We can now consider the radiated energy d E per particle traversed distance d x particle . It can be expressed through the electromagnetic energy flow P a through the surface of an infinite cylinder of radius a around the path of the moving particle, which is given by the integral of the Poynting vector S = c / ( 4 π π ) [ E × × H ] over the cylinder surface: ( d E d x particle ) rad = 1 v P a = − − c 4 π π v ∫ ∫ − − ∞ ∞ ∞ ∞ 2 π π a B 3 E 1 d x