A reference frame K' is moving with uniform velocity **v **= v**i**
with respect to reference frame K.

(a) In K, a plane wave with angular
frequency ω is traveling in the **i** direction. What is its frequency in K'?

(b) In K, a plane wave with angular frequency ω is traveling in the
**j
**direction. What is its frequency in K'?

(c) In K, a plane wave with
angular frequency ω is traveling in a direction making an angle of 45^{o}
with respect to the **i** direction and the **j **direction. What is its
frequency in K'?

Consider an infinite sheet of charge with uniform charge density ρ = σδ(x) in
the y-z plane.

(a) An observer moves on a trajectory **r**(t) = (x_{0}, 0, vt).
Calculate the 4-vector current density and electromagnetic fields **E** and
**B** in the rest frame of this observer.

(b) Calculate the 4-vector current density and electromagnetic fields
**E**
and **B** in the rest frame of an observer moving along the x-axis in the
positive x-direction with speed v with respect to the sheet of charge.

Calculate the force, as observed in the laboratory, between two electrons moving side by side along parallel paths 1 mm apart, if they have a kinetic energy of 1 eV and 1 MeV.

(a) A fast electron (kinetic energy = 5*10^{-17
}Joule) enters a region of space containing a
uniform electric field of magnitude **E** = -**i** 1000 V/m. The field is parallel to the electron's motion
and in a direction such as to decelerate it. How far does the electron travel
before it is brought to rest? Neglect radiation losses.

(b) Now assume that the initial velocity of the electron is **v** = v_{0}**j**,
perpendicular to the direction of the electric field. Assume that at t = 0
the electron moves through the origin. Find the speed of the electron, its
position, and the work done by the field on the electron as a function of time.

(c) Find the trajectory of the electron, x(y).

Starting with the transformation of the electromagnetic fields under a
Lorentz transformation show that

(a) if **E** is normal to **B** in an inertial frame, it must be true in
all other inertial frames, and

(b) if |E| > c|B| in an inertial frame, it must be true in all other inertial
frames.