Presentation 2011-09-04
Transcript of Presentation 2011-09-04
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
i iii
ii
. , .
I ii i i. ..
i i
i,
c. , . , I2011, i, i iii ii
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwells equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Maxwells equations
div E = 4, div B = 0,
rot E = 1
c
B
t, rot B =
4
cj +
1
c
E
t;
E = EC + EF : rot EC = 0, div EF = 0;
boundary conditions: E = 0, Bn = 0, (rot B) = 0.
magnetoquasistatics (MQS):E
t= 0 divj = 0;
electroquasistatics (EQS): Bt
= 0 t
+ divj = 0;
Darwin model (DW):EF
t= 0
t+ divj = 0.
see, e.g.,: J. Larsson, Am. J. Phys. 75, pp. 230239 (2007).
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwells equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Comparison of quasistatic approximations
E = EC + EF : rot EC = 0, div EF = 0.Faradays law Ampere-Maxwell law
Maxwells equations rot EF = 1
c
B
trot B =
4
cj +
1
c
E
t
MQS (E/t = 0) rot
EF =
1
c
B
t rotB =
4
cj
EQS (B/t = 0) rot EF = 0 rot B =4
cj +
1
c
ECt
DW (EF/t = 0) rot EF = 1
c
B
trot B =
4
cj +
1
c
ECt
Poyntings theorem: wt
+ div S = j E; S = c4
E B.
Maxwells equations MQS EQS DW
8w E2 + B2 B2 E2CE2C +
B2 + 2 EC EF
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwells equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Electric field components
EDW(x, t) = EC(x, t) + EF(x, t), where
EC|r = qGD(x; x)
r, EC| = q
GD(x; x)
r, EC|z = q
GD(x; x)
z;
EF(x, t) =
EF(x, t) +
E
a
F(x, t) EF(x, t) + O(a) :
EF|r =q
4c2
vz(v
) (v (v ))
z+
vr
(v )z
z
S
GD(x; x)
4rV
GD(x,z; x)
zGD(x; x)d3xd2x
(v (v ))z +
vr
(v )
S
GN (x; x)
rGN(x,z; x)d
2x
,
c. , . , I2011, i, i iii ii
Q i i i i d D i d l M ll i d i f i i i i
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwells equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Electric field components (continued)
EF| =q
4c2
vz(v
) (v (v ))
z
+ v
r(v )z
z
S
GD(x; x)
4r
V
GD(x,z; x)
zGD(x; x)d3xd2x+
+
(v (v ))z +
vr
(v )
S
GN (x; x)
rGN(x,z; x)d
2x
,
EF|z =q
4c2vz(v ) (v (v ))
z+
vr
(v )z
z
V
GD(x; x)GD(x; x)d3x; where
t (v );
x = (r(t)cos (t), r(t)sin (t), z(t)); vr =dr(t)
dt, v = r(t)
d(t)
dt, vz =
dz(t)
dt.
c. , . , I2011, i, i iii ii
Q asistatic a ro imations and Dar in model Ma ells eq ations and com arison of q asistatic a ro imations
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwell s equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Electric field components (final)
EaF|r = qc2
[az (a ) z
]S
GD(x; x
)
4r
V
GD
(x, z; x
)4z
GD(x; x)d3xd2x (a)z
S
GN (x; x)
4rGN(x, z; x)d
2x
,
EaF| =q
c2
[az (a
)
z]
S
GD(x; x)
4r
V
GD(x, z; x)
4z
GD(x; x)d3xd2x+ (a)z
S
GN (x; x)
4rGN(x, z; x)d
2x,EaF|z =
q
c2[az (a
)
z]
V
GD(x; x)
4GD(x; x)d3x;
where ar =dvr(t)
dt
v2(t)
r(t), a =
dv(t)
dt+
vr(t)v(t)
r(t), az =
dvz(t)
dt.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin model Maxwells equations and comparison of quasistatic approximations
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwell s equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Magnetic field components
BDW|r = q4c
[vz (v )
z]S
GD(x; x
)
rGD(x, z; x)d2x
+
+(v)z
z
S
GN (x; x)
rGN(x, z; x)d
2x
,
BDW| =q
4c
[vz (v
)
z]
S
GD(x; x)
rGD(x, z; x)d
2x
(v)z
zS
GN (x; x)
r
GN(x, z; x)d2x,
BDW|z =q
c(v)zG
N(x; x); where for u = (u, z) it is defined
u = ur
r+ u
r, (u)z = ur
r u
r.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin model Maxwells equations and comparison of quasistatic approximations
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Maxwell s equations and comparison of quasistatic approximationsElectric field componentsMagnetic field componentGreens functions
Greens functions
GD(x;x) = 4a+q=1
exp[0q||]
20qJ0(0q)J0(0q
)
J21(0q)
+
++
n,q=1
exp[nq||]
nq
Jn(nq)Jn(nq)
J2n+1
(nq)cos[n()]
,
GN (x;x) =4a
+
q=1
exp[0q||]
20q
J0(0q)J0(
0q
)
J20(0q
)+
++
n,q=1
nq exp[nq||]
2nqn2
Jn(nq)Jn(
nq
)
J2n(nq)
cos[n()]
;
GD(x;x) = 8
+
q=1
J0(0q)J0(0q)
220qJ21 (0q)
++
n,q=1
Jn(nq)Jn(nq)
2nqJ2n+1
(nq)cos[n()]
,
GN (x;x) = 8+
q=1
J0(0q)J0(
0q
)
22
0qJ2
0(
0q)
++
n,q=1
Jn(nq)Jn(
nq
)
(2
nqn2)J2
n(
nq)
cos[n()];where =
x2 + y2/a, = arcsin[y/r] and = z/a are dimensionless coordinates, a
is drift-tube radius; nq and nq are qs roots to equations Jn(x) = 0 and Jn(x) = 0.
G. Gorbik, K. Ilyenko, T. Yatsenko, Telecomm. Radio Eng. 67, pp. 11771188 (2008).
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelV l f G f l
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Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Vector analogue of Greens formulaEM fields in Darwin modelExample of induced charge and current densities
Vector analogue of Greens formula
Using vector identityV
[ Grotrot F Frotrot G]dV =
S
[ Frot G Grot F]ndS,
where n is internal normal to drift-tube surface S V, G = G(x; x)a[G(x; x) := 1/|x x| is scalar Greens function of free space ],
F = EDW EC + EF [ or F = BDW ], and a is arbitrary constantvector, one gets
EDW(t, x) = EinDW(t, x) +
EscDW(t, x),
BDW(t, x) = BinDW(t, x) +
BscDW(t, x);
EinDW(t, x) andBinDW(t, x) are fields induced by point-like charge in free
space, and EscDW(t, x) andBscDW(t, x) can be deduced with aid ofknown
solutions of Darwin model equations presented in preceding slides.
see, e.g.,: J.A. Stratton, L.J. Chu, Phys. Rev. 56, pp. 99107 (1939).
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVe to n lo e of G een fo m l
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Q ppVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin modelSummary
Vector analogue of Greens formulaEM fields in Darwin modelExample of induced charge and current densities
Proof Kirchhoff formula
Fig. 1.
Sphere with radius r0 withcenter in point x is threw out
Denoted V = V, S = S.
V
[G(x; x)a rotx
rotx
EF
(t, x)
EF(t, x) rot xrot x(G(x; x
)a)]dV =
=
S
([ EF(t, x) rot x(G(x; x
)a)]
[(G(x; x)a) rot x EF(t, x)]) ndS.
rot xrot x(G(x; x)a) = x(xG(x; x
) a)
EF(t, x) x(xG(x; x
) a) =
= div x [EF(t, x)(a xG(x; x))].
V
div x [ EF(t, x)(a xG(x; x
))]dV =
= a
S
(n EF(t, x))xG(x; x
)dS
.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
Vector analogue of Green s formulaEM fields in Darwin modelExample of induced charge and current densities
Proof Kirchhoff formula (continued)
S
[(G(x; x)a) rot xrot x EF(t, x) EF(t, x) rot xrot x(G(x; x)a)]dV =
= a
V
4
c2G(x; x)
jr(t, x)
tdV + a
S
(n EF(t, x))xG(x; x
)dS.
Because
[ EF(t, x) rot x(g(x; x
)a)] n = [[n EF(t, x)] xG(x; x
)] a,
[a rot x EF(t, x)] n =
1
c[a
B(t, x)
t] n =
1
ca [n
B(t, x)
t].
and a is arbitrary vector
V
4
c2G(x; x)
jr(t, x)
tdV =
S
1
cG(x; x)[n
B(t, x)
t]
[[n EF(t, x)] xG(x; x
)] (n EF(t, x))xG(x; x
)dS.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
Vector analogue of Green s formulaEM fields in Darwin modelExample of induced charge and current densities
Proof Kirchhoff formula (continued)
1
c
G(x; x)[nB(t, x)
t
]+[[n EF(t, x)]xG(x; x
)]+(n E(t, x))xG(x; x
)dS =
=
V
4
c2g(x; x)
jr(t, x)
tdV +
S
1
cG(x; x)[n
B(t, x)
t]+
+[[n EF(t, x)]xG(x; x
)]+(n EF(t, x))xG(x; x
)dS, xG(x; x
)| = 1
r20
n.
1
cG(x; x)[n
B(t, x)
t]+[[n EF(t, x
)] xG(x; x)]+(n EF(t, x
))xG(x; x)
dS
=
r20
1
c
1
r0[n
B(t, x)
t]
1
r20[[n EF(t, x
)] n] 1
r20(n EF(t, x
))n
d =
=
r20
1
c
1
r0[n
B(t, x)
t]
1
r20
EF(t, x)
d =
=
r0
c[n
B(t, x)
t] EF(t, x
)
d = 4
r0
c[n
B(t, x)
t] EF(t, x)
,
is averaging on sphere.c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelV Ki hh ff f l d EM fi ld l i i D i d l
Vector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
Vector analogue of Green s formulaEM fields in Darwin modelExample of induced charge and current densities
Proof Kirchhoff formula (final)
limr00
1
c G(x; x)[n
jr(t, x)
t ]+
+[[n EF(t, x)]xG(x; x
)]+(n EF(t, x))xG(x; x
)
dS = 4 EF(t, x)
Perfect metal n E = 0 and n B = 0,
EF(t, x) =1
4
V
4
c2G(x; x)
jr(t, x)
t
dV+
1
4
S
1
cG(x; x)[n
B(t, x)
t]+
+[[n EF(t, x)] xG(x; x
)] + (n EF(t, x))xG(x; x
)dS,EF(t, x) =
V
1
c2g(x; x)
jr(t, x)
t
dV+
S
1
c2
t+ Fx
G(x; x)dS,
where =c
4[n B]|S F =
1
4(n EF)|S .
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVe tor Kir hhoff form l nd EM field ol tion in D r in model
Vector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
gEM fields in Darwin modelExample of induced charge and current densities
EM fields in free space
Ein
DW(t, x) Ein
C (t, x) + Ein
F (t, x) :
div EinC = 4, rotrotBinDW =
4
crotj,
rotrot EinF =4
c2
tj +1
4
EinCt ;
EinC (t, x) =
V
(t, x)G(x; x)d3x, where G(x; x) =1
|x x|,
EinF (t, x) = 1
c2V
j(t, x)t +
(x x)
2
2(t, x)
t2G(x; x)d3x,
BinDW(t, x) = 1
c
V
j(t, x) G(x; x)d3x.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
gEM fields in Darwin modelExample of induced charge and current densities
EM fields in free space: point-like charge
For point-like charge of value q(t, x) = q(x x(t)) j(t, x) = qv(t)(x x(t))
[x(t) and v(t) = dx(t)/dt are its trajectory and instantaneous velocity].
