The magnetic vector potential can be written in dimensionless
form as
q A
˜
A = −
.
(2.113)
mc
The velocity, space charge density, and space current density can
be written as
˜
v =
v
c
,
˜
ρ = −
q
mc 2
ρ
f 0
,
˜ j = −
q
mc
µ 0 j,
˜ j = ˜
ρ ˜
v,
(2.114)
respectively, where ˜
ρ ≤ 0, regardless of sign of charge q.
The rest energy plus the kinetic energy is given in dimensionless
units by
γ = 1 + φ = 1 + p 2 ,
(2.115)
where the rest energy mc
2 is unity in these units. Solving this for
the scalar kinetic momentum p, we obtain (2.115)
p = 2φ + φ 2 ,
(2.116)
where φ and p can be regarded as functions of the coordinates
x j only. This is due to the fact that the zero of potential energy
is fixed (2.111). In the nonrelativistic limit, the kinetic energy is
small relative to the rest mass, as follows:
φ « 1.
(2.117)
50
Chapter 2. Geometrical optics
In the following discussion, we will not make this approximation,
but rather retain the full relativistic form throughout.
All quantities are dimensionless except coordinates and time,
which retain their SI units of meters and seconds, respectively.
One can easily return to SI units at any point in a calculation by
inverting the above transformations. Many calculations seek position, such as the path of a ray, or the deviation of the path from its
paraxial or Gaussian approximation. In such cases, it is not necessary to convert back to SI units for the result to be practical. We
form as
q A
˜
A = −
.
(2.113)
mc
The velocity, space charge density, and space current density can
be written as
˜
v =
v
c
,
˜
ρ = −
q
mc 2
ρ
f 0
,
˜ j = −
q
mc
µ 0 j,
˜ j = ˜
ρ ˜
v,
(2.114)
respectively, where ˜
ρ ≤ 0, regardless of sign of charge q.
The rest energy plus the kinetic energy is given in dimensionless
units by
γ = 1 + φ = 1 + p 2 ,
(2.115)
where the rest energy mc
2 is unity in these units. Solving this for
the scalar kinetic momentum p, we obtain (2.115)
p = 2φ + φ 2 ,
(2.116)
where φ and p can be regarded as functions of the coordinates
x j only. This is due to the fact that the zero of potential energy
is fixed (2.111). In the nonrelativistic limit, the kinetic energy is
small relative to the rest mass, as follows:
φ « 1.
(2.117)
50
Chapter 2. Geometrical optics
In the following discussion, we will not make this approximation,
but rather retain the full relativistic form throughout.
All quantities are dimensionless except coordinates and time,
which retain their SI units of meters and seconds, respectively.
One can easily return to SI units at any point in a calculation by
inverting the above transformations. Many calculations seek position, such as the path of a ray, or the deviation of the path from its
paraxial or Gaussian approximation. In such cases, it is not necessary to convert back to SI units for the result to be practical. We