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Basic Maths and Physics for Accelerators.Pdf Basic Maths and Physics for Accelerators C. Biscari Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 Physical constants Constant Symbol Value Speed of light in vacuum c 2.99792458 108 m/s Electric charge unit e 1.60217653 10-19 C 2 Electron rest energy mec 0.51099818 MeV 2 Proton rest energy mpc 938.27231 MeV Fine structure constant a 1/137036 Avogadro’s number A 6.021415 1023 1/mol -15 Classical electron radius rc 2.8179433 10 m Planck’s constant h 4.1356675 10-15 eV s l of 1 eV photon ħc/e 12398.419 Ȧ -12 Permittivity of vacuum o 8.85418782 10 C/(Vm) -6 Permeability of vacuum o 1.25663706 10 Vs/(Am) SI measurement system Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 2 Vector calculus Remember for example: Gradient of a scalar function For a vector 휑 푥, 푦, 푧, 푡 휕휑 휕휑 휕휑 퐹 = 퐹1, 퐹2, 퐹3 훻휑 = , , 휕푥 휕푦 휕푧 Divergence Curl 훻 × 퐹 휕퐹3 휕퐹2 휕퐹1 휕퐹3 휕퐹2 휕퐹1 휕퐹1 휕퐹2 휕퐹3 = − , − , − 훻 ∙ 퐹 = + + 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 휕푥 휕푦 휕푧 A vector potential is a vector field whose curl is a given vector field. A scalar potential is a scalar field whose gradient is a given vector field Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 3 Maxwell’s Equations Relate electric and magnetic fields generated by charge and current distributions Put together in 1863 results known from work of Gauss, Faraday, Ampere, Biot, Savart and others 훻 ∙ 퐷 = 휌 훻 ∙ 퐵 = 0 휕푩 훻 × 퐸 = − 휕푡 휕푫 훻 × 퐻 = 푗 + 휕푡 2 In vacuum 퐷 = 휖0 퐸, 퐵 = 휇0퐻, 휖0휇0푐 = 1 Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 4 1st Maxwell equation Equivalent to Gauss’s Flux Theorem, or Coulomb’s Law 휌 1 푄 훻 ∙ 퐸 = ↔ 훻 ∙ 퐸푑푉 = 퐸 d푆 = 휌푑푉 = 휖0 휖0 휖0 푉 푆 푉 The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface Field generated by a point charge: 푞 푟 푞 1 퐸 = 3 ↔ 퐸푟 = 2 4휋휖0 푟 4휋휖0 푟 푞 푑푠 푞 퐸 d푆 = 2 = 4휋휖0 푟 휖0 푠푝ℎ푒푟푒 푠푝ℎ푒푟푒 Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 5 2nd Maxwell equation Gauss Law for Magnetism 훻 ∙ 퐵 = 0 퐵푑푆 = 0 The net magnetic flux out of any closed surface is zero. Surround a magnetic dipole with a closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole. If there were a magnetic monopole source, this would give a non-zero integral. The magnetic field can be derived from a vector potential A defined by B = ∇×A. Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 6 3rd Maxwell equation Relates electric field and non constant magnetic field Equivalent to Faraday’s law of Induction 휕푩 훻 × 퐸 = − 휕푡 휕푩 훻 × 퐸 ∙ 푑푆 = − 푑푆 휕푡 푆 푆 푑 푑Φ 퐸 ∙ 푑푙 = − 퐵 ∙ 푑푆 = − 푑푡 푑푡 퐶 푆 Faraday’s Law is the basis for electric generators, inductors and transformers. The electromotive force round a circuit is proportional to the rate of change of flux of magnetic field through the circuit. Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 7 4th Maxwell equation Relates magnetic field and not constant electric field. Equivalent to Faraday’s law of Induction 휕푫 1 1 휕 휖퐸 훻 × 퐻 = 푗 + ↔ 훻 × 퐵 = 휇 푗 + 휕푡 휇 표 푐2 휕푡 From Ampere’s (Circuital) Law : 훻 × 퐵 = 휇0푗 Type equation here. 퐵 ∙ 푑푙 = 훻 × 퐵 ∙ 푑푆 = 휇0 푗 ∙ 푑푆 = 휇0퐼 퐶 푆 푆 Satisfied by the field for a steady line current (Biot-Savart Law, 1820): 휇0퐼 푑푙 × 푟 퐵 = 4휋 푟3 휇0퐼 Which for a straight line current 퐵 = 2휋푟 Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 8 Example Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 9 Example: Field in a permeable dipole • Cross section of dipole magnet Integration loop g 1 B g H dl B dl gap C steel path in steel 0 Bgapg Ienclosed 0 m m steel gap N I B 0 turns gap g Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 10 10 Relativity For the most part, we will use SI units, except – Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J] – Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2] – Momentum: eV/c [proton @ b=.9 = 1.94 GeV/c] v pc b c E 1 1 b 2 momentum p mv total energy E mc2 kinetic energy K E mc2 2 E 2 mc2 pc2 Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 