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Announcements • Reading for Review chapters 1 and 2 A proton (mass 1.6e-27 kg) is travelling at 0.99 c. How much Kinetic energy does it have? • HW 4 Due Wed. noon. Some of the topics on this will be on the exam! Don’t be late. K = (γ-1)mc2 a) 6e-11 Joules • First Midterm is on the 16th. Will cover relativity b) 2e-10 Joules – I can put some other practice exams (not multiple choice up if it will help). c) 9e-10 Joules d) 1e-9 Joules • Extra Office Hours. Duane F619 e) 4e-8 Joules – Today 1-4pm – Tomorrow 10-12pm (I will also be in the help room 34:30) Today’s class Review: Relativistic mechanics The good and bad of nuclear energy: • • • • Momentum and energy review Applications Massless particles Review (more next time) Or: Or: Relativistic force: F dr mu dt proper dp d m u dt dt Etot = γmc2 = K + mc2 These definitions fulfill the momentum and energy conservation laws. And for u<<c the definitions for p, F, and K match the classical definitions. But we found that funny stuff happens to the proper mass ‘m’. Equivalence of Mass and Energy We use conservation of energy because it often simplifies calculations. Remember the charged particle (charge -e) with mass m in a uniform electric field ℇ ? Plot the velocity u of the particle as a function of time t (assume the particle is released at t = 0). Potential energy: U = -eV Therefore: pm Total energy of a particle with mass ‘m’: Example simpler: Energy Relativistic Kinetic Energy: K ≡ Relativistic momentum: (γ-1)mc2 m Conservation of the total energy requires that the final energy Etot,final is the same 2 as the energy Etot, before 2 thetot collision. final Therefore: initial E =γ Mc = 2K + 2mc We find that the total mass M of the final system is larger than the sum of the masses of the two parts! M>2m. qV = (γ-1)mc2 qV /mc2 +1= γ -v Etot,final = Mc2 ≡ 2K + 2mc2 = Etot,initial 0-qV = K-0 v m 1 1 Then just solve for u. No integration necessary. u2 c2 Potential energy inside an object contributes to its mass!!! This argument isn’t a “proof”. But let’s see if it’s correct. 1 Example: Rest energy of an object with 1kg Turns out this is very useful E0 = mc2 = (1 kg)·(3·108 m/s )2 = 9·1016 J 9·1016 J = 2.5·1010 kWh = 2.9 GW · 1 year This is a very large amount of energy! (Equivalent to the yearly output of ~3 very large nuclear reactors.) Enough to power all the homes in Colorado for a year! E=mc2 Convert mass to energy? Atomic cores are built from neutrons and protons. There are very strong attractive forces between them. The potential energy associated with the force keeping them together in the core is called the binding energy E B. We now know that the total rest energy of the particle equals the sum of the rest energy of all constituents minus the total binding energy EB (the binding energy is what counters repulsion and lowers internal potential energy): Mc2 = Σ(mi c2) – EB Mc2 = Σ(mi c2) – EB Or in terms of Mass per nucleon 2 Example: Deuterium fusion Example: Deuterium fusion Isotopes of Hydrogen: Isotope mass: Deuterium: 2.01355321270 u Helium 4: 4.00260325415 u (1 u ≈ 1.66·10-27 kg) 1kg of Deuterium yields ~0.994 kg of Helium 4. Energy equivalent of 6 grams: E0 = mc2 = (0.006 kg)·(3·108 m/s )2 = 5.4·1014 J Enough to power ~20,000 American households for 1 year! Rest energy is real: Why does the sun shine? Rest Energy is real: Particle creation The bubble chamber shows trajectories of charged particles. Hans Bethe first proposed this mechanism 4 Hydrogen → 1 Helium 4 Important comments In this bubble chamber picture, a photon hits an electron so hard that two new massive particles, an electron and a “positron”, are created from the extra energy. The remaining energy shows up as kinetic energy of motion. Chain reactions Modern experiments probe higher and higher energies. Initial mass of a neutron and a 235U nucleus. More and more particles! Intermediate nucleus is unstable 236U Particle tracks from LHC. Final products have less mass, but much more kinetic energy. Conversion of mass to kinetic energy. As an experimental result: These quantities (momentum and energy as above) ARE conserved. Seen over and over again in high energy experiments. Oh yes, and more neutrons, so the reaction can run wild (chain reaction!). 