Lecture1-2

Its nal momentum is 1000 mevc what was the magnitude

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Unformatted text preview: ht- cone u༇  Spacelike vectors have posiCve norm o  (Dot product with itself) Timelike vectors have negaCve norm. u༇  Lightlike vectors are on the light- cone and have zero norm. u༇  49 Example: Norm of a 4- vector u༇ Dot the 4- vector with itself x = ( ct, x, y, z ) = ( 4, 3, 2, 1) norm = x ⋅ x = −16 + 9 + 4 + 1 = −2 this is a "future pointing" timelike vector u༇ Norm>0, spacelike u༇ Norm<0, Cmelike u༇ Norm=0, lightlike 50 QuesCon: Units u༇  To make the laws of physics simple and to make rotaCons in 4D most easily understandable, what should the speed of light be? A)3X108 m/s B)3X1010 cm/s C)1 m/s D)1 cm/s E)1 51 4- vectors u༇ 4- vectors transform under rotaCons u༇ The dot product of 4- vectors is a scalar and is invariant under rotaCons u༇ The norm of a vector can be posiCve, negaCve or zero o  Spacelike, Cmelike or lightlike respecCvely u༇ Transforming to an inerCal frame moving in the x direcCon is a rotaCon in the x- t plane. 52 The SpaceCme Interval Δs u༇ The spaceCme interval is the 4D distance between two events. o  spacelike if >0 o  Cmelike if <0 o  invariant under transformaCons (dot product) 53 RotaCons in 4D (Not on tests) RotaCon in the xy plane. β= γ= All 4- vectors transform the same way. v c 1 1− β2 cosh θ = γ tanh θ = β sinh θ = βγ RotaCon in the xt plane A rotaCon in the xt plane is a “boost” along the x direcCon to another inerCal frame 54 RotaCon Symmetry in 4D u༇ Includes 3D rotaCons o  RotaCons in xy, yz, xz planes u༇ Plus symmetry that all the laws of physics are the same in any inerCal frame. o  RotaCon in xt plane is boost in the x direcCon o  RotaCon in yt plane is boost in the y direcCon o  RotaCon in zt plane is boost in the z direcCon u༇ All 4- vectors transform with the save matrix 55 Velocity, Momentum 4- vectors u༇  The spaceCme (posiCon) 4- vector xµ is defined. u༇  To get a velocity 4- vector (or momentum) we cannot differenCate with respect to Cme. o  Cme is a component of a vector! o  we need something like a scalar which is invariant under transformaCons. o  The proper ;me τ is defined in every frame and thus an invariant. o  vx=dx/dτ= γ dx/dt o  px=γ m dx/dt u༇  This is the conserved momentum. 56 Compute 4- momentum u༇ Use proper Cme τ to get 4- velocity u༇ 4- velocity square is invariant: check u༇ 4=momentum = mass Cmes 4- velocity u༇ This will give us the Energy equaCon 57 Force EquaCon u༇  Since it is the 4- momentum that is conserved, we will conCnue to define Force in an inerCal frame to be the Cme derivaCve of the space component of the 4- momentum. u༇  We may also conCnue to define the 3- velocity and 3- acceleraCon in the frame in terms of the simple Cme derivaCve. 58 Example: Constant Force u༇ An electron is accelerated by a constant Electric field. Find the acceleraCon as a funcCon of the electron’s velocity. 59 Example: proton circles in constant B- field u༇  A proton with momentum p move in a circle in a constant B- field perpendicular to the moCon. What is the radius of the circle? (the radius of curvature of the proton path) o  Work at the point where the proton is at its minimum x, the velocity is in the +y direcCon, and let the B field be in the z direcCon. 60 Derive Important...
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