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'?
In reference frame K a long, straight, neutral wire with a circular cross
sectional area A = πR2 lies along the x-axis and carries a steady current
I in the +x direction.
The current density is j(x, y, z) = I/A i for r = (y2 +
z2)1/2 < R and zero everywhere else./
We write j = j- i = ρ-<v->i,
since the current is carried by the electrons. Both ρ- and <v->
are negative numbers, j- is a positive number.
In general j = (j- + j+). ρ = ρ- + ρ+.
In reference frame K j+ = 0, ρ- = -ρ+.
(a) Consider the negative charge and current density and the positive
charge and current density separately.
In a frame K' moving with velocity vi with respect to K, find the ρ-',
j-' and ρ+', j+' and ρ', j' by transforming the
appropriate 4-vector current densities.
Write your answers in terms of ρ- and <v->. Is
the wire neutral in K'? If not, is it positively or negatively
charged? What is the direction of the current flow in K'?
Hint: The y and z components of all current densities are zero in K and K'.
The cross sectional area of the wire is the same in K and K'. Inside the
wire the current densities are constant in K and K'. There is no need to
transform the coordinates.
(b) Now assume that v = <v->, i.e. K' moves in the
negative x-direction with a velocity equal to the drift velocity of the
electrons.
Write down the expressions for ρ-', j-' and ρ+',
j+' and ρ', j' for this special case. Is the wire neutral
in K'? If not, is it positively or negatively charged? Can you
argue why this should be the case by considering length contraction?
A point charge of magnitude q moves on the trajectory
z(t) = vt,
y(t) = x(t) = 0.
(a) Find the electric and
magnetic fields at an arbitrary point (x,y,z) as a function of the time
t.
(b) Find the energy current density (Poynting vector) at each point as a function of
time.
(c) What is the total amount of electromagnetic energy, which is radiated out of the
surface of a very large sphere of radius R centered at the origin?
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 particle accelerator accelerates
electrons to a speed of 0.999 999 999 7 c, which is very nearly equal to the
speed of light.
(a) What is the rest energy (in MeV) of the electrons?
(b) Find the magnitude of the relativistic momentum of the electrons.
Comparing it with the non-relativistic value, is it bigger, smaller? By what
factor?
(c) Compute the electron's total energy (in GeV).
(d) Assume the electrons collide with the nuclei of "stationary" hydrogen atoms
in a gas cell. Assume the collision products are a proton, an electron, and a
particle-antiparticle pair. What is the rest energy of the most massive
particle (or antiparticle) that can be created in such a collision?