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, q>0)
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 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

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 y0=0
Where y was found from the solution for the Larmor orbit (here y0=0):

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 Fieldq r Rc
A magnetic field has field lines of constant curvature radius Rc
Maxwell’s 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 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

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: