Gauge transformation between retarded and multipolar gauges

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P R A M A N A @ Indian Academy of Sciences Vol. 56. No. 5

- - journal of May 200 l

physics pp. 685 689

G a u g e t r a n s f o r m a t i o n b e t w e e n r e t a r d e d a n d m u l t i p o l a r g a u g e s

A M STEWART

Department of Appliecl Mathematics, Research Sct~ool of Physical Sciences and Engineerh~g, The Australian National University, Canberra ACT 0200, Australia

Email: andrew.stewart @ mm.edu.au

MS received 24 July 2000; revised 9 January 200t

Abstract. The gauge function, expressed in terms of the sources, required for a gauge nansforma tiol] between the retarded electromagnetic gauge and the three-vector version of tlne rnultipolar gauge is obtained.

Keywords. Gauge; transformation; retarded; multipolar.

PACS Nos 03.50.De; 11.15.q

The retarded sohltiOllS to the inhomogeneous wave equations for tile electromagnetic scalar and vector potentials ~ ( r , t) arid A ( r , l)

c92A/0(c~)~ - V2A + V ( V . A + ~,-'-'#)~,/&) = #0a

₍₁₎ and

v ~ b + ( # / a t ) v . A = -p/~o (2)

are

and

~h (~', ~) -- 4~-~03 ./' P(~"' ~')<;'[~' - lre-+-- It-,.,] ~"l/c}clr'de' (3)

A.~.Qr, m> : #o J' aQr'= ~:')~-{-~' - m + I~" r'l/c}&"d-~'

4~ I t - r'l ' (4)

where d is the Dirac delta function and c is the velocity of Jight. They describe the potentials at position r and time t arising '~!rom charge ttnd current densities/) and J at position r ' and time F [1~ 21] and satisfy the Lorentz gauge condition V . A I + c

~)(/q/igt

= 0. The electromagnetic fields E ( r , t) anti B (r, t) are obtai~md from the potentials by the relations

i3 V x A a,l"~(] E : : -V(/) - OA/~)/:~ (5)

685

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A M Stewart

where V is the spatial gradient operator with respect to r. In consequence, if the potentials are transformed to

A - + A ' = A + V x ,rod ( l S - > c / / = ~ b - D X / 0 t , (6) the electromagnetic fields are unchat:ged. The gauge function x ( r , t) is required to satisfy the conditiorl { ( O / O i ) ( O / O j ) - (O/Oj)(O/Oi)};V - O, where i a n d j are any pair of the co- ordinates z, ?], z and t. The principle of gauge invariance requires all observable quantities, such as the fields E and B, to be independent of the gauge function [3,4].

In the retarded gauge above, the potentials, denoted by the subscripts t are described in terms of the charge and current source densities p and J at the retarded time t~ -- - ]r - r ' l / c , Another gauge that is of interest in the theory of magnetism [5,6] and in semiclassical electrodynamics which describes the interaction of atoms with radiation [6]

is the three-vector version of the multipolar gauge [5,7-9]. In this gauge the potentials, denoted by the subscript 2, are described in terms of the instantaneous but non-local values of the fields 173 and 13

as

q52 (r, t) : - r - E(wr, ~)dzt and A 2 ( r , t ) = - r x

a/0

^I'B('ttr,t)ttd'u.

(7) The multipolar gauge satisfies the condition r . A2 (r, t) = 0 and is obtained f r o m a gauge functiot~ that is essentially given by - fff r . A ( u r , t)d't, [9].

It is the purpose of this note to obtain the gauge function x ( r , t) that effects a gauge transformation between these two important gauges by means of the relations A 2 = A1 q- VX and r = 4)1 - OX/Ot. The procedure that is used is to get B1 (r, t). and El (r, t) fiom eqs (3)-(5) and substitute them in eq. (7) to get A~(r, t) and &2(r, t). A gauge function x ( r , t) is then found that relates the two sets of potentials.

First we get I3! by taking the curl of A1 with respect to r. Noting that J is a function of r' but not of r this gives

B l ( ' t ~ r ~ t ) - @ r . # ~ x V.,,.~.h(ur),

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where h('ur) = a(t' - .t + I'ur - r ' ] ) / c / I ' ~ r - r' I and the gradient is taken with respect to the parameter ~r. Hence

A ~ ( r , t ) = /-to 4-7 d r ' d r ~ t x { a f t ' , x (9) The triple vector product may be expanded as

x { a ( r ' , ~ ' ) • V~,.[~('~,')} = a ( r ' , e ) { r . V,,,-h,(',,r)}

- { a ( r ' , e ) - r } V , d , , ( , * ~ ' ) (10)

and this gives rise to two terms in (9). When the relations between derivatives uV~,,.h('ar) = Vh(t~r) ancl (r 9 V ) h ( u r ) = "t~(Oh('ur)/Ot 0 derived in the appendix are used, where h is any function of tt, r, the first term in the integrand becomes 'u(Oh/Ot,) = (c3/cg,tt)(,uh) - It and so ( 9 ) is

686 Pramana - j. Phys., Vol. 56, No. 5, May 2001

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Retarded cmd m~tltipolar gauges

/

A~ (r,/~) = #o

W

d { d { J ( { , ~')

g d

• d,.{[a/~4.,O~(~,,~')) - h,O,,~')] -.~V,.~<~r)}, (l l)

where in the tast term the vector dot product is between j and r. The integral of,t~ over tile perfect differential can be carried out to give

= - - [ j ' d r ' d { a ( r ' , t') h,(r) - d't~{h(,~,r) k -rV,.h,(,u,r)}

A~(r, ~) /~Lo 47r

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The first term can be recognized to be A t (i', t) of eq. (4) so,

/

A2 (r, g) - Az (r, f) #o d r ' d f , ' J ( r ' , ( ) d'u.{h,(v,r) + -rVrh(v,r)}.

471-,

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Next we calculate q52(r,~), From eq. (5), E is the sum of two parts, denoted bythe superscripts a and b, which, from eq. (7), give rise to two te,ms in the potential. The first, involving the gradient, is

~ , t)

⁼

d.~(r. %,.)(/,:~ (~r, t), (14)

so using the resuks in the appendix, r , V.~.tz(q) = O h / O u we obtain d2~ (r, t) - (hi (r, t) = --cgh (0, t). The other term is ~eb (i; t) = r . fo 1 (O/FJt)AL (ur, ~)du and leads to

clrld{

{1..

_J

(r,,

{ ) }

s ~. # { ~ '

^- ^{~ §} I,~,,~ ~ - r ' l / c }

( 1 5 )

where 6' is the derNative of the delta function with respect to its argument.

Consider now the scalar function x ( r , t) = f ( r , ~) + .q(~) where

"dr'dt;'{r ~ 1

.f(r, t) ~0 [ . a ( / , t')} d,~t,,(v,i,) (16)

47r J and

/

1 d r ' d { p ( r ' , t')O{~' - ~ +

b'l/c}/lr'l (17)

g(~) - 47re0 ,

and 0(m) is the function which is I for x > 0 and zero otherwise; its derivative is the delta function. The gradieat of X, which is the gradient of f , is obtained by noting that V{,'. S ( r ' , t')/<~,~-) } = h(',.,,r)V{r, a(~", ~,')} + { r - a ( ~ " , ~')}V/<a~-) = / + ~ , ' ) a (r', t,') + {1"-a(r', t')} W,.(,,r) si,loe V { a ( r ' , ~')-,'} = a(,.': ~:'), t~m~oe

Pl=mana - J. Plays,., gol. 56~ No. 5, May 200] 687

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A MSsewas't

, ]/s

V X - 4rr#~ d r ' d t ' J ( r ' , t ' ) . du{h,('ta') + -rVh('u,r)}. (18) T h e time deriwltNe DX~/0/; has two terms one c o m i n g from f and one from 9.

/ ~i 2 d ' { t ' - t + l m ' - r " / c }

Of~at

= #0 d r ' d / , ' { r - a ( r ' , t ' ) } d~u, ( i 9 )

and ,

@ / o r , 1 dr'de'p(r', e')a{i' - t, ⁺Ir'l/~}/k'l 47rco

= r (0, t). (20)

By c o m p a r i n g eqs (18) and (20) with (13) and ( t 5 ) it c a n be seen that the gauge function X" = f + g is indeed able to transform the retarded gauge into the three-vector version o f the multipolar gauge.

The integral over ~z in eq. (16) can be simplified. U s i n g the standard relation

6"[,t'(u)] = E~(T['8-

,t~q/lcgf/~)zLI,m,

⁽²¹⁾

where

.f[v, 'i]

= 0, in this case with ,/'[tL] -- t' - t, + ]'{zr -

r'[/c,

we obtain the roots t~ ~ from (u~- - r ' ) ~ - c'-' (e - t ' ) ~ = 0 to be

,-,[ j ... ]

' ~ t + - = - COS~9•

C2('t tl)2/~'12--si]22qO ,

(22)

7"

where r and r ~ are the lengths of the vectors r and r ~ and ~p is the angle between them. F r o m the reIatioa c 2 (~ - t') ~ =

v.",r ,2 + r '2 - 2'c~r'P

cos ~ it follows that e '2 (t - ~ ) " - ~.~2 sin 2 99 = (t,r' - r' c o s qo) 2 _> 0 so the square root is always real9 Next, it is straightforward to show that

oqf

_ r - ( ~ l " -- r ' )

d)'~ c]~,r - r q (23)

and that

8 O

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s . <,, sin:~ , s

d . m ~ ( . , ~ ) = c d < a ( . < , , - , , + ) + a(.,.,.- ,~ )],

i

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as the ]~,r r'[ terms i~ the n u m e r a t o r and d e n o m i n a t o r cancel ancl where '8 q aud i t - are given by eq. (22) so

f(~',

~1 . . . /:,2 / '

drq.tt'd~',

a ( , . ' , ~')}

9 4 r r . rX/c.~(t, _ tO e ,r,: ~ sin 2 ~

.///'

d'.,[,$('~,. 'u. +)

-t ~('~

- v ,

)!- (26)

h: carrying out the hltegrations over r ' and t' in (26) tb.e integral over '~ gives phts o~ae whe~ u + a~3cl '~.t- calculated from ec 1. (22) lie hetweeli zero and unity and zero od3erwise.

688

Pramcma -d. Phys.,

Vol. 56, No. 5> M a y 2t}01

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Retarded a n d m u l @ o l a r gauges

A p p e n d i x

We show that r . V/~(q) :t~(0h/:),/z) where 27 is the gradient operator with respect to r, q = 't~r and h is any :function o f the vector q. Noting that Ocfi/~);c j -- :tl.dij and Oq~/O't.~, = :c i we iliad that O]~/O~a: ~ = ~(Oh,/OC_l ~) so Vh('t~r) = '~zV,~rh,(~n'). Also Ot~/L')t~ = 2i:c:(~h/OCl i) so 'tt.Oh,(q)/O'~z E~t~:~:iOh/0q i. Next, r , V/z,(q) = ~:~:~D/O:~:~h(q) : Ei~tz~Oh/~)q ' so it follows that r . Vh.(q) = v,(Oh,/O'u) and r V.,,.h,(q) = Oh/Ott.

R e f e r e n c e s

[1] J D Jackson, Classical electrodyrmmics (Wiley, New York, 1975)

[2] W K H Panofsky and M Philips, Classical elec~rici O, and magnetism (AddisonoWesley, Cam- bridge, Mass., 1955)

[3] A M Stewart, J. Phys. A29, 141! (1996) [4] A M Stewart, Aust. Z Phys. 50, 1061 (1997) [5] A M Stewart, J. Phys, A32, 6091 (1999) [6] A M Stewart, Aust. J. Phys. 53, 613 (2000) [7] J G Valatin, Proc. R. Sac. London A222, 93 (1954) [8] R G Woolley, Z Phys. B6; L97 (1973)

[9] D H Kobe, Am. ,L Phys. 50, i28 (1982)

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