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

- 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 . - 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} - 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.- in a frame moving at a velocity
*u*in the direction; - in a frame moving at a velocity
*u*in the direction. - 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.

- in a frame moving at a velocity

2006-01-29