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Unformatted text preview: 1 Gyration in a uniform magnetic field Of fundamental importance in space physics is how individual particles behave in the presence of electric and magnetic fields. The resulting particle orbits can usually be thought of as the sum of two parts, gyration and drift . Gyration is the periodic motion of a charged particle about a magnetic field line, and usually represents the fastest time scale of the particle orbit. In addition, the particle may also move steadily in a specified direction, and this is known as particle drift. Gyration is just the tendency of a charge particle to describe a circular orbit about a magnetic field. To illustrate this consider a point particle of charge q and mass m in which-→ B = B b z ,-→ E = 0, and where B is a constant. It follows that the Lorentz Equation m d dt-→ v = q ‡-→ E +-→ v ×-→ B · (1) can be written as m d dt v x = q v y B m d dt v y =- q v x B (2) and m d dt v z = 0 (3) Equation (3) yields v z = v k = constant; i.e., the velocity parallel to the magnetic field is a constant of the motion. Equation (2), the motion perpendicular to the magnetic field, can be solved to yield v x = v ⊥ sin( ω c t + α ) v y = v ⊥ cos( ω c t + α ) (4) The speed v ⊥ ≡ p v 2 x + v 2 y perpendicular to the field is another constant of the motion, ω c ≡ qB/m is called the cyclotron frequency , and α is the initial phase of the wave. Figure 1 (“Gyro Motion”) shows a sketch of the expected particle orbit. Note that ω c is a signed quantity, and depends on the sign of the charge q . 2 The ExB (E Cross B) Drift Now consider the case of a particle in a uniform magnetic field-→ B = B b z and a non-zero but constant electric field-→ E . The equation of motion along the magnetic field becomes m d dt v z = q-→ E · b z = q E k , (5) 1 Figure 1: Illustration of particle gyro-motion, bounce motion, and drift motion in the mag- netosphere. From http://www-ssc.igpp.ucla.edu/personnel/russell/papers/magsphere where E k = constant is the component of the electric field parallel to the magnetic field. It follows that v z = v k + qE k m t, (6) where v k is the parallel velocity at t = 0. Thus, the particle accelerates without bound. In reality, the presence of such a parallel electric field will cause a current to flow that will act to short-circuit the electric field. Given the high mobility of the electrons, the time scale for such a process is extremely small, so it is normally quite difficult to maintain parallel electric fields for even short periods of time. In order to analyze the motion perpendicular to the magnetic field, define-→ v ( t ) in terms of two new variables; i.e.,-→ v ( t ) =-→ v 1 ( t ) +-→ V D , (7) where V D is a constant. Since we have already determined the parallel motion, in what follows assume that-→ E ·-→ B = 0 ( E k = 0). We get m d dt ‡-→ v 1 ( t ) +-→ V D · = q h-→ E + ‡-→ v 1 ( t ) +-→ V D · ×-→ B i (8) which can be rewritten as m d dt-→ v...
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This note was uploaded on 09/10/2010 for the course ECE 107 taught by Professor Fullterton during the Spring '07 term at UCSD.
- Spring '07