In reference frame K a long,
straight, neutral wire with a circular cross sectional area A =
πr02 lies centered on the x-axis and carries a current with uniform current
density j i. For r > r0, choose the scalar potential to
be Φ = 0 and the vector potential to be
A = -[μ0I/(2π)] ln(r)
i.
In a frame K' moving with velocity vi with respect to K,
find the ρ', j', Φ' and A' at the point
P on the z-axis a distance z = z' > r0 from the wire.
Determine the E and B fields of an electric dipole moving along the z-axis with a velocity v = v k relative to the laboratory using the Lorentz transformation of the fields. The dipole moment in its rest frame is p = p0 k.
(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 = v0j,
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).
A reference frame K' is moving with uniform velocity v = vi
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 45o
with respect to the i direction and the j direction. What is its
frequency in K'?