General Physics I. Lecture 19: Relativistic Energy and Momentum. Prof. WAN, Xin ( 万歆 )

Transcription

1 General Physics I Lecture 19: Relativistic Energy and Momentum Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn

2 Outline Minkowski diagram Relativistic velocity, momentum, and energy The mass-energy equivalence Nuclear energy

3 Lorentz Transformation The complete transformation 1) t = γ(t ' + x ' v /c 2 ) y y ' v 2) x = γ(x ' + vt ') 3) 4) y = y ' z = z ' z Frank x z ' Mary x ' Or, 1) 2) 3) 4) ct = γ[(ct ') + β x '] x = γ[β(ct ') + x '] y = y ' z = z ' β = v /c γ = = 1 1 v 2 /c β 2

4 Space-Time Squeezing in (ct ', x ') = (1,0) (ct, x) = (γ, γ β) (ct ', x ') = (0,1) (ct, x) = (γ β, γ) Unit lengths for the ct' and x' axes change to ( ct x ) = ( γ γ β γ β γ ) ( ct ' x ' ) = γ ( 1 β β 1 ) ( ct ' x ' ) γ 1 + β 2 x X x ' X Frank = 1 + β2 1 β 2 ct ct ' X x ' X Mary v ct '

5 Minkowski Diagram x x ' What does now mean? What does here mean? q ct ' Well, they depend on the choice of frame. q ct tan θ = β

6 Minkowski Diagram x x ' now What does now mean? What does here mean? here ct ' In the rest frame K, now means t = const, and here means x = const. ct

7 Minkowski Diagram x x ' now here What does now mean? What does here mean? ct ' ct In the moving frame K', now means t' = const, and here means x' = const.

8 Length Contraction, Again right end same time in K C B left end x O q A same time in K' x ' tan θ = β ct ' ct Suppose the length of the stick is 1 in the K' frame. This means the length of OA is OA = 1 + β2 1 β 2 The length measured by Frank in the K frame is the length of OB. OC = OAcos θ BC = AC tanθ AC = OAsinθ OB =?

9 Length Contraction, Again same time in K x same time in K' x ' Therefore, the length of the stick measured in the K frame is OB = OC BC right end C B q A ct ' = 1 β 2 which is the standard length contraction result. Note that you can figure out left end O tan θ = β ct cosθ = β 2

10 Length Contraction, Again same time in K x same time in K' x ' Suppose the length of the stick is 1 in the K frame, i.e. Therefore, OA = 1 right end left end A O q E tan θ = β ct ' ct OE = OA/cosθ = 1 + β 2 The length measured by Mary in the K' frame is in units of the x' axis, or OE 1 + β 2 / 1 β 2 = 1 β 2

11 Rotation in Space ( x y ) = ( cosθ sin θ sinθ cosθ ) ( x ' y ' ) y y ' x ' Show that if we define Frank x Rose tanh ϕ = β = v/c The Lorentz transformations in the matrix form become ( ct x ) = ( cosh ϕ sinh ϕ sinh ϕ cosh ϕ ) ( ct ' x ' )

12 Velocity Transformation The complete transformation 1) t ' = γ(t vx/c 2 ) y y ' v 2) x ' = γ(x vt) 3) 4) y ' = y z ' = z z Frank x z ' Mary x ' Therefore, Recognize the velocity addition rule?

13 Velocity Transformation The complete transformation 1) t ' = γ(t vx/c 2 ) y y ' v 2) x ' = γ(x vt) 3) 4) y ' = y z ' = z z Frank x z ' Mary x '

14 Linear Momentum Linear momentum p must be conserved in all collisions. v -v v p xi = 0 p xf = 0 -v v 2v 1 + v 2 /c 2 v p ' xi = 2mv 1 + v 2 /c 2 p ' xf = 2mv

15 Linear Momentum Linear momentum p must be conserved in all collisions. The relativistic value calculated for p must approach the classical value mu as u approaches zero. Please verify the momentum conservation in the previous example of collision. Newton's second law becomes

16 Relativistic Energy We generalize energy from the work-kinetic energy theorem. We assume that the particle is accelerated from rest to some final speed u.

17 Relativistic Total Energy Evaluating the integral We can define Rest energy E 0 = mc 2 Total energy E = γ mc 2

18 Relativistic Kinetic Energy We can define the relativistic kinetic energy K At low u/c,

19 A Comparison

20 Mass-Energy Equivalence A small mass corresponds to an enormous amount of energy. This is a concept fundamental to nuclear and elementary-particle physics. Together with m 2 c 2 = ( E/c) 2 p 2 another invariant

21 The Most Famous Equation The release of atomic energy has not created a new problem. Albert Einstein on Atomic War or Peace

22 Fission of Uranium How much energy is released by the fission of uranium-235 to form barium-142 and krypton-91? The particle masses are 235U ( u) 142Ba ( u) 91Kr ( u) 1n ( u) 1n U 142 Ba + 91 Kr n The discovery of nuclear fission by Otto Hahn and Fritz Strassmann opened up a new era in human history. It seems to me that what makes the science behind this discovery so remarkable is that it was achieved by purely chemical means. Lise Meitner

23 Fission of Uranium 1. Determine the change in mass for one U-235 atom: Δ m = u 2. This means that for each mole of U-235 atoms ( g), the corresponding energy by Einstein's equation is E = mc 2 = J = kwh (Note that 1 kwh» 0.5 RMB) The experimental apparatus with which Otto Hahn and Fritz Strassmann discovered nuclear fission in 1938.

24 Otto Hahn's Notebook

25 Chain Reaction Otto Hahn Farmhall n U 142 Ba + 91 Kr n...he had contemplated suicide, when he first recognized the possible military use of fission; now, with the blame of its realization drawn squarely upon his shoulders, suicide again seemed a way to escape his desolation... Never has social responsibility hit a scientist with such impact.

26 Fukushima Nuclear Disaster The plant comprised six separate boiling water reactors originally designed by General Electric and maintained by the Tokyo Electric Power Company. Units 2 through 6 were BWR-4, while Unit 1 was the slightly older BWR-3 design. The Fukushima Nuclear Accident Independent Investigation Commission found the nuclear disaster was "manmade" and that its direct causes were all foreseeable.

27 Light Water Reactors Two main varieties of light water reactors are: the pressurized water reactor (PWR) and the boiling water reactor (BWR). The main difference between a BWR and PWR is that in a BWR, the reactor core heats water, which turns to steam and then drives a steam turbine. In a PWR, the reactor core heats water, which does not boil. This hot water then exchanges heat with a lower pressure water system, which turns to steam and drives the turbine.

28 Pressurized Water Reactor 中国首台也是世界首台第三代先进压水堆 AP1000 型核电机组 浙江三门核电站 1 号机组, 计划于 2015 年年底并网发电

29 Fission Reactor Philippsburg Nuclear Power Plant, Germany

30 Heavy Water Reactor Qinshan Phase III Units 1 & 2, located in Zhejiang China ( N E): Two CANDU 6 reactors, designed by Atomic Energy of Canada Limited (AECL), owned and operated by the Third Qinshan Nuclear Power Company Limited.

31 Deuteron Binding Energy A deuteron, which is the nucleus of a deuterium atom, contains one proton and one neutron and has a mass of u. The mass of proton is u and the mass of neutron u. Note that 1 u = kg. This total deuteron mass is not equal to the sum of the masses of the proton and neutron. The mass difference is therefore u. The energy equivalence of the mass difference is called the binding energy of the nucleus

32 Fusion In the nuclear reaction called fusion, two atomic nuclei combine to form a single nucleus. The fusion reaction in which two deuterium nuclei fuse to form a helium nucleus is of major importance in the development of controlledfusion reactors. The decrease in mass that results from the creation of one helium nucleus from two deuterium nuclei is kg. Hence, the corresponding excess energy that results from one fusion reaction is J = 23.9 MeV. If 1 g of deuterium is converted to helium, the energy released is about J!

33 Sustainable Energy Technologies that promote sustainable energy include renewable energy sources, such as hydroelectricity, solar energy, wind energy, wave power, geothermal energy, bioenergy, tidal power and also technologies designed to improve energy efficiency.