Larmor Orbits

1
Larmor Orbits

• The general solution of the harmonic oscillator
  equation

by choosing the initial conditions
can be rewritten as (for a positive charge q)
2
Larmor Orbits (II)

• Since the equation for vy was

then

• The position of the particle is found by
  integrating the equations

3
Larmor Orbits (III)

• A simple integration yields

and

• The coordinates of the particle are clearly
  describing a circular motion of radius (again,
  qgt0)

called Larmor radius or gyroradius
4
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

5
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

6
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
7
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

8
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
9
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

10
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

11
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

12
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

13
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

14
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)

15
Charge Motion in a Gravitational Field

• For a gravitational field Fmg, 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

16
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

17
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
18
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

19
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 y00

• Where y was found from the solution for the
  Larmor orbit (here y00)

20
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

21
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

22
Curvature Drift due to Curved Magnetic Field
q
r
Rc

• A magnetic field has field lines of constant
  curvature radius Rc
• Maxwells equations in a vacuum region prescribe
  that

23
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

24
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
25
Curvature Drift due to Curved Magnetic Field (V)

• The gradB drift due to the curved magnetic field
  will be then

or
26
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 centers 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

27
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