3 Relationship of Energy and momentum Recall: Total Energy: E Momentum: γmc2 = p = γmu Application: Massless particles From the momentum-energy relation E2 = p2c2 + m2c4 we obtain for mass-less particles (i.e. m=0): Therefore: p2c2 = γ2m2u2c2 = γ2m2c4 · u2/c2 use: u2 2 1 2 c2 E = pc , (if m=0) p2c2 = γ2m2c4 – m2c4 p=γmu and E=γmc2 p/u = E/c2 =E2 This leads us the momentum-energy relation: E2 = or: (pc)2 (mc2)2 + E2 = (pc)2 + E02 Using E=pc leads to: u=c , (if m=0) Massless particles travel at the speed of light!! … no matter what their total energy is!! Example: Electron-positron annihilation Do neutrinos have a mass? Positrons (e+, aka. antielectron) have exactly the same mass as electrons (e-) but the opposite charge: me+ = me-= 511 keV/c2 (1 eV ≈ 1.6·10-19J) Neutrinos are elementary particles. They come in three flavors: electron, muon, and tau neutrino (e,μ, τ). The standard model of particle physics predicted such particles. The prediction said that they were mass-less. E1, p1 eBAM! e+ E2, p2 At rest, an electron-positron pair has a total energy E = 2 · 511 keV. Once they come close enough to each other, they will annihilate one other and convert into two photons. Conservation momentum: 1 = -ptwo 2. What can of you tell about pthose photons? Conservation of energy: E1+E2 = 2mc2 , E1 = E2 = 511 keV Do neutrinos have a mass? (cont.) Bruno Pontecorvo predicted the ‘neutrino oscillation,’ a quantum mechanical phenomenon that allows the neutriono to change from one flavor to another while traveling from the sun to the earth. The fusion reaction that takes place in the sun (H + H He) produces such e. The standard solar model predicts the number of e coming from the sun. All attempts to measure this number on earth revealed only about one third of the number predicted by the standard solar model. What about those faster than light neutrinos? The OPERA collaboration (hundreds of scientists) has recently reported the discovery of faster than light neutrino’s What did they do? Why would this imply that the neutrinos have a mass? Massless particles travel at the speed of light! i.e. γ ∞, and therefore, the time seems to be standing still for the neutrino: ΔtEarth = γ ∙ Δtneutrino(“proper”) They shot a beam of neutrinos from CERN Measured their arrival time in Italy In the HW: muon or pion experiments. The half-live time of the muons/pions in the lab-frame is increased by the factor γ. 4 What about those faster than light neutrinos? What about those faster than light neutrinos? This is a simple of a measurement one can do. Measure the Time of Flight. velocity = distance/time Same result with simple proton distribution When the protons (and hence neutrion’s) left CERN When neutrino’s detected What about those faster than light neutrinos? With neutrino’s getting there 60nS before light would Summary SR • Classical relativity Galileo transformation • Special relativity (consequence of 'c' is the same in all inertial Is this result right? We know that neutrino’s can travel at sub luminal speeds (supernova 1987a) We know that other particles are governed by the constancy of speed of light. What could have gone wrong? What are the implications? frames; remember Michelson-Morley experiment) – Time dilation & Length contraction, events in spacetime Lorentz transformation – Spacetime interval (invariant under LT) – Relativistic forces, momentum and energy – Lot's of applications (and lot's of firecrackers) … Everything we have discussed to this point will be part of the first mid-term exam (including reading assignments and homework.) If you have questions ask as early as possible!! 5