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Introduction to Accelerators Unit 1 – Lecture 2 Preliminaries: Maxwell’S Equations & Special Relativity

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Introduction to Accelerators Unit 1 – Lecture 2 Preliminaries: Maxwell’S Equations & Special Relativity Introduction to Accelerators Unit 1 – Lecture 2 Preliminaries: Maxwell’s Equations & Special Relativity William A. Barletta Director, United States Particle Accelerator School Dept. of Physics, MIT US Particle Accelerator School Mechanics, Maxwell’s Equations & Special Relativity US Particle Accelerator School Relativity describes transformations of physical laws between inertial frames y y What is an inertial frame? x v x How can you tell? z z In an inertial frame free bodies have no acceleration US Particle Accelerator School Postulate of Galilean relativity Under the Galilean transformation x = x Vxt y = y v x = vx Vx z = z t'= t the laws of physics remain invariant in all inertial frames. Not true for electrodynamics ! For example, the propagation of light US Particle Accelerator School Observational basis of special relativity Observation 1: Light never overtakes light in empty space ==> Velocity of light is the same for all observers For this discussion let c = 1 World line t of physicist moving at velocity v World line Space-time diagrams of physicist v = c = 1 at rest x US Particle Accelerator School Relativistic invariance Observation 2: All the laws of physics are the same in all inertial frames This requires the invariance of the space-time interval 2 2 2 2 2 2 2 2 ()ct x y z = ()ct x y z t World line of physicist t moving at velocity v v = c = 1 World line of physicist at rest x x US Particle Accelerator School The Galilean transformation is replaced by the Lorentz boost Where Einstein’s relativistic factors are US Particle Accelerator School Thus we have the Lorentz transformation x vt t (v /c 2 )x x = , t = 1 v 2 /c 2 1 v 2 /c 2 y = y , z = z Or in matrix form ct 00 ct x 00 x = y 0010 y z 0001 z US Particle Accelerator School Proper time & length We define the proper time, , as the duration measured in the rest frame The length of an object in its rest frame is Lo As seen by an observer moving at v, the duration, T , is T = > v 2 1 2 c And the length, L, is L = Lo/ US Particle Accelerator School Four-vectors Introduce 4-vectors, w, with 1 time-like and 3 space-like components ( = 0, 1, 2, 3) x = (ct, x, y, z) [Also, x = (ct, -x, -y, -z) Note Latin indices i =1, 2, 3 1/2 2 2 2 2 1/2 Norm of w is w = (w w ) = (wo w1 w2 w3 ) 2 μ w = gμ w w where the metric tensor is 10 0 0 0 10 0 g = μ 0010 00 01 US Particle Accelerator School Velocity, energy and momentum For a particle with 3-velocity v, the 4-velocity is dx u = (c,v) = d The total energy, E, of a particle is its rest mass, mo, plus kinetic energy, T 2 2 E = moc + T = moc The 4-momentum, p, is μ p = (cm0,m0v) 2 2 2 p = mo c US Particle Accelerator School Doppler shift of frequency Distinguish between coordinate Harvard Yale rows Light at rest past at transformations and observations cone velocity v K>1 Yale sets his signal to flash at a constant interval, t' Harvard sees the interval Light foreshortened by K(v) as Yale cone K<1 approaches Harvard see the interval stretched by K(-v) as Yale moves away Homework: Show K(v) = K-1(-v) & for large find K() US Particle Accelerator School Head-on Compton scattering by an ultra-relativistic electron 2 E=mc out in What wavelength is the photon scattered by 180°? US Particle Accelerator School Particle collisions Two particles have equal rest mass m0. Laboratory Frame (LF): one particle at rest, total energy is E. Centre of Momentum Frame (CMF): Velocities are equal & opposite, total energy is Ecm. Exercise: Relate E to Ecm US Particle Accelerator School The Basics - Mechanics US Particle Accelerator School Newton’s law We all know d F = p dt The 4-vector form is dm dp dpμ F μ = c , = dt dt d 2 2 2 Differentiate p = m o c with respect to μ 2 dp μ d(mc ) pμ = pμ F = F o v = 0 d dt The work is the rate of changing mc2 US Particle Accelerator School Harmonic oscillator Motion in the presence of a linear restoring force F = kx k x˙˙ + x = 0 m x A sin t where = k = o o m It is worth noting that the simple harmonic oscillator is a linearized example of the pendulum equation 3 x˙˙ 2 sin(x) x˙˙ 2 (x x ) 0 + o + o 6 = that governs the free electron laser instability US Particle Accelerator School Solution to the pendulum equation Use energy conservation to solve the equation exactly Multiply 2 by to get x˙˙ + o sin(x) = 0 x˙ 1 d d x˙ 2 2 cos x = 0 2 dt o dt Integrating we find that the energy is conserved 1 2 2 x˙ cos x = constant = energy of the system = E 2o With x= US Particle Accelerator School Stupakov: Chapter 1 Non-linear forces Beams subject to non-linear forces are commonplace in accelerators Examples include Space charge forces in beams with non-uniform charge distributions Forces from magnets high than quadrupoles Electromagnetic interactions of beams with external structures • Free Electron Lasers • Wakefields US Particle Accelerator School Properties of harmonic oscillators Total energy is conserved p2 m 2 x 2 U = + o 2m 2 If there are slow changes in m or , then I = U/o remains invariant U o = o U This effect is important as a diagnostic in measuring resonant properties of structures US Particle Accelerator School Hamiltonian systems In a Hamiltonian system, there exists generalized positions qi, generalized momenta pi, & a function H(q, p, t) describing the system evolution by dq H dpi H q {} q1,q2,....,qN i = = dt pi dt qi p {} p1, p2,...., pN H is called the Hamiltonian and q & p are canonical conjugate variables For q = usual spatial coordinates {x, y, z} & p their conjugate momentum components {px, py, pz} H coincides with the total energy of the system H = U +T = Potential Energy + Kinetic Energy Dissipative, inelastic, & stochastic processes are non-Hamiltonian US Particle Accelerator School Lorentz force on a charged particle Force, F, on a charged particle of charge q in an electric field E and a magnetic field, B 1 F = q E + v B c E = electric field with units of force per unit charge, newtons/coulomb = volts/m. B = magnetic flux density or magnetic induction, with units of newtons/ampere-m = Tesla = Weber/m2. US Particle Accelerator School A simple problem - bending radius Compute the bending radius, R, of a non-relativistic particle particle in a uniform magnetic field, B. Charge = q Energy = mv2/2 v mv 2 F = q B = F = Lorentz c centripital mvc pc = = qB qB p 1 1 T ()m = 3.34 1 GeV/c q B US Particle Accelerator School The fields come from charges & currents Coulomb’s Law r 1 q 1,2 1 ˆ F12 = q2 2 r1 2 = q2E1 4o r1,2 Biot-Savart Law i1dl1 r1,2 i2dl2 μ0 (i1dl1 rˆ12 ) dF12 = i2dl2 2 = i2dl2 B2 4 r1,2 US Particle Accelerator School Electric displacement & magnetic field In vacuum, The electric displacement is D = oE, The magnetic field is H = B/o Where -12 -7 o = 8.85x10 farad/m & o= 4 x10 henry/m. US Particle Accelerator School Maxwell’s equations (1) Electric charge density is source of the electric field, E (Gauss’s law) •E = Electric current density J = u is source of the magnetic induction field B (Ampere’s law) E B = μ J + μ o 0 o t If we want big magnetic fields, we need large current supplies US Particle Accelerator School Maxwell’s equations (2) Field lines of B are closed; i.e., no magnetic monopoles. •B = 0 Electromotive force around a closed circuit is proportional to rate of change of B through the circuit (Faraday’s law). B E = t US Particle Accelerator School Maxwell’s equations: integral form US Particle Accelerator School Exercise from Whittum Exercise: A charged particle has a kinetic energy of 50 keV. You wish to apply as large a force as possible. You may choose either an electric field of 500 kV/m or a magnetic induction of 0.1 T. Which should you choose (a) for an electron, (b) for a proton? US Particle Accelerator School Boundary conditions for a perfect conductor, = 1. If electric field lines terminate on a surface, they do so normal to the surface a) any tangential component would quickly be neutralized by lateral motion of charge within the surface. 2. Magnetic field lines avoid surfaces a) otherwise they would terminate, since the magnetic field is zero within the conductor. US Particle Accelerator School Lorentz transformations of E.M. fields Bz = Bz E z = E z v E = E vB Bx = Bx + E y x ()x y c 2 E y = ()E y + vBx v B = B E y y c 2 x Fields are invariant along the direction of motion, z US Particle Accelerator School The vector potential, A The Electric and magnetic fields can be derived from a four-vector potential, Aμ = ( , ) E = B = A Aμ transforms like the vector (ct, r) = () vAz Ax = Ax Ay = Ay v A = A z z c 2 US Particle Accelerator School Energy balance & the Poynting theorem The energy/unit volume of E-M field is The Poynting vector, S = E H = energy flux The Poynting theorem says rate of change of EM energy flow EM energy due to = - work done by E - through the interaction on moving charges enclosing surface with moving charges US Particle Accelerator School Stupakov: Ch.
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