Reaction forces on a relativistic point charge moving above a dielectric or a metallic half-space
D. Schieber and L. Scha
¨
chter
Department of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
Received 28 July 1997; revised manuscript received 15 December 1997
We investigate the forces that act on a particle as it moves at a height h above a half-space of dielectric or
lossy material. The expressions for the longitudinal and transverse forces are calculated numerically and
analytic expressions are presented for limit cases. In a dielectric it is shown that the longitudinal force is zero
at velocities below the Cherenkov velocity and it reaches a constant value at high energies. The transverse
force below the Cherenkov velocity is the result of a superposition of evanescent waves only; in this regime it
increases with the momentum. Above the Cherenkov velocity propagating waves add their contribution to the
transverse force. For relatively low energies their contribution tends to increase the total force. At high energies
the total transverse force decays as 1/ . In the case of a metallic medium , the longitudinal force is
proportional to ( )
3
/
0
h when this quantity is smaller than unity and it reaches a constant value when
this parameter is much larger than unity. The transverse attraction force decreases monotonically with the
relativistic factor . S1063-651X9804205-6
PACS numbers: 41.75.-i, 41.60.-m, 41.20.-q
INTRODUCTION
It is well known that if a charged particle moves in the
vicinity of a dielectric at a velocity higher than the phase
velocity of a plane wave in the corresponding material, then
Cherenkov radiation is emitted; its spectrum is a topic that
appears in many textbooks; see, e.g., 1. This radiation
comes at the expense of the kinetic energy of the particle. In
a previous study 2 we calculated the deceleration force that
acts on a particle as it moves in a cylindrical vacuum channel
bored in an otherwise infinite dielectric material. In a similar
way we calculated 2 the force that acts on the charge when
the dielectric medium was replaced by a metal 3. In all
these cases the reaction force decelerates the particle. It was
also shown 4,5 that the force becomes accelerating if the
medium is active.
The motion of electrons above dielectric or metallic sur-
faces is of interest since this may become one of the attract-
ing ways to generate millimeter and submillimeter wave ra-
diation without excitation of multiple modes in the system.
In particular, it is important to estimate the decelerating lon-
gitudinal force that acts on the moving particle due to its
proximity to material, as well as the attracting force in the
transverse direction. The advantages of open structures can
be utilized in the case of particle accelerators, e.g., those
which rely on the Smith-Purcell effect 6–10 or planar qua-
siopen structures manufacture using very-large-scale inte-
grated technology, which recently has been attracting atten-
tion.
An extensive amount of work has been dedicated in the
past to the motion of charged particles in the vicinity of a
metallic half-space 11–18. The main goal of these studies
is electron spectroscopy; in other words, electrons scattered
by a solid-state surface may provide important information
about the characteristics of the medium. Specifically, excita-
tion of plasmons and/or phonons 12,15,16,18 by grazing
electrons has been investigated extensively. The assumptions
common to all these studies is that the particles are not rela-
tivistic and for an effective ‘‘interaction’’ their distance from
the surface is of the order of nanometers. The system envi-
sioned in this study consists of relativistic even ultrarelativ-
istic particles and their height above the surface is not
smaller than a few micrometers. The goal of this study is to
investigate the longitudinal and transverse forces on a bunch
of charged particles as it moves above a metallic or dielectric
half-space.
DEFINITION OF THE MODEL
Consider a charge -q moving at a velocity v parallel to a
half space of dielectric material
r
. A Cartesian coordinate
system is introduced: Its x coordinate is parallel to the mo-
tion of the particle and its y coordinate is transverse to the
direction our particle moves, but parallel to the plane of in-
terface between the dielectric ( z 0) and the vacuum ( z 0)
as illustrated in Fig. 1. The electromagnetic field generated
by this moving charge, as measured in the laboratory frame
of reference in the absence of the dielectric half-space can be
derived from the x component of the magnetic vector poten-
tial A
x
( p )
=
( p )
and the scalar electric potential, which
reads
p
=
-q
4
0
1
2
-
dk
x
dk
y
e
- j k
x
x -v t +k
y
y -| z -h| k
x
2
+k
y
2
1
k
x
2
+k
y
2
. 1
The presence of the dielectric half-space causes additional
secondary potentials in the upper region superscripts s and
u , which are given by
A
x
s , u
A
y
s , u
s , u
=
-q
4
0
1
2
-
dk
x
dk
y
v
c
2
R
x
v
c
2
R
y
R
x
+
k
y
k
x
R
y
e
- j k
x
x -v t +k
y
y - k
x
2
+k
y
2
z
.
2
PHYSICAL REVIEW E MAY 1998 VOLUME 57, NUMBER 5
57 1063-651X/98/575/60088/$15.00 6008 © 1998 The American Physical Society