Download Preliminary version Particle motion in a uniform magnetic field The

Preliminary version
Particle motion in a uniform magnetic field
~ on a particle with charge q, mass m and velocity
The Lorentz force exerted by the magnetic field B
~υ is
d~υ
q
~ .
(1)
=
~υ × B
dt
γm
γ is the Lorentz factor, i.e. the ratio of the energy W of the particle to the rest energy mc2 . Since
the acceleration is perpendicular to both the magnetic field vector and the velocity vector, the
momentum or kinetic energy,a nd therefore the Lorentz factor, are constant. To see this, multiply
the above equation by ~υ . The motion along the magnetic field has constant speed. The particle’s
trajectory is hence helicoidal: a circle, superposed on a uniform motion along the magnetic field.
For a uniform magnetic field that is constant in time, we can hence identify a reference frame that
travels at constant speed along the magnetic field, within which the particle has a circular motion
around the origin. This origin is called the guiding centre. In the reference frame of the guiding
centre the velocity vector is perpendicular to the magnetic field. We will consider this situation in
the following, i.e. υ will denote the particle speed perpendicular to the magnetic field.
Integration of the equation of motion yields
~υ = ~r ×
~
qB
~c .
= ~r × Ω
γm
The angular frequency of the circular motion is Ωc =
(cyclotron radius or Larmor radius) is hence
rc =
~
|q||B|
γm .
(2)
The radius of the circular orbit
υ
βγmc p 2
mc
γmυ
=
= γ −1
.
=
~
~
~
Ωc
|q||B|
|q||B|
|q||B|
Here β is the ratio of the particle speed to the speed of light, and γ = √
(3)
1
.
1−β 2
If the magnetic field
is given in nanoTesla (nT), the cyclotron radius per unit charge is
rc = 3.1 × 109 (
B −1 p 2
)
γ − 1 [m].
1 nT
(4)
The cyclotron radius decreases with increasing charge - a natural consequence of the fact that
the Lorentz force is proportional to charge. We can express it in a charge-independent way by
introducing the magnetic rigidity: since the momentum of the particle is p = γmυ,
rc =
γmυ
p
R
=
=
,
~
~
~
|q||B|
|q||B|
c|B|
(5)
pc
where R := |q|
is called the magnetic rigidity. This quantity measures the cyclotron radius in
a given magnetic field, and is therefore an indicator of the sensitivity of the particle, whatever its
1
Table 1: Cyclotron radii in the terrestrial environment
Magnetic
rigidity
[GV]
1
5
20
Kinetic
energy
[GeV]
0.43
4.1
19.1
Speed
[% of c]
73
98
99.8
Corona
(10 mT)
330 m
1.65 km
6.60 km
Cyclotron radius
IP (1 AU)
Earth surface
(5 nT)
(5µT)
8
5
6.6 × 10 m ' 100 RE 1.1 × 10 m ' 0.017 RE
3.3 × 109 m ' 520 RE 5.5 × 105 m ' 0.086 RE
1.3 × 1010 m ' 2100 RE
2.2 × 106 m ' 0.34 RE
charge or mass, to the magnetic field. The trajectory of a charged particle is the more strongly
curved by the magnetic field, the lower its magnetic rigidity. If the magnetic rigidity is given in
giga-Volt (GV), as is typical for cosmic rays detected by neutron monitors, the cyclotron radius is
rc = 3.3 × 109 (
B −1 R
) (
) [m].
1 nT
1 GV
(6)
The magnetic rigidity is related to the energy of the particle by
rc =
1p 2
pc
=
W − m2 c4 .
q
q
2
(7)