2.3.2 Drift in crossed E × B fields
(
%
Y Y !
FIGURE 2.4
Drift in crossed E×B fields.
While we are on this topic, let’s consider the case of uniform
E and B fields that are perpendicular — a situation often encountered when dealing with plasma and beams.
Qualitatively, if a particle is initially at rest in these
crossed fields, it is initially pulled by the electric field, and
then, due to emerging velocity, the magnetic field turns it
around. When the direction of the particle’s motion reverses,
the electric field slows it down and eventually stops the particle some distance away from its initial position. After that,
the aforementioned motion begins anew. The resulting trajectory of the particle resembles Fig. 2.4. As a result, these
equations of motion predict a particle drift with constant velocity, which is perpendicular to both E and B and with its
value given by
E B
E B
SI : v d =
×
Gaussian : v d = c
×
(2.15)
B 2
B 2
The efficiency of the Gaussian system of units continues to
astound us! Not only does it give us an intuitively clear and
beautiful formula, but it also immediately shows that the
above equation is valid only in the assumption that the electric field is much smaller than the magnetic field E « B.
2.3.3 Motion in quadrupole fields
Let’s consider motion in a quadrupole magnet (see Fig. 2.5)
where, ideally, the fields depend linearly on the distance from
the center of the magnet:
B x = Gz and
B z = Gx
where G is the gradient of the quadrupole (and we use z ≡ y
in this section). We will rewrite the equation of motion as
dγ v
m 0
= q v B
dt
×
and using Cartesian coordinates:
q
q
x ¨ = −
G sx ˙ , z ¨ =
Gsz ˙
m 0 γ
m 0 γ
q
and s ¨ =
G(xx ˙ − zz ˙ )
m 0 γ
Changing the independent variable from time to path length,
F
and considering only the case of small deviations from the
axis, reduces these equations to
a
q
x
''
− Kx = 0 and
''
z + Kz = 0
(2.16)
q ∂B z q
where K =
= G
(2.17)
p ∂x
p
This brings us to the discussion of linear betatron motion.
\
% \
1
%
6
[
[
6
1
\
)
\
1
6
) [
[
6
1
IGURE 2.5
agnetic fields and forces
cting on a particle in a
uadrupole.
M
transverse dynamics 25
(
%
Y Y !
FIGURE 2.4
Drift in crossed E×B fields.
While we are on this topic, let’s consider the case of uniform
E and B fields that are perpendicular — a situation often encountered when dealing with plasma and beams.
Qualitatively, if a particle is initially at rest in these
crossed fields, it is initially pulled by the electric field, and
then, due to emerging velocity, the magnetic field turns it
around. When the direction of the particle’s motion reverses,
the electric field slows it down and eventually stops the particle some distance away from its initial position. After that,
the aforementioned motion begins anew. The resulting trajectory of the particle resembles Fig. 2.4. As a result, these
equations of motion predict a particle drift with constant velocity, which is perpendicular to both E and B and with its
value given by
E B
E B
SI : v d =
×
Gaussian : v d = c
×
(2.15)
B 2
B 2
The efficiency of the Gaussian system of units continues to
astound us! Not only does it give us an intuitively clear and
beautiful formula, but it also immediately shows that the
above equation is valid only in the assumption that the electric field is much smaller than the magnetic field E « B.
2.3.3 Motion in quadrupole fields
Let’s consider motion in a quadrupole magnet (see Fig. 2.5)
where, ideally, the fields depend linearly on the distance from
the center of the magnet:
B x = Gz and
B z = Gx
where G is the gradient of the quadrupole (and we use z ≡ y
in this section). We will rewrite the equation of motion as
dγ v
m 0
= q v B
dt
×
and using Cartesian coordinates:
q
q
x ¨ = −
G sx ˙ , z ¨ =
Gsz ˙
m 0 γ
m 0 γ
q
and s ¨ =
G(xx ˙ − zz ˙ )
m 0 γ
Changing the independent variable from time to path length,
F
and considering only the case of small deviations from the
axis, reduces these equations to
a
q
x
''
− Kx = 0 and
''
z + Kz = 0
(2.16)
q ∂B z q
where K =
= G
(2.17)
p ∂x
p
This brings us to the discussion of linear betatron motion.
\
% \
1
%
6
[
[
6
1
\
)
\
1
6
) [
[
6
1
IGURE 2.5
agnetic fields and forces
cting on a particle in a
uadrupole.
M
transverse dynamics 25