2. Consider the charged-particle in problem 1b. In all space (inside and outside the beam), find in the laboratory frame:
a. The E and B fields.
b. The energy density and momentum density. Griffiths equations 8.5 and 8.29 remain relativistically correct in the frame where fields are measured.
In all space (just the section S is different):
The electric filed is already (from Q1,b)
$E_y=(γ^2 λ)/(2πϵ_0 y^2 )$ $E_x=0$ (assume xy plane)
The magnetic field of an infinite current
$B_z=(μ_0 I)/2πr=(μ_0 j S)/2πy=(μ_0 λS^2 v_x)/2πy$
The energy density is
$u=1/2*(ϵ_0 E^2+B^2/μ_0 )=$
$=1/2*((γ^4 λ^2)/(4π^2 ϵ_0 y^4 )+(μ_0 λ^2 S^4 v^2)/(4π^2 y^2))=λ^2/(8π^2 y^2 ) (γ^4/ϵ_0 y^2 +μ_0 S^4 v^2 )$
The momentum density is
$g=ϵ_0 (E ⃗ ×B ⃗ )$ and thus $g_x=(μ_0 γ^2 λ^2 S^2 v)/(4π^2 y^3 )$