Physics D Lecture Slides Jan 15 Vivek Sharma UCSD Physics
Relativistic Momentum and Revised Newton s Laws and the Special theory of relativity: Example : p= mu Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform Watching an Inelastic Collision between two putty balls S P = mv mv = 0 P = 0 Before 1 v v V=0 1 After S ' v1 v ' v v v V v v1 = = 0, v = =, V ' = = v vv 1 vv 1 Vv 1 1 v 1 1+ c c c c ' ' ' mv ' pbefore = mv1+ mv =, p ' after = mv = mv v 1+ c p ' before ' after v 1 =0 1 V 1 v Before After p
Definition (without proof) of Relativistic Momentum mu With the new definition relativistic p = =γ mu 1 ( u/ c) momentum is conserved in all frames of references : Do the exercise New Concepts Rest mass = mass of object measured In a frame of ref. where object is at rest γ = 1 1 ( u/ c) u is velocity of the object NOT of a reference frame!
Nature of Relativistic Momentum m u mu p = =γ mu 1 ( u/ c) With the new definition of Relativistic momentum Momentum is conserved in all frames of references Good old Newton
p mu = =γ mu 1 ( u/ c) Relativistic Force And Acceleration Reason why you cant quite get up to the speed of light no matter how hard you try! Relativistic Force & Acceleration dp d mu d du d F = = use = dt dt 1 ( u/ c) dt dt du m mu 1 u du F = + ( )( ) 3/ 1 ( u/ c) ( 1 ( u/ c) ) c dt mc mu + mu du F = 3/ c ( 1 ( u/ c) ) dt m du F = : Relativistic Force 3/ ( 1 ( u/ c) ) dt du Since Acceleration a =, dt F 3/ a = 1 ( / ) m u c Note: As u / c 1, a 0!!!! Its harder to accelerate when you get closer to speed of light
A Linear Particle Accelerator - F q E d V + E= V/d F=eE Charged particle q moves in straight line in a uniform electric field E with speed u accelarates under f orce F=qE du F u qe u a = = 1 = 1 dt m c m c larger the potential difference V across 3/ 3/ plates, larger the force on particle Under force, work is done on the particle, it gains Kinetic energy New Unit of Energy 1 ev = 1.6x 10-19 Joules
A Linear Particle Accelerator ee a= 1 ( u/ c) m 3/ PEP-II accelerator schematic and tunnel view
Magnetic Confinement & Circular Particle Accelerator Classically F = v m r V B qvb = v m r F B r dp d( γ mu) du F = = = γ m = qub dt dt dt du u = (Centripetal accelaration) dt r u γm = qub γmu = qbr p = qbr r
Charged Form of Matter & Anti-Matter in a B Field
Circular Particle Accelerator: LEP @ CERN, Geneve
Magnets Keep Circular Orbit of Particles
Inside A Circular Particle Accelerator @ CERN
Accelerating Electrons Thru RF Cavities
Test of Relativistic Momentum In Circular Accelerator mu p = = γ mu 1 ( u/ c) γ mu = qbr γ = qbr mu
Relativistic Work Done & Change in Energy x 1, u=0 X, u=u x x dp W = F. dx =. dx dt p x 1 1 = = 3/ u dt u 1 1 W = W u 0 x du m mu dp dt c du m dt u 1 c udt 3/ (change in var x u mudu mc = = mc = 3/ 1/ 0 u u 1 1 c c Work done is change in Kinetic energy K γ K = mc mc or Total Energy E= γ mc c = K + mc, substitute in W, u) γ mc mc
Why Can s Anything go faster than light? Lets accelerate a particle from rest, particle gains velocity & kinetic energy mc mc K = mc 1/ ( K + mc ) = 1/ u u 1 1 c c u = + c 4 1 mc K mc u = c K 1 ( + 1) mc K (Parabolic in u Vs ) mc Non-relativistic case: K = 1 K mu u = m
Relativistic Kinetic Energy
When Electron Goes Fast it Gets Fat E = γ mc v As 1, γ c Apparent Mass approaches
Relativistic Kinetic Energy & Newtonian Physics Relativistic KE = γ mc mc 1 u 1 u When u<< c, 1-1 c c +...smaller terms 1 u 1 c so K mc [1 ] mc = mu (classical form recovered) Total Energy of a Particle E = γ mc = KE + mc For a particle at rest, u = 0 Total Energy E= mc
E = γ mc E = γ m c 4 Relationship between P and E p 4 E p c = γ m c γ m u c = γ m ( c u ) For = γ mu pc = γ mc u 1 c 4 4 = ( c u ) = ( c u ) = m c muc ( )...important relation E = p c + mc mc c u particles with zero rest mass like photon (EM waves) E E= pc or p = (light has momentum!) c Relativistic Invariance : E p c = m c 4 : In all Ref Frames Rest Mass is a "finger print" of the particle
Mass Can Morph into Energy & Vice Verca Unlike in Newtonian mechanics In relativistic physics : Mass and Energy are the same thing New word/concept : Mass-Energy It is the mass-energy that is always conserved in every reaction : Before & After a reaction has happened Like squeezing a balloon : If you squeeze mass, it becomes (kinetic) energy & vice verca! CONVERSION FACTOR = C
Mass is Energy, Energy is Mass : Mass-Energy Conservation Examine Kinetic energy Before and After Inelastic Collision: Conserved? S K = mu K=0 Before 1 v v V=0 1 After Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction E = E before mc mc m + = Mc M = > m u u u 1 1 1 c c c Kinetic energy has been transformed into mass increase after K M = M -m = = mc c c u 1 c mc Kinetic energy is not lost, its transformed into more mass in final state
Conservation of Mass-Energy: Nuclear Fission M M 1 + M M 3 + Nuclear Fission Mc Mc Mc Mc = + + u u u 1 1 1 c c c 1 3 1 3 M > M + M + M 1 3 < 1 < 1 < 1 Loss of mass shows up as kinetic energy of final state particles Disintegration energy per fission Q=(M (M 1 +M +M 3 ))c = Mc 36 9 U Cs+ Rb+3 n 143 55 90 9 1 0-8 1 10 kg 165.4 MeV m=0.177537u=.947 = What makes it explosive is 1 mole U = 6.03 x 10 3 Nuclei!!
Relativistic Kinematics of Subatomic Particles Reconstructing Decay of a π Meson