Ein
DW(t, x) =Ein
DW(t, x) +Ea
DW(t, x) Ein
C (t, x) +Ein
F (t, x) + O(a) :
EinDW(t, x) =q
R2
R
R
1
v2(t)
c2+
3
2
v2(t)c2
v(t)
c
R
R
2,
BinDW(t, x) =q
R2v(t)
c
R
R and R(t, x) = x x(t),EaDW(t, x) =
q
2c2R
a(t) +
a(t)
R
R
RR
.
It is important that, although acceleration is present, asymptotic of Poyntingsvector is O(R3) for large values ofR, i.e. these EM fields are non-propagating.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formula
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
EM fields in Darwin modelExample of induced charge and current densities
EM fields of induced charge and current densities
EM fieldsEscDW(t, x) = EscC (t, x) +
EscF (t, x) and BscDW(t, x) are
produced by charges and currents induced on drift-tube wall bycharged-particles moving inside the tube
EscDW(t, x) = [C(t, x
) + F(t, x
)]
(x x)
2c2
2C(t, x)
t2+
+1
c2
(t, x)
t
G(x; x)d
2x,
BscDW(t, x) = 1
c
(t, x)G(x; x)d
2x; where
C(t, x) =
1
4n EC(t, x)
x=x
, F(t, x) =
1
4n EF(t, x)
x=x
,
(t, x) =c
4n BDW(t, x)
x=x
;
n = er and x = (a cos
, a sin , z).
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formulaEM fi ld i D i d l
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Vector Kirchhoff formula and EM field solutions in Darwin modelDynamics calculations in Darwin model
Summary
EM fields in Darwin modelExample of induced charge and current densities
Induced charge and current densities: point-like charge
C(t, x) = q4
GD
(x; x)r
x=x
, F(t, x) = q162c2
S
GD
(x; x)
4r
V
GD(x , z; x)
z
vz(v
)(v(v))
z+
vr
(v)z
z
GD(x ; x)d3x
GN (x; x )
r
(v(v))z+
vr
(v)
GN(x , z; x)
d2x
(x,z)=x
;
r(t, x) = 0, (t, x) =q
4(v )zG
N(x; x),
z(t, x) = q162
vz(v)
z S
GD
(x; x
)r
GD(x, z; x)d2x
(v )z
z
S
GN (x; x)
rGN(x, z; x)d
2x
(x,z)=x
.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formulaEM fi ld i D i od l
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Dynamics calculations in Darwin modelSummary
EM fields in Darwin modelExample of induced charge and current densities
Remaining moderately relativistic contributions to EM fields
2C(t, x)
t2=
q
4
(v (v ))
vr
(v )z
GD(x; x)
r
x=x
,
(t, x)
t
=q
4(v (v ))z +
v
r
(v )GN(x; x),
z(t, x)
t=
q
162
vz(v
)
(v (v
))vr
(v )z
z
S
GD(x; x)
r
GD(x,z; x)d2x
(v (v ))z +
vr
(v )
z
S
GN(x; x)
rGN(x,z; x)d
2x
(x,z)=x
.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formulaEM fields in Darwin model
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Dynamics calculations in Darwin modelSummary
EM fields in Darwin modelExample of induced charge and current densities
Contributions to induced surface-charge density
Non-Relativistic Contribution
Fig. 1. Normalized surface-charge density(2a2C/q) as a function of and .Blue, green, and red surfaces (represented bynumbers 1, 2 and 3) are plotted for the normali-zed radial distance (r0/a) equal to 0.3, 0.4, and0.5, respectively (a is the drift-tube radius).
