Physics 2D Lecture Slides. Oct 8. UCSD Physics. Vivek Sharma

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2 Physics D Lecture Slides Oct 8 Vivek Sharma UCSD Physics

3 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 1 γ = 1 ( u/ c) u is velocity of the object NOT of a referen ce frame!

4 Nature of Relativistic Momentum m u mu p = mu 1 ( u/ c) =γ With the new definition of Relativistic momentum Momentum is conserved in all frames of references

5 mu p= =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 F F F F dp d mu = = dt dt 1 ( u/ c) m mu 1 u du = + ( )( ) 3/ 1 ( u/ c) ( 1 ( u/ c) ) c dt + c ( 1 ( u/ c) ) mc mu mu du = 3/ m = 3/ ( 1 ( u/ c) ) du dt dt d u Since A ccel e r a tion a =, d t F 3/ a = 1 ( / ) m u c Note: As u / c 1, a 0!!!! Its harder to accelerate when closer to speed of light use d du d = dt dt du : Relativistic For you get ce

6 A Linear Particle Accelerator - F q E d V + Parallel Plates 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 a cross 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 = 1.6x10-19 Joules 1 MeV = 1.6x10-13 Joules 1 GeV = 1.6x10-10 Joules

7 A Linear Particle Accelerator ee a= 1 ( u/ c) m 3/ PEP-II accelerator schematic and tunnel view

8 Magnetic Confinement & Circular Particle Accelerator Classically v F = m r v qvb = m r F B V 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 = qb r r

9 Charged Form of Matter & Anti-Matter in a B Field

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11 Accelerating Electrons Thru RF Cavities

12 Circular Particle Accelerator: CERN, Geneve

13 Magnets Keep Circular Orbit of Particles

14 Inside A Circular Particle CERN

15 Test of Relativistic Momentum In Circular Accelerator mu p = = γ mu 1 ( u / c ) γ mu = qb r = p γ = qb r mu

16 Relativistic Work Done & Change in Energy x 1, u=0 X, u=u x x dp W = F. dx =. dx dt p x x 1 1 = = W = W u dt u 1 1 c c u 0 1 3/ du m mu dp dt du m dt u c udt 3/ (change in var x u u mudu m c = = mc = 3/ 1/ 0 u u 1 1 c c Work d one is change in Kinetic energy K, substitute i n W, γ ) mc m c K = mc mc γ or Total Energy E= γ mc = K + mc

17 But Professor Why Can s ANYTHING go faster than light? mc mc K = mc 1/ ( K + mc ) = 1/ u u 1 1 c c u 4 1 mc K mc = + c u = c K K 1 ( + 1) (Parabolic in u Vs ) mc mc 1 K Non-relativistic case: K = mu u = m

18 Relativistic Energy

19 A Digression: How to Handle Large/Small Numbers Example: consider very energetic particle with very large Energy E γ E mc + K K = = = + mc mc mc 1 Lets Say γ = 3x10 11, Now calculate u from 1 = 1 c γ Try this on your el-cheapo calculator, you will get u/c =1, u=c due to limited precision. In fact u c but not exactly!, try to get this analytically γ 1 1 = = 1 β (1 β)(1 + β) u Since β = 1, 1 + β = c 1 γ 1 β β = = 5 10, u = βc γ u = c!! Such particles are routinely produced in violent cosmic collisions u 1/ In Quizzes, you are Expected to perform Such simple approximations

20 When Electron Goes Fast it Gets Fat E = γ mc v As c 1, γ Apparent Mass approaches

21 Relativistic Kinetic Energy & Newtonian Physics Relativistic KE = γ mc mc 1 u 1 u When u<< c, smaller terms c + + c 1 u c 1 so K mc [1 + ] mc = mu (classical form recovered) Total Energy of a Pa r ticle E = γ mc = KE + mc For a particle at rest, u = 0 Total Energy E= m c

22 E = γ mc E = γ m c 4 Relationship between P and E p = γ mu pc = γ muc 4 E p c = γ m c γ m u c = γ m c ( c u ) F or mc mc 4 4 = ( c u ) = ( c u ) u c u 1 c E = p c + ( m particles with zero rest mass like pho c ) = mc...important relation ton (EM waves) E E= pc or p = (light has momentu c m!) Relativistic Invariance : E p c = m c 4 : In all Ref Frames Rest Mass is a "finger print" of the particle

23 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, just like spacetime 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 This exchange rate never changes!

24 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 E = E be mc mc Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction f ore after + = Mc u u 1 1 c c Kinetic energy has been transformed m M = > m u 1 c into mass increase K mc M = M - m = = mc c c u 1 c Kinetic energy is not lost, its transformed into more mass in final state