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E&M and Relativity

This lecture explores the fundamentals of Electromagnetism (E&M) and Relativity, focusing on concepts like total charge, currents, and the significance of free charges. It covers the dynamics of charged particles in magnetic fields, the Lorentz force law, and the cyclotron frequency. Key relationships such as energy units and 4-vectors are discussed to aid understanding. Additionally, the lecture delves into multipole expansions and the effects of magnetic fields, including the roles of dipoles, quadrupoles, and octupoles in beam dynamics and synchrotron motion.

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E&M and Relativity

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  1. E&M and Relativity Eric Prebys, FNAL

  2. Maxwell’s Equations Local effects of media Lecture 2 - Basic E&M and Relativity In terms of total charge and current In terms of free charge an current

  3. Example: Field in a permeable dipole Integration loop g Lecture 2 - Basic E&M and Relativity Cross section of dipole magnet

  4. Electrodynamics and Electrodynamic Potentials Lecture 2 - Basic E&M and Relativity We can write the electric and magnetic fields in terms of Vector and Scalar potentials Particle dynamics are governed by the Lorentz force law

  5. Cyclotron (1930’s) top view side view • A charged particle in a uniform magnetic field will follow a circular path of radius “Cyclotron Frequency” Red box = remember! For a proton: Accelerating “DEES” Lecture 2 - Basic E&M and Relativity

  6. Relativity Some Handy Relationships (homework) Lecture 2 - Basic E&M and Relativity • Basics • A word about units • 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]

  7. 4-Vectors and Lorentz Transformations Lecture 2 - Basic E&M and Relativity We’ll use the conventions Note that for a system of particles We’ll worry about field transformations later, as needed

  8. Some Handy Relationships Lecture 2 - Basic E&M and Relativity Know all of these by heart because you’re going to use them over and over!

  9. Synchrotrons and beam “rigidity” • The relativistic form of Newton’s Laws for a particle in a magneticfield is: • A particle in a uniform magnetic field will move in a circle of radius • In a “synchrotron”, the magnetic fields are varied as the beam accelerates such that at all points , and beam motion can be analyzed in a momentum independent way. • It is usual to talk about he beam “rigidity” in T-m Booster: (Br)~30 Tm LHC : (Br)~23000 Tm Lecture 2 - Basic E&M and Relativity

  10. Thin lens approximation and magnetic “kick” Lecture 2 - Basic E&M and Relativity If the path length through a transverse magnetic field is short compared to the bend radius of the particle, then we can think ofthe particle receiving a transverse “kick”and it will be bent through small angle In this “thin lens approximation”, a dipole is the equivalent of a prism in classical optics.

  11. Field multipole expansion Magnetic field is the gradient of a scalar… …which satisfies Laplace’s equation Lecture 2 - Basic E&M and Relativity Formally, in a current free region The general solution in two dimensions

  12. Lecture 2 - Basic E&M and Relativity • Solving for B components • Combining

  13. Lecture 2 - Basic E&M and Relativity Symmetry properties of mulitpoles The phase angle δmrepresents a rotation of each component about the axis. Set all δm=0 for the moment

  14. “normal” “skew” Lecture 2 - Basic E&M and Relativity Back to Cartesian Coordinates. Differentiate both sides n times wrt x And we can rewrite this as “Normal” terms always have Bx=0 on x axis. “Skew” terms always have By=0 on x axis. Generally define

  15. quadrupole sextupole dipole sextupole quadrupole octupole dipole Lecture 2 - Basic E&M and Relativity Expand first few terms… Note: in the absence of skew terms, on the x axis

  16. Application of Multipoles • A positive particle coming out of the page off center in the horizontal plane will experience a restoring kick Lecture 2 - Basic E&M and Relativity Dipoles: bend Quadrupoles: focus or defocus

  17. SextupolesOctupoles • Sextupole magnets have a field(on the principle axis) given by • One common application of this is to provide an effective position-dependent gradient. • In a similar way, octupoles have a field given by • So high amplitude particles will see a different average gradiant Lecture 2 - Basic E&M and Relativity

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