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Download Preliminary version Particle motion in a uniform magnetic field The
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Preliminary version Particle motion in a uniform magnetic field ~ on a particle with charge q, mass m and velocity The Lorentz force exerted by the magnetic field B ~υ is d~υ q ~ . (1) = ~υ × B dt γm γ is the Lorentz factor, i.e. the ratio of the energy W of the particle to the rest energy mc2 . Since the acceleration is perpendicular to both the magnetic field vector and the velocity vector, the momentum or kinetic energy,a nd therefore the Lorentz factor, are constant. To see this, multiply the above equation by ~υ . The motion along the magnetic field has constant speed. The particle’s trajectory is hence helicoidal: a circle, superposed on a uniform motion along the magnetic field. For a uniform magnetic field that is constant in time, we can hence identify a reference frame that travels at constant speed along the magnetic field, within which the particle has a circular motion around the origin. This origin is called the guiding centre. In the reference frame of the guiding centre the velocity vector is perpendicular to the magnetic field. We will consider this situation in the following, i.e. υ will denote the particle speed perpendicular to the magnetic field. Integration of the equation of motion yields ~υ = ~r × ~ qB ~c . = ~r × Ω γm The angular frequency of the circular motion is Ωc = (cyclotron radius or Larmor radius) is hence rc = ~ |q||B| γm . (2) The radius of the circular orbit υ βγmc p 2 mc γmυ = = γ −1 . = ~ ~ ~ Ωc |q||B| |q||B| |q||B| Here β is the ratio of the particle speed to the speed of light, and γ = √ (3) 1 . 1−β 2 If the magnetic field is given in nanoTesla (nT), the cyclotron radius per unit charge is rc = 3.1 × 109 ( B −1 p 2 ) γ − 1 [m]. 1 nT (4) The cyclotron radius decreases with increasing charge - a natural consequence of the fact that the Lorentz force is proportional to charge. We can express it in a charge-independent way by introducing the magnetic rigidity: since the momentum of the particle is p = γmυ, rc = γmυ p R = = , ~ ~ ~ |q||B| |q||B| c|B| (5) pc where R := |q| is called the magnetic rigidity. This quantity measures the cyclotron radius in a given magnetic field, and is therefore an indicator of the sensitivity of the particle, whatever its 1 Table 1: Cyclotron radii in the terrestrial environment Magnetic rigidity [GV] 1 5 20 Kinetic energy [GeV] 0.43 4.1 19.1 Speed [% of c] 73 98 99.8 Corona (10 mT) 330 m 1.65 km 6.60 km Cyclotron radius IP (1 AU) Earth surface (5 nT) (5µT) 8 5 6.6 × 10 m ' 100 RE 1.1 × 10 m ' 0.017 RE 3.3 × 109 m ' 520 RE 5.5 × 105 m ' 0.086 RE 1.3 × 1010 m ' 2100 RE 2.2 × 106 m ' 0.34 RE charge or mass, to the magnetic field. The trajectory of a charged particle is the more strongly curved by the magnetic field, the lower its magnetic rigidity. If the magnetic rigidity is given in giga-Volt (GV), as is typical for cosmic rays detected by neutron monitors, the cyclotron radius is rc = 3.3 × 109 ( B −1 R ) ( ) [m]. 1 nT 1 GV (6) The magnetic rigidity is related to the energy of the particle by rc = 1p 2 pc = W − m2 c4 . q q 2 (7)