Charged-Particle in Helical Motion
r(t) r0 = a0,(t) = 0 + vt/r0,z(t) = z0 + vt;
vr(t) = 0,v(t) v = c,
vz(t) v = c.
Fig. 2. Normalized correction (2a2F/q) tosurface-charge density of point-like charge inmoderately relativistic motion (v = 0.1c,v
= 0.5c) as function of and .
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Vector analogue of Greens formulaEM fields in Darwin model
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Dynamics calculations in Darwin modelSummary
EM fields in Darwin modelExample of induced charge and current densities
Components of induced surface-current density
Fig. 3. Normalized components, 2a2/(cq) and 2a2z/(cq), of surface-current
density of point-like charge in moderately relativistic helical motion (v = 0.1c,v = 0.5c) as a function of and (c is the speed of light). The blue, green,and red surfaces (represented by numbers 1, 2 and 3) are plotted for the normalizedradial distance (r0/a) equal to 0.3, 0.4, and 0.5, respectively (a is the drift-tube
radius).c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
D i l l i i D i d lRelativistic equations of motionFi N i i i
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Dynamics calculations in Darwin modelSummary
First post-Newtonian approximation
Relativistic equations of motion
mqid(ivi)
dt= qi
Ei(t, xi) +
vic Bi(t, xi)
,
dxidt
= vi;
dvidt
=qi
mqi1i
Ei(t, xi)+
vic Bi(t, xi)
vi
c2(vi E
i(t, xi))
,
where i = 1/
1 v2i /c2;
Ei(t, xi) EiQS(t, xi) +
Ecohrad(t, xi) +Eext(t, xi),
Bi(t, xi) BiQS(t, xi) +
Bcohrad(t, xi) +Bext(t, xi).
Fig. 4. Problem geometry.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
D i l l ti i D i d lRelativistic equations of motionFi t t N t i i ti
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Dynamics calculations in Darwin modelSummary
First post-Newtonian approximation
First post-Newtonian approximation
Neglecting completely the radiation field and for no external fields, we
finally obtain the system of equations, which describe dynamics of intense(high-current) charged-particle beams in the Darwin model:
dxidt
= vi,
dvidt
qimqi
1
v2i2c2
EiC(t, xi)
vic2 (vi E
iC(t, xi))+
EiF(t, xi)+
vic B
iDW(t, xi)
;
EiC(t, xi) =EscCi(t, xi) +
Nk=i,k=1
EinCk(t, xi) +
EscCk(t, xi)
,
EiF(t, xi) =EscFi(t, xi) +
Nk=i,k=1
EinFk(t, xi) +
EscFk(t, xi)
,
BiDW(t, xi) =BscDWi(t, xi) +
Nk=i,k=1
BinDWk(t, xi) +
BscDWk(t, xi)
.
c. , . , I2011, i, i iii ii
Quasistatic approximations and Darwin modelVector Kirchhoff formula and EM field solutions in Darwin model
Dynamics calculations in Darwin model
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Dynamics calculations in Darwin modelSummary
Summary
In the moderately relativistic the first post-Newtonian approximation (the Darwin model), we worked out a procedure forobtaining electric and magnetic fields that are induced by a point-likecharged-particle moving arbitrarily in a cylindrical drift-tube
Applying the vector Kirchhoff formula in the Darwin model, we showhow to calculate electromagnetic field of surface-charge andsurface-current densities, which are induced by charged-particles thatmove arbitrarily in the cylindrical drift-tube, in the net forceexpressions defining self-consistent particle dynamics in the drift-tube
In the first post-Newtonian (moderately relativistic) approximation,we formulate the equations of motion for point-like charged-particlesin the cylindrical drift-tube that account for not only potential butalso rotational space-charge field of these charged-particles
c. , . , I2011, i, i iii ii
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