Problem 1 [20 points]
Consider making a Lorentz transformation L1 from inertial reference frame K to the inertial reference frame K', where K' moves with velocity v1 in the x direction with respect to K. Then make a second Lorentz transformation L2 from frame K' to frame K", where K" moves with velocity v2 in the x direction with respect to K'. Find the velocity v3 with which K" is moving with respect to the initial frame K. This gives the relativisitic law of addition of velocities!
To find the answer, use the result that the matrix of the combined transformation L3 = L2L1 from K to K" is given by the product of the matrices for the transformations L2 and L1, i.e.
aμν(L3) = ∑σ aμσ(L2)aσν(L1)
Compute the above matrix and identify the velocity v3 by comparing it to the known form for a Lorentz transformation with velocity v3 in the x direction.
Problem 2 [30 points]
Consider an infinitely long straight cylindrical wire of radius R, centered about the x axis. Suppose that in an inertial frame K there is a total current I flowing down the wire in the x direction and the wire also has a net charge per unit length λ. Assume the current I and the charge λ are uniformly distributed over the cross-sectional area of the wire.
a) We know that the 4-current jμ = (j, icρ) is a 4-vector, where j is the current density and ρ is the charge density. For the geometry described above, argue why Iμ = (Iex, icλ) also transforms like a 4-vector with respect to Lorentz transformations directed along the x axis. Does Iμ also transform like a 4-vector with respect to Lorentz transformations in directions perpendicular to the x axis? You must explain your answer, not just say "yes" or "no". Here ex is the unit vector in the x direction.
b) If I and λ are the current and charge per length seen in frame K, what are the current I′ and charge per length λ′ as seen in a frame K′ that moves with velocity vex with respect to frame K? In what situation is it possible to transform to a frame K′ in which I′ = 0? In what situation is it possible to transform to a frame K′ in which λ′ = 0?
c) Compute the electric and magnetic fields E(r, t) and B(r, t) in frame K from the current I and charge λ. Then compute the electric and magnetic fields E′(r′, t′) and B(r′, t′) in frame K′ by applying a Lorentz transformation to the fields E(r, t) and B(r, t) in frame K. Now compute the fields E′ and B′ in frame K′ directly from the current I′ and charge λ′ that you found in (b). Show that these two approaches give the same result for the fields in K′. For this part you may assume that the radius of the wire is vanishingly small.