THE UNIVERSITY OF BRITISH COLUMBIA

Physics 401 Assignment # 5:

RELATIVISTIC ELECTRODYNAMICS

Wed. 1 Feb. 2006 - finish by Wed. 8 Feb.

This is a relatively [pun intended] short Assignment, since the first Midterm Exam is on Monday February 6 (in class, 50 minutes). Nevertheless it will count the same as other Assignments, if you choose to tackle it. For the exam, you may bring your own 1-page summary sheet with any hard-to-remember equations etc. The exam will cover all of Chapters 7 and 8, sections 10.1 of Chapter 10, and all of Chapter 12. There will not be anything on the Midterm explicitly about (second problem below) but applications of Eqs. (12.108) are fair game.

1. If we stay in the Lorentz gauge ( ), is a genuine 4-vector and therefore is a Lorentz scalar. Use this fact to show that, in any frame, , where is the charge density in the frame where .

2. Use the formal definition of the FIELD TENSOR in Eq. (12.118)1 and the rule for its Lorentz transformation,2  , to derive the equations analogous to Eqs. (12.108) describing the transformation of and under a "boost" into a reference frame moving at velocity u (with the usual corresponding definitions of and ) in the positive direction.3

3. A capacitor made from two square parallel plates a on a side and d apart is given a charge -Q on the upper plate and +Q on the lower plate. Let the origin of coordinates be in the centre of the capacitor, the edges of the plates parallel to and and the gap in the direction.

Using Lorentz transformations, find and inside the capacitor
1. in a frame moving at a velocity u in the direction;

2. in a frame moving at a velocity u in the direction.

3. For the former case, compare your result with what you would expect from simply transforming the dimensions of the plates into the moving frame and treating their motion as sheets of current.

Jess H. Brewer
2006-01-29