Particle in uniform B-field 1/26/2018 Particle in uniform B-field Equation of motion of a particle of charge qs = se (s = ±1, for s = i, e) in a given uniform magnetic field, B0, pointing in the z-direction Dot v into both sides to show that v2 = const. Magnetic field can do no work on particles. Can’t change their kinetic energy, but can change their momentum. Decompose particle velocity, v, into components parallel and perpendicular to B: v = v|| + v Parallel and perpendicular motions are independent Parallel velocity, v|| is constant. Parallel motion is free-streaming. Since v2 is constant, so is the magnitude v = |v| Perpendicular motion is circular around B (looking down on B). Both r and v rotate at the cyclotron frequency, s Ions move clockwise, electrons counterclockwise Combined motion is spiraling parallel or antiparallel to B Lorenz force provides centripetal acceleration in perp plane z, B0 x y z, B0 x y Ri Re ve vi ions electrons

1/26/2018 Gyromotion and Drifts Constant magnetic field, B in z-direction: v||, |v| = const Constant B and constant force, F Electrons drift opposite from ions unless F = seE. In the electric force they both drift the same way: Inhomogeneous straight magnetic field lines Example of B = (0, 0, B(y)), (note, ·B = 0 ) Constant drift, vD, smaller than local gyrospeed, u, is ap-proximate solution when gradient scale length, Ly, is large compared to local gyroradius and drift motion is time-averaged

Drifts and Adiabatic Invariants 1/26/2018 Drifts and Adiabatic Invariants Curvature drift Let Rc be the radius of curvature of a magnetic field line with constant magnitude, B. Let v|| be the parallel velocity (curving) of particles gyrating along that field line. The curvature drift, vc, is orthogonal to both the local B and the local radius of curvature vector Rc (i.e. is azimuthal for a cylindrically symmetric set of curved field lines) and given by the following expression: Magnetic moment The magnetic moment provides a simple way to understand the effective parallel force on the guiding center of a particle s = e, i gyrating along a magnetic field line (in z-direction) when surrounding field lines are converging or diverging. The relation between the force and the gradient of the magnetic field is: F||eff = - s∂zBz or more generally, F|| = -s·B The magnetic moment is an approximate (adiabatic) invariant of the motion of the guiding center when the timescale for motion of the guiding center is slow compared to the gyrofrequency (parallel force small compared to perpendicular force causing gyromotion) smsv2/2B B Rc

Ring current is due to drifts 1/26/2018 Ring current is due to drifts

1/26/2018 Van Allen Belt trapped electrons NASA Website: http://www-istp.gsfc.nasa.gov/Education/wtrap1.html Charged particles--ions and electrons--can be trapped by the Earth's magnetic field. Their motions are an elaborate dance--a blend of three periodic motions which take place simultaneously: 1. A fast rotation (or "gyration") around magnetic field lines, typically thousands of times each second. 2. A slower back-and-forth bounce along the field line, typically lasting 1/10 second. 3. A slow drift around the magnetic axis of the Earth, from the current field line to its neighbor, staying roughly at the same distance. Typical time to circle the Earth--a few minutes. View from North Pole Because positive ions and negative electrons drift in opposite directions, that motion will create an electric current that circulates clockwise around the Earth when viewed from north. The current is aptly named the ring current. The magnetic field produced by the ring current contributes (rather slightly) to the magnetic field observed at the surface of the Earth. There are however times when the population of trapped particles is greatly reinforced. The ring current then becomes stronger and its magnetic effect at Earth may grow 10-fold or more: that is known as a magnetic storm.

Magnetic mirror and toroidal device drifts 1/26/2018 Magnetic mirror and toroidal device drifts