Larmor Orbits The general solution of the harmonic oscillator equation by choosing the initial conditions can be rewritten as (for a positive charge q)

Larmor Orbits (II) Since the equation for vy was then The position of the particle is found by integrating the equations

Larmor Orbits (III) A simple integration yields and The coordinates of the particle are clearly describing a circular motion of radius (again, q>0) called Larmor radius or gyroradius

Larmor Orbits (IV) By taking the real part of the general expressions for x, y, vx, vy the Larmor orbit solution is found as: The center of the orbit (x0,y0) remains fixed in a plane perpendicular to the magnetic field (unless other forces act on the charge) The motion along the magnetic field is not affected by the field itself: the 3D motion of a charged particle in a uniform magnetic field is then in general a helix

Plasma Diamagnetism Charged particles in a magnetic field execute circular orbits perpendicularly to the field itself These orbital motions are electric currents in circular paths The magnetic field generated by these currents tends to cancel (is opposed to) the external magnetic field that generated the orbits themselves This effect is called diamagnetism of the plasma

4.2.2 Charge in Uniform Electric and Magnetic Fields Charged particles in a magnetic field execute circular orbits perpendicularly to the field itself If an electric field is added, the particle will feel a push in the direction given by the electric force This push will produce an initial velocity in the direction of the electric force that will be then affecting the Larmor orbit E B

Charge in Uniform Electric and Magnetic Fields (II) The governing equation for the general motion of a particle in a electromagnetic field is then An electric field in the x-z plane in considered The magnetic field is still in the x-y plane The equation of the particle motion along z is simply The particle is then subject to a constant acceleration along z

Charge in Uniform Electric and Magnetic Fields (III) The equation of the particle motion along x will be The solution for vy is found by differentiation and substitution: or

Charge in Uniform Electric and Magnetic Fields (IV) The solution for the x,y velocity components will be then This solution shows that the Larmor orbit has a superimposed drift velocity in the y direction, perpendicular then to the electric field component that acts in the orbit plane The center of the Larmor orbit is called guiding center: this guiding center is subject to a drift velocity

Charge in Uniform Electric and Magnetic Fields (V) The general expression for the drift velocity is found by solving the equation for a generic E and B. The solution will be the sum of a gyration (Larmor) velocity plus some drift velocity: The gyration velocity is the solution of

Charge in Uniform Electric and Magnetic Fields (VI) The equation can be rewritten as and simplified using the definition of vw: The previous analysis shows that the drift velocity is a constant, therefore it must be:

Charge in Uniform Electric and Magnetic Fields (VII) It was also shown that the drift is perpendicular to the magnetic field By taking the cross product with the direction of the magnetic field only the perpendicular components of the equation are retained Vector identities:

Charge in Uniform Electric and Magnetic Fields (VIII) By applying the double cross product vector identity it comes Since the drift velocity is perpendicular to B its scalar product with B is zero. Finally then: The drift velocity vE is independent on q, m and on the initial conditions

4.2.3 Charge in Uniform Force Field and Magnetic Field The general expression for the drift velocity of a particle in a magnetic field B and a force field F is found by solving the equation The solution analogous to the one for the electric force is found by replacing E/q with F: Now this drift depends on the charge (and its sign)

Charge Motion in a Gravitational Field For a gravitational field F=mg, where g is the (local) gravity acceleration The drift is therefore Now this drift depends both on the charge (and its sign) and on the particle mass. The gravitational drift however is normally negligible

4.2.4 Charge Motion in Non Uniform Magnetic Field Non uniformities in the magnetic field make the solution based on the orbit theory much more complex In general the problem requires numerical solution Particle trajectory codes (and self-consistent particle simulation codes) are in general sufficient for the orbit analysis

Grad-B Perpendicular to the Magnetic Field The intensity of the magnetic field is changing only in the plane perpendicular to the field itself y B x gradB

Grad-B Perpendicular to the Magnetic Field (II) Orbit-averaged solution of The orbit-averaged force is computed as there was no drift (undisturbed orbit approximation) Force due to the magnetic field Since B is uniform along the x-axis the average of Fx is zero

Grad-B Perpendicular to the Magnetic Field (III) Orbit-averaging Fy requires vy and Bz From the solution for the Larmor it was found Taylor expansion of the magnetic field along y with y0=0 Where y was found from the solution for the Larmor orbit (here y0=0):

Grad-B Perpendicular to the Magnetic Field (IV) Then the force due to the magnetic field along y can be approximated as: The orbit average of the first term is zero while cos2 averages to ½ over one orbit period. Then:

Grad-B Drift Perpendicular to the Magnetic Field (V) Then average force due to the gradient will produce a drift according to the general relation that in this case will become: The general expression for an arbitrary direction of gradB and arbitrary charge q can be readily inferred as:

Curvature Drift due to Curved Magnetic Field q r Rc A magnetic field has field lines of constant curvature radius Rc Maxwell’s equations in a vacuum region prescribe that

Curvature Drift due to Curved Magnetic Field (III) For a magnetic field directed along q with variations only along r (gradB is directed along r) curlB is directed along z: Therefore a magnetic field with curved field lines will not have constant magnitude, that is it will have a finite gradient The particles in a curved magnetic field will be then always subjected to a gradB drift

Curvature Drift due to Curved Magnetic Field (IV) The gradB drift due to the curved magnetic field will be found from where gradB, is directed along r, can be estimated from and then

Curvature Drift due to Curved Magnetic Field (V) The gradB drift due to the curved magnetic field will be then or

Curvature Drift due to Curved Magnetic Field (VI) The guiding centers of the particle orbits moving along the field lines will feel a centrifugal force (present in the guiding center’s frame of reference) A guiding center with average velocity v// along the magnetic field will be subjected to a centrifugal force Then centrifugal force due to the gradient will produce a drift according to the general relation

Curvature Drift due to Curved Magnetic Field (VII) Then centrifugal force drift will be then The total drift due to the curved magnetic field will be then the sum of the gradB drift and of the centrifugal force drift: