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Enhanced electromagnetic radiation recoil
Volume 60A, number 3
PHYSICS LETTERS
21 February 1977
ENHANCED ELECTROMAGNETIC RADIATION RECOIL* Fl. COOPERSTOCK and D.W. HOBILL Department of Physics, University of Victoria, Victoria, B.C., Canada V8W 2Y2 Received 2 November 1976 The interference between two separated electric dipole radiators leads to a lower order linear momentum flux and hence recoil than that which can be achieved by a single system.
If the electromagnetic radiation from a system of accelerating charges carries away linear momentum in a preferred direction, the system must recoil in the opposite direction. This effect has been calculated from the Lorentz force on the radiation field back on the
that the aforementioned analogy is complete: a system of two separated electric dipole radiators can interfere constructively to induce a recoil force of lower order than that of eq. (1). Consider a system of two electric dipoles d1, d2
current density [1] and from the rate at which the elec. tromagnetic field carries momentum across an infinite sphere [2]. For a single system, the recoil force is [2]
separated by a distance L along the z axis. The charges in each of the dipoles are assumed to move non.rela. tivistically and hence the characteristic wavelength,
)i~c~ ~ ÷~ (d X ni)0 ~~ (1) 4irc where d and m are the electric and magnetic dipole moments respectively and D0~is the electric quadrupole moment. It has been noted that this force is of higher order in 1/c than the radiation damping forces which correspond to an energy loss rate bilinear in the elec. tric dipole moment. A completely analogous situation prevails in the theory of gravitational radiation. Because of momenturn conservation, there can be no dipole gravitational radiation. The lowest order energy loss is quadrupole, and the rate is bilinear in the mass quadrupole moment. For a single system, gravitational radiation recoil results from the interference between the quadrupole and octupole radiation. The gravitational and electro-
X, of the emitted is much greater than the characteristic size,radiation i~(e = 1,2) of each sub-system. From the energy-momentum conservation laws, the rate of change of linear momentum of the system P~ is simply expressed in terms of the momentum flux density [2]: 0 = d 1 T~0d V ~)CTaP dS~ (2) P C where the momentum flux density or Maxwell stress tensor is T0~= (1/4~)[_E0E~ H~HP+ 1 6013(E2 + H2)]. (3)
F0 =
—
,
—
,
2
Since the flux in eq. (2) will be found in the wave zone where E and H are orthogonal, eqs. (2) and (2) yield .
P0
=
—
4__fH2nbR~d~l. magnetic phenomena are shifted relative to each other by one order. For the gravitational case, it has been demonstrated [3—5]that under the correct circumstances, two Separated systems can induce a lower order radiation recoil by virtue of the interference from the quadrupole radiation alone. In this letter, it is demonstrated
where H~= (A~X n~)/c
*
and
.
.
Supported by National Research Council of Canada Grant A5340 and University of Victoria Faculty Research Grant 08584.
168
(4)
The total magnetic field is a linear superposition of contributions from each dipole: —
,
(6)
1 R A~=—~--fJ(t__~!) dV~
(7)
,
c = 1, 2
.
Volume 60A, number 3
PHYSICS LETFERS
21 February 1977
4
It is to be noted that to the required accuracy, the hR0 factor is common to both vector potential con-
d~=
tributions, where R0 represents the distance from the origin of one of the dipoles (let it be d1) to a field point. However, the different retarded times in the current densities must be taken into account. For the dipole order field, we take / R~\ / R0 L \ J1r~J1 ~2~’2 ~ (8)
where y is a phase factor, a~are real amplitudes and the real parts of eqs. (14) correspond to the physical dipoles. From eqs. (13) and (14), the time.averaged recoil force is 4 (15) ~ ..~L.fcos(J~Ln~ ~ 4irc where k w/c. Since there are odd numbers of unit normals which multiply the amplitudes in the integrand of eq. (15), only the odd part in n3 of cos (kLn3 + 7), namely —sin kLn3 sin 7, will contribute upon integration over solid angle. The required integrals are given in the Appendix of ref. [4]. The result is
~—-~--~
‘
.
From eqs. (6)—(8) and the relationship between dipole moment and current density, d =f.JdV,
(9)
e~(~ t+y),
the magnetic field contributions are 1 / 2R H1=—-— di ~t c 0 2R d2 H2=~ c 0
R0\ __)Xn, C
(~_~
C
4
(10)
(P
•
j
(P
+
13
[(aia2+aia2)
w. [32 siny (a1a2 L
)—
1
3
—~——-~.
P
P
sinp
+
23/1 a1a2) i—2 \p
3~. p4JI sinp
——
cos p
~
P / -~+ -~-) cos p sin 7 ra~a~(,,— C4 L ~ P3/ / 2 3 ~ +1_——~siflp
3) = (P
4 — -~—
Pa=_~.fH 1.H2n0R~d~~, (12) which can contribute to dipole-dipole order. From eqs. (10) and (12), the interference yields a recoil force 1 .. •. = ~ (13) 2~ where d 1 and d2 are evaluated at the retarded times as indicated in eq. (10). It is to be noted that the integral in eq. (13), containing odd products of unit normals, would vanish if the two dipoles had the same time retardation. Thus, its potential as a source of lower order recoil is easily overlooked, As a simple example, consider two simple harmonic oscillators
_
(14)
cos~)]
+~ .~,
where the common unit normal, n=R 0/R0, (11) has been used for the required accuracy. Corrections to both hR0 and n would not affect the computation of the momentum flux over the distant sphere. From eqs.momentum (4) and (5),flux. we see that there three terms in the However, it isare only the interference term
a~e~~t
‘31
>=—~-Sm7 C
+~n3~ Xn, C /
=
2
16
( )
p4.’
\p + a~a~(!_ ~°
2-)
cos p
+
(— -±+ 2-) sin p ]
p3
p2 p4 where p kL. For kL ~ 1, the negative powers of kL, in the brackets multiplying the amplitudes in eqs. (16), cancel out leaving a leading term proportional to kL. The result is then of the same order as in eq. (1). The case kL ~- 1 is likewise uninteresting. The enhanced recoil occurs for kL 1. The brackets are then of order unity and the recoil is oflower order than that of eq. (1). It is to be emphasized that the non-relativistic velo. city condition imposes the constraint k1~ 1 for both components 1 and 2 of the composite system. However, the separation L is entirely free and hence the ‘~
169
Volume 60A, number 3
PHYSICS LETTERS
three possibilities for kL, mentioned above, can be arranged [5]. The enhanced recoil can occur when the spacing is of the order of the characteristic wavelength. It would be of interest to test the effect experimentally and to consider possible technological applications.
References [1] A. Peres, Phys. Rev. 128 (1962) 2471.
170
21 February 1977
[2] L.D. Landau and E.M. Lifshitz, The classical theory of fields, 4th revised English edition (Pergamon Press, Oxford, 1975). [31 F.I. Cooperstock, Phys. Rev. 165 (1968) 1424. [4] F.I. Cooperstock and D.J. Booth, Phys. Rev. 187 (1969) 1796. [5] F.I. Cooperstock, to be published in Ap. J. Part I (April 1977).
PHYSICS LETTERS
21 February 1977
ENHANCED ELECTROMAGNETIC RADIATION RECOIL* Fl. COOPERSTOCK and D.W. HOBILL Department of Physics, University of Victoria, Victoria, B.C., Canada V8W 2Y2 Received 2 November 1976 The interference between two separated electric dipole radiators leads to a lower order linear momentum flux and hence recoil than that which can be achieved by a single system.
If the electromagnetic radiation from a system of accelerating charges carries away linear momentum in a preferred direction, the system must recoil in the opposite direction. This effect has been calculated from the Lorentz force on the radiation field back on the
that the aforementioned analogy is complete: a system of two separated electric dipole radiators can interfere constructively to induce a recoil force of lower order than that of eq. (1). Consider a system of two electric dipoles d1, d2
current density [1] and from the rate at which the elec. tromagnetic field carries momentum across an infinite sphere [2]. For a single system, the recoil force is [2]
separated by a distance L along the z axis. The charges in each of the dipoles are assumed to move non.rela. tivistically and hence the characteristic wavelength,
)i~c~ ~ ÷~ (d X ni)0 ~~ (1) 4irc where d and m are the electric and magnetic dipole moments respectively and D0~is the electric quadrupole moment. It has been noted that this force is of higher order in 1/c than the radiation damping forces which correspond to an energy loss rate bilinear in the elec. tric dipole moment. A completely analogous situation prevails in the theory of gravitational radiation. Because of momenturn conservation, there can be no dipole gravitational radiation. The lowest order energy loss is quadrupole, and the rate is bilinear in the mass quadrupole moment. For a single system, gravitational radiation recoil results from the interference between the quadrupole and octupole radiation. The gravitational and electro-
X, of the emitted is much greater than the characteristic size,radiation i~(e = 1,2) of each sub-system. From the energy-momentum conservation laws, the rate of change of linear momentum of the system P~ is simply expressed in terms of the momentum flux density [2]: 0 = d 1 T~0d V ~)CTaP dS~ (2) P C where the momentum flux density or Maxwell stress tensor is T0~= (1/4~)[_E0E~ H~HP+ 1 6013(E2 + H2)]. (3)
F0 =
—
,
—
,
2
Since the flux in eq. (2) will be found in the wave zone where E and H are orthogonal, eqs. (2) and (2) yield .
P0
=
—
4__fH2nbR~d~l. magnetic phenomena are shifted relative to each other by one order. For the gravitational case, it has been demonstrated [3—5]that under the correct circumstances, two Separated systems can induce a lower order radiation recoil by virtue of the interference from the quadrupole radiation alone. In this letter, it is demonstrated
where H~= (A~X n~)/c
*
and
.
.
Supported by National Research Council of Canada Grant A5340 and University of Victoria Faculty Research Grant 08584.
168
(4)
The total magnetic field is a linear superposition of contributions from each dipole: —
,
(6)
1 R A~=—~--fJ(t__~!) dV~
(7)
,
c = 1, 2
.
Volume 60A, number 3
PHYSICS LETFERS
21 February 1977
4
It is to be noted that to the required accuracy, the hR0 factor is common to both vector potential con-
d~=
tributions, where R0 represents the distance from the origin of one of the dipoles (let it be d1) to a field point. However, the different retarded times in the current densities must be taken into account. For the dipole order field, we take / R~\ / R0 L \ J1r~J1 ~2~’2 ~ (8)
where y is a phase factor, a~are real amplitudes and the real parts of eqs. (14) correspond to the physical dipoles. From eqs. (13) and (14), the time.averaged recoil force is 4 (15) ~ ..~L.fcos(J~Ln~ ~ 4irc where k w/c. Since there are odd numbers of unit normals which multiply the amplitudes in the integrand of eq. (15), only the odd part in n3 of cos (kLn3 + 7), namely —sin kLn3 sin 7, will contribute upon integration over solid angle. The required integrals are given in the Appendix of ref. [4]. The result is
~—-~--~
‘
.
From eqs. (6)—(8) and the relationship between dipole moment and current density, d =f.JdV,
(9)
e~(~ t+y),
the magnetic field contributions are 1 / 2R H1=—-— di ~t c 0 2R d2 H2=~ c 0
R0\ __)Xn, C
(~_~
C
4
(10)
(P
•
j
(P
+
13
[(aia2+aia2)
w. [32 siny (a1a2 L
)—
1
3
—~——-~.
P
P
sinp
+
23/1 a1a2) i—2 \p
3~. p4JI sinp
——
cos p
~
P / -~+ -~-) cos p sin 7 ra~a~(,,— C4 L ~ P3/ / 2 3 ~ +1_——~siflp
3) = (P
4 — -~—
Pa=_~.fH 1.H2n0R~d~~, (12) which can contribute to dipole-dipole order. From eqs. (10) and (12), the interference yields a recoil force 1 .. •. = ~ (13) 2~ where d 1 and d2 are evaluated at the retarded times as indicated in eq. (10). It is to be noted that the integral in eq. (13), containing odd products of unit normals, would vanish if the two dipoles had the same time retardation. Thus, its potential as a source of lower order recoil is easily overlooked, As a simple example, consider two simple harmonic oscillators
_
(14)
cos~)]
+~ .~,
where the common unit normal, n=R 0/R0, (11) has been used for the required accuracy. Corrections to both hR0 and n would not affect the computation of the momentum flux over the distant sphere. From eqs.momentum (4) and (5),flux. we see that there three terms in the However, it isare only the interference term
a~e~~t
‘31
>=—~-Sm7 C
+~n3~ Xn, C /
=
2
16
( )
p4.’
\p + a~a~(!_ ~°
2-)
cos p
+
(— -±+ 2-) sin p ]
p3
p2 p4 where p kL. For kL ~ 1, the negative powers of kL, in the brackets multiplying the amplitudes in eqs. (16), cancel out leaving a leading term proportional to kL. The result is then of the same order as in eq. (1). The case kL ~- 1 is likewise uninteresting. The enhanced recoil occurs for kL 1. The brackets are then of order unity and the recoil is oflower order than that of eq. (1). It is to be emphasized that the non-relativistic velo. city condition imposes the constraint k1~ 1 for both components 1 and 2 of the composite system. However, the separation L is entirely free and hence the ‘~
169
Volume 60A, number 3
PHYSICS LETTERS
three possibilities for kL, mentioned above, can be arranged [5]. The enhanced recoil can occur when the spacing is of the order of the characteristic wavelength. It would be of interest to test the effect experimentally and to consider possible technological applications.
References [1] A. Peres, Phys. Rev. 128 (1962) 2471.
170
21 February 1977
[2] L.D. Landau and E.M. Lifshitz, The classical theory of fields, 4th revised English edition (Pergamon Press, Oxford, 1975). [31 F.I. Cooperstock, Phys. Rev. 165 (1968) 1424. [4] F.I. Cooperstock and D.J. Booth, Phys. Rev. 187 (1969) 1796. [5] F.I. Cooperstock, to be published in Ap. J. Part I (April 1977).