Classical Electrodynamics PHY5347

First Midterm ExamInstructor: Jorge Piekarewicz

(February 19, 2013)

DO NOT START WORKING ON THE EXAM UNTIL YOU HAVEREAD CAREFULLY THE FOLLOWING INSTRUCTIONS. The midtermconsists of 2 problems. You have exactly one hour and fifteen minutes (75 minutes)to complete the exam. After this time is over, sign below attesting that you haveupheld the Academic Honor Code. Please write clearly and dont forget to write yourname. Best of luck!

Academic Honor Code: Students are expected to uphold the Academic Honor Codepublished in the Florida State University Bulletin and the Student Handbook. The firstparagraph reads: The Academic Honor System of Florida State University is based onthe premise that each student has the responsibility (1) to uphold the highest standardsof academic integrity in the students own work, (2) to refuse to tolerate violationsof academic integrity in the University community, and (3) to foster a high sense ofintegrity and social responsibility on the part of the University community.

Students Name:

Students SignatureI have upheld the Academic Honor Code.

PROBLEM 1 (40 points)

Three men (A,B, and C) are riding on a train of length L0; A is in front, B is in themiddle, and C is on the rear. A fourth man (O) is standing on the platform and seesthe train moving at a constant velocity v to the right (i.e., v= vx). At the momentthat B passes in front of O, two light signals coming from A and C reach them at thevery same instant (use these light signals to synchronize the clocks of B and O).

(a) According to passenger B, what are the space-time coordinates of the emissionof the light signals by A and C, namely, xA =(ct

A, x

A) and x

C =(ct

C , x

C)?

(b) Using the information obtained in part (b), compute the space-time coordinatesof the emission of both light signals as measured by O, namely, xA = (ctA, xA)and xC =(ctC , xC)?

(c) As seen by O, how much earlier did C emit the light signal than A?

PROBLEM 2 (60 points)

For an observer located in the laboratory frame the electromagnetic four-vector po-tential A = (,A) is given by the following expression:

(r, t)0 and A(r, t)=B0y x .(a) Compute the electromagnetic fields E and B in the laboratory frame.

(b) Using the results obtained in part (a), write down the electromagnetic field tensorF in the laboratory frame.

(c) Compute the electromagnetic four-vector potential A=(,A) as measured byan observer moving relative to the laboratory frame with a velocity v=vx.

(d) Using the expression obtained in part (c), compute the electromagnetic fields E

and B in the moving frame.(e) Starting with the expression for F obtained in part (b), obtain the electromag-

netic field tensor F in the moving frame.(f) Could you identify at least one combination of electromagnetic fields that remains

invariant?

You may find the following information useful:

F = A A =

0 Ex Ey Ez

Ex 0 Bz ByEy Bz 0 BxEz By Bx 0

.