function of time. Use ω = qB/m. (b) In particular, prove that the particle takes a path along a cycloid. That is, the particle moves as though it were a spot on the rim of a wheel rolling along the xaxis. It satisfies the equation of a circle of radius R whose center (vt, R, 0) travels along the x-direction at constant speed v, (x − vt)2 + (y − R)2 = R2 . (4) Determine the radius R and speed v in terms of E, B, q, and m. Show that v = ωR. 2. (30 points.) The path of a relativistic particle moving along a straight line with constant (proper) acceleration α is described by the equation of a hyperbola x2 − c2 t2 = x20 , x0 = c2 . α (5) This is the motion of a particle ‘dropped’ from x = x0 at t = 0 in a region of constant (proper) acceleration. (a) Will a photon dispatched to ‘chase’ this particle at t = 0 from x = 0 ever catch up with it? If yes, when and where does it catch up? 1 (b) Will a photon dispatched to ‘chase’ this particle at t = 0 from 0 < x < x0 ever catch up with it? If yes, when and where does it catch up? (c) Will a photon dispatched to ‘chase’ this particle, at t = 0 from x < x0 ever catch up with it? If yes, when and where does it catch up? 3. (30 points.) The Poincaré formula for the addition of (parallel) velocities is v= va + vb va vb , 1+ 2 c (6) where va and vb are velocities and c is speed of light in vacuum. (For the following assume that the Poincaré formula holds for all speeds, subluminal (vi < c), superluminal (vi > c), and speed of light.) Analyse what is obtained if you add a subluminal speed to a superluminal speed? That is, is the resultant speed subluminal or superluminal. Hint: Analyse the case va c (7) = − ± δ, c vb for infinitely small δ > 0. 2