11 Incremental relationships between energy, velocity and momentum Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 12 Particle velocity as a function of kinetic energy v b b 1 Particle at light velocity c c 1 1 0.8 0.8 electrons electrons 0.6 0.6 protons protons b b 0.4 0.4 0.2 0.2 0 0 0 1 2 3 4 5 0 5 10 Energy (MeV) Energy (GeV) Electrons are ultrarelativistic at few MeV Protons at few GeV (mass 2000 times electron mass) Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 Electrodynamic Potentials We can write the electric and magnetic fields in terms of Vector and Scalar potentials B Ar,t A E t dp dv F eE v B m ; for v c dt dt dp ; relativistically correct dt Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 14 14 Lorentz’s Force Particle dynamics are governed by the Lorentz force law F qE v B dv F m for v c dt dp F for any v dt Acceleration Bending and focusing E electric field B magnetic field Electric field Beam Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 15 Particle in a magnetic field The magnetic force always acts at right angles to the charge motion, the magnetic force can do no work on the charge. The B-field cannot speed up or slow down a moving charge; it can only change the direction in which the charge is moving. The component of velocity of the charged particle that is parallel to the magnetic field is unaffected, i.e. the charge moves at a constant speed along the direction of the magnetic field. Negatively charged particles circulate in the opposite direction as positively charged particles. The direction can be found using the right hand rule applied to the perpendicular component of the velocity Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 16 Motion in constant electric field 푑 푚 훾푣 표 = q 퐸 푑푡 푞퐸 훾푣 = t 푚표 2 2 훾푣 푞퐸푡 훾2 = 1 + 훾 = 1 + 푐 푚표푐 If the electric field is constant: 퐸 = 퐸, 0,0 It can be demonstrated that 2 푚표푐 훾 − 1 = 푞퐸(푥 − 푥표) Uniform acceleration in straight line Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 17 Motion in a constant magnetic field • A charged particle moves in circular arcs of radius r with angular velocity w: dp d dv F mv m qv B dt dt dt dr v w r qv B mw mw v dt mv2 qvB r v qB w r m Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 18 Example: Cyclotron (1930’s) side view • A charged particle in a uniform top view magnetic field will follow a B circular path of radius B r mv r (v c) qB v f 2r “Cyclotron Frequency” qB (constant!!) 2m qB 2f s m For a proton: fC 15.2 B[T] MHz Accelerating “DEES” Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 Rigidity Relation between radius and momentum 푝 퐵휌 = 푞 How hard (or easy) is a particle to deflect? • Often expressed in [T-m] (easy to calculate B) • Be careful when q≠e!! 퐺푒푉 푝[ ] 퐵휌[푇푚] ≈ 3.33 푐 푞[푒] Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 20 Electrons and protons (now in MeV) 퐸표 = 0.511 푀푒푉 표푟 938.27 푀푒푉 퐸푡표푡 = 퐸푘푛 + 퐸표 퐸푡표푡 1 훾 = , 훽 = 1 − 2 퐸표 훾 푝 = 훽퐸푡표푡 −3 퐵휌 = 3.33 10 푝 Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 Ions Kinetic energy per nucleon: Ekin Total charge = Qe 퐸푒 = 0.511 푀푒푉 퐸푝 = 938.27 푀푒푉 퐸푛 = 939.57푀푒푉 푁푒, 푍, 푁 ⇒ 퐴 = 퐴푡표푚푖푐 푚푎푠푠 푍 + 푁 퐸표 = 푍퐸푝 + 푁퐸푝 + 푁푒퐸푒 − 퐴 ∗ 0.8 퐸푡표푡 = 퐴 퐸푘푛 + 퐸표 퐸푡표푡 1 훾 = , 훽 = 1 − 2 퐸표 훾 푝 = 훽퐸푡표푡 −3 퐵휌 = 3.33 10 푝/Q Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 Example: particle spectrometer Identify particle momentum by measuring bending angle from a calibrated magnetic field B Accelerator Physics - UAB 2015-16 Basic Maths & Physics C. Biscari - Lecture 2 23 Simplified Particle Motion Design trajectory • Particle motion will be expanded around a design trajectory or orbit • This orbit can be over linacs, transfer lines, rings Separation of fields: Lorentz force • Magnetic fields from static or slowly-changing magnets transverse to design trajectory • Electric fields from high-frequency RF cavities in direction of design trajectory • Relativistic charged particle velocities Slide from T.
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