Chapter 3
Fields, Potentials and Equations of
Motion
For the study of transfer maps of particle optical systems, first it is necessary
to undertake a classification of the possible fields that can occur. All fields
are governed by Maxwell’s equations, which in SI units have the form
div
B = 0, curl
H = j +
∂
D
∂t
,
div
D = ρ, curl
E = −
∂
B
∂t
.
(3.1)
In the case of particle optics, we are mostly interested in cases in which
there are no sources of the fields in the region where the beam is located, so
in this region we have ρ = 0 and j = 0. Of course any beam that is present
would represent a ρ and a j, but these effects are usually considered separately.
In the following, we want to restrict ourselves to time independent situations, and neglect the treatment of elements with quickly varying fields
including cavities. This limitation in very good approximation also includes
slowly time varying fields like the magnetic fields that are increased during
the ramping of a synchrotron.
So, Maxwell’s equations simplify to
div
B = 0, curl
H = 0,
div
D = 0, curl
E = 0,
(3.2)
where
B = μ 0
H,
D = ε 0
E.
Because of the vanishing curl, we infer that
E and
B have scalar potentials
V E and V B such that
E = −
∇V E ,
B = −
∇V B .
Note that here even the magnetic field is described by a scalar potential,
and not by the vector potential
A that always exists. From the first and third
equations of (3.2), we infer that both scalar potentials V E and V B satisfy
Laplace’s equation, and we thus have
ΔV E = 0, ΔV B = 0.
49
DOI:10.1201/b12074-3
Fields, Potentials and Equations of
Motion
For the study of transfer maps of particle optical systems, first it is necessary
to undertake a classification of the possible fields that can occur. All fields
are governed by Maxwell’s equations, which in SI units have the form
div
B = 0, curl
H = j +
∂
D
∂t
,
div
D = ρ, curl
E = −
∂
B
∂t
.
(3.1)
In the case of particle optics, we are mostly interested in cases in which
there are no sources of the fields in the region where the beam is located, so
in this region we have ρ = 0 and j = 0. Of course any beam that is present
would represent a ρ and a j, but these effects are usually considered separately.
In the following, we want to restrict ourselves to time independent situations, and neglect the treatment of elements with quickly varying fields
including cavities. This limitation in very good approximation also includes
slowly time varying fields like the magnetic fields that are increased during
the ramping of a synchrotron.
So, Maxwell’s equations simplify to
div
B = 0, curl
H = 0,
div
D = 0, curl
E = 0,
(3.2)
where
B = μ 0
H,
D = ε 0
E.
Because of the vanishing curl, we infer that
E and
B have scalar potentials
V E and V B such that
E = −
∇V E ,
B = −
∇V B .
Note that here even the magnetic field is described by a scalar potential,
and not by the vector potential
A that always exists. From the first and third
equations of (3.2), we infer that both scalar potentials V E and V B satisfy
Laplace’s equation, and we thus have
ΔV E = 0, ΔV B = 0.
49
DOI:10.1201/b12074-3