E and B Fields (Homework 9-323)

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

$j ⃗=(j_x,0,0)=(ρv_x,0,0)=(λSv_x,0,0)$


$B_z=(μ_0 I)/2πr=(μ_0 j S)/2πy=(μ_0 λS^2 v_x)/2πy$ 


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 $g_x=(μ_0 γ^2 λ^2 S^2 v)/(4π^2 y^3 )$