2
Transverse Dynamics
2.1 Maxwell equations Let’s begin our discourse about the basics of accelerator
and units
21 physics with the topic of the transverse dynamics of charged
2.2 Simplest
particles. This will lead into a discussion of the basics of synaccelerator
22 chrotron radiation and of acceleration in the following chap2.3 Equations of
ters, intermediated by a dialogue on the synergies between
motion
24 accelerators, lasers and plasma.
2.4 Matrix formalism
27
2.5 Phase space
35
2.6 Dispersion and
2.1 Maxwell equations and units
tunes
36 We start by recalling the Maxwell equations with an empha2.7 Aberrations and
sis on their systems of units, focusing in particular on SI and
coupling
38 Gaussian-cgs systems. The microscopic Maxwell equations
(i.e., equations in vacuum) expressed in SI units, in both differential and integral form, are
This Maxwell equation explains the universality of the
inventive principle of chang-
ρ
1
E =
or
E dS =
ρdV
(2.1)
ing the volume-to-surface ra-
∇ ·
ε 0
∂Ω
·
ε 0 Ω
tio.
∇ · B = 0 or
B · dS = 0
(2.2)
∂Ω
Permittivity of free space
ε
−1
0 ≈ 8.85 F/m or A 2 s 4 kg m −3
∂B
d
∇ × E = −
or
E dt =
B dS
(2.3)
∂t
·
− dt
·
Vacuum permeability μ 0 =
∂Σ
Σ
1/(c 2 ε 0 ) ≈ 1.26 × 10 −6 N ·
A −2
∂E
∇ × B = μ 0
(
J + ε 0
)
or
∂t
d
B · dt = μ 0
J · dS + μ 0 ε 0
E · dS (2.4)
∂Σ
Σ
dt Σ
Speed of light in vacuum c
2.99 × 10 8
≈
m/s
The Lorentz force acting on a charged particle in an electric
and magnetic field in SI units is expressed as
F = q (E + v × B)
(2.5)
While SI is the standard, the Gaussian system is more natural for electromagnetism. The differential Maxwell equations
expressed in Gaussian-cgs units are
21
DOI: 10.1201/b18696-2
Transverse Dynamics
2.1 Maxwell equations Let’s begin our discourse about the basics of accelerator
and units
21 physics with the topic of the transverse dynamics of charged
2.2 Simplest
particles. This will lead into a discussion of the basics of synaccelerator
22 chrotron radiation and of acceleration in the following chap2.3 Equations of
ters, intermediated by a dialogue on the synergies between
motion
24 accelerators, lasers and plasma.
2.4 Matrix formalism
27
2.5 Phase space
35
2.6 Dispersion and
2.1 Maxwell equations and units
tunes
36 We start by recalling the Maxwell equations with an empha2.7 Aberrations and
sis on their systems of units, focusing in particular on SI and
coupling
38 Gaussian-cgs systems. The microscopic Maxwell equations
(i.e., equations in vacuum) expressed in SI units, in both differential and integral form, are
This Maxwell equation explains the universality of the
inventive principle of chang-
ρ
1
E =
or
E dS =
ρdV
(2.1)
ing the volume-to-surface ra-
∇ ·
ε 0
∂Ω
·
ε 0 Ω
tio.
∇ · B = 0 or
B · dS = 0
(2.2)
∂Ω
Permittivity of free space
ε
−1
0 ≈ 8.85 F/m or A 2 s 4 kg m −3
∂B
d
∇ × E = −
or
E dt =
B dS
(2.3)
∂t
·
− dt
·
Vacuum permeability μ 0 =
∂Σ
Σ
1/(c 2 ε 0 ) ≈ 1.26 × 10 −6 N ·
A −2
∂E
∇ × B = μ 0
(
J + ε 0
)
or
∂t
d
B · dt = μ 0
J · dS + μ 0 ε 0
E · dS (2.4)
∂Σ
Σ
dt Σ
Speed of light in vacuum c
2.99 × 10 8
≈
m/s
The Lorentz force acting on a charged particle in an electric
and magnetic field in SI units is expressed as
F = q (E + v × B)
(2.5)
While SI is the standard, the Gaussian system is more natural for electromagnetism. The differential Maxwell equations
expressed in Gaussian-cgs units are
21
DOI: 10.1201/b18696-2