Classical Electrodynamics — PHY5347

HOMEWORK 15

(January 15, 2013)

Due on Thursday, February 7, 2013

PROBLEM 43

Three men (A,B, and C) are riding on a train; A is in front, B is in the middle, andC is on the rear. A fourth man (O) is standing on the platform and sees the trainmoving at a constant velocity v to the right. At the moment that B passes in frontof O, two light signals coming from A and C reach them at the very same instant.

(a) According to passenger B, who emitted the light signal first (A or C)? Evaluatequantitatively the difference between the times of emission of the two flashes.

(b) According to the stationary observer O, who emitted the light signal first (Aor C)? Evaluate quantitatively the difference between the times of emission ofthe two flashes. Using simple arguments explain the qualitative nature of youranswer.

PROBLEM 44

Consider the case of constant electric and magnetic fields given by the following simpleexpressions:

E = E0y and B = B0z .

Assume without loss of generality that B0>E0>0.

(a) Can you find a system of reference in which the magnetic field vanishes? If so,show explicitly the Lorentz transformation that accomplishes this and the valueof the electric field in the new system of reference.

(b) Can you find a system of reference in which the electric field vanishes? If so,show explicitly the Lorentz transformation that accomplishes this and the valueof the magnetic field in the new system of reference.

PROBLEM 45

Consider a particle of charge q and mass m.

(a) Assume that the particle, initially at rest, is placed in a uniform electric fieldgiven by E= zE. Compute the velocity and position of the particle as a functionof time. Make a plot of your results and of those obtained in the non-relativistic(β�1) limit. Plot your results in units of t0 =mc/qE.

(b) Assume that the particle is now placed in a uniform magnetic field given byB = zB. Compute the velocity and position of the particle as a function oftime. Assume the following initial conditions: x(0) =−a, y(0) = v and y(0) =z(0)= x(0)= z(0)=0.

(c) In the large hadron collider protons circulating in opposite directions within a27 km ring will collide with an energy of 7 TeV each. Using the equations ob-tained in part (b), estimate the magnitude of the magnetic field that is requiredto maintain the protons in such a circular orbit.