Presentation 2011-09-04

7/31/2019 Presentation 2011-09-04

1/23

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
http://find/

7/31/2019 Presentation 2011-09-04

2/23



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).

c. , . , I2011, i, i iii ii
http://find/

7/31/2019 Presentation 2011-09-04

3/23




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

c. , . , I2011, i, i iii ii
http://find/

7/31/2019 Presentation 2011-09-04

4/23




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
http://find/

7/31/2019 Presentation 2011-09-04

5/23




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
http://find/http://goback/

7/31/2019 Presentation 2011-09-04

6/23



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
http://find/

7/31/2019 Presentation 2011-09-04

7/23




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
http://find/

7/31/2019 Presentation 2011-09-04

8/23




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
http://find/

7/31/2019 Presentation 2011-09-04

9/23



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
http://find/

7/31/2019 Presentation 2011-09-04

10/23

Q ppVector Kirchhoff formula and EM field solutions in Darwin model


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
http://find/

7/31/2019 Presentation 2011-09-04

11/23

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

7/31/2019 Presentation 2011-09-04

12/23


Summary


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


7/31/2019 Presentation 2011-09-04

13/23


Summary


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

http://find/

7/31/2019 Presentation 2011-09-04

14/23


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



7/31/2019 Presentation 2011-09-04

15/23


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



7/31/2019 Presentation 2011-09-04

16/23


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


Vector analogue of Greens formulaEM fi ld i D i d l
http://find/

7/31/2019 Presentation 2011-09-04

17/23


Summary


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


Vector analogue of Greens formulaEM fi ld i D i od l

7/31/2019 Presentation 2011-09-04

18/23



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


Vector analogue of Greens formulaEM fields in Darwin model

7/31/2019 Presentation 2011-09-04

19/23



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


Vector analogue of Greens formulaEM fields in Darwin model
http://find/

7/31/2019 Presentation 2011-09-04

20/23



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


D i l l i i D i d lRelativistic equations of motionFi N i i i
http://find/

7/31/2019 Presentation 2011-09-04

21/23


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


D i l l ti i D i d lRelativistic equations of motionFi t t N t i i ti
http://find/

7/31/2019 Presentation 2011-09-04

22/23




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


Dynamics calculations in Darwin model
http://find/

7/31/2019 Presentation 2011-09-04

23/23


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
http://find/

Documents

Presentation 2011-09-04