# Zimmar Physics Accelarator Notes-13.pdf - 1.5.5 Vector...

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1.5.5Vector Analysisa·(b×c) =b·(c×a) =c·(a×b)a×(b×c) = (a·c)b(a·b)c(a×b)·(c×d) = (a·c)(b·d)(a·d)(b·c)∇ × ∇ψ= 0∇ ·(∇ ×a) = 0(φψ) =φψ+ψφ∇ ×(∇ ×a) =(∇ ·a)− ∇2a∇ ·(ψa) =a· ∇ψ+ψ∇ ·a∇ ×(ψa) =ψ×a+ψ∇ ×a(a·b) = (a· ∇)b+ (b· ∇)a+a×(∇ ×b) +b×(∇ ×a)∇ ·(a×b) =b·(∇ ×a)a·(∇ ×b)∇ ×(a×b) =a(∇ ·b)b(∇ ·a)+(b· ∇)a(a· ∇)ba×(b×c) +b×(c×a) +c×(a×b) = 01.5.6RelativityLetFbe the stationary laboratory frame withspace time coordinates(x, t). LetFwith(x , t)be a frame moving with velocityV=withrespect toF. Lorentz transformations:Coordinates :x=x+γβγγ+1β·xctt=γ(t1cβ·x)Velocity :v=v+γVγγ+1v·Vc21γ1V·vc2Energy-momentum :P=P+γβγγ+1β·P1cEE=γ(E·P)EM fields :E=γ(E+×B)γ2γ+1(β·E)βB=γ(B1cβ×E)γ2γ+1(β·B)βWhenV=Vˆx, the above becomesCoordinates :x=γ(xV t),t=γ(tV xc2)y=y,z=zVelocity :vx=vxV1βvxcvy=vy1βvxc,vz=vz1βvxcEnergy-momentum :Px=γ(PxβcE),E=γ(EcβPx)Py=Py,Pz=PzEM fields :Ex=Ex,Bx=BxEy=γ(EycβBz),By=γ(By+βcEz)Ez=γ(Ez+cβBy),Bz=γ(BzβcEy)1.6GLOSSARY OF ACCELERATORTYPES1.6.1Antiproton SourcesK. Gollwitzer, J. Marriner, FNALAntiproton (¯p) sources are complete acceleratorcomplexes utilizing many accelerator technolo-gies [1]–[4].A primary proton beam is used toproduce¯p’s on a target. The production processis inefficient, and the secondary¯pbeam is several