A recent article in this journal by Narayan1 illustrated that magnetic forces do not obey Newton's third law of action and reaction via the example of two “point” electric dipoles whose moments pi=p0,ieλtx̂i grow exponentially with time. He correctly found that F12+F21+dPEM/dt=0, where PEM=E×BdVol/4πc (in Gaussian units, with c as the speed of light in vacuum) is the electromagnetic-field momentum associated with the two dipoles.2 This is an example of a general result by Page and Adams3 published in this journal many years ago, in the so-called Darwin approximation4–6 that keeps terms only to order v2/c2 (which ignores electromagnetic radiation), where v is velocity. Narayan's demonstration is nice in that it holds to all orders of v/c.

However, Narayan mistakenly claimed that “there are no electromagnetic fields in the far field zone,” and hence, “no radiation is emitted” in his example, based on the appearance of his Eqs. (2) and (3). While those forms are convenient for later computations, they do not well display the characteristics of electromagnetic fields of a time-dependent dipole. This was the topic of the appendix to a paper in this journal by Berman,7,8 where the magnetic field of a time-dependent electric dipole (at the origin) was shown to be B(r,t)=[p¨]×r̂/c2r+[ṗ]×r̂/cr2 with [p]=p(tr/c). The term that varies as 1/r is a “radiation” field, nonzero except in the limit t when the dipole moment is zero by construction. The radiation pattern of Narayan's two dipoles can be computed using Eq. (66.6) of Landau,5dI/dΩ=cr2[Brad]2/4π, and Fig. 1 of Narayan, where x̂1=x̂ and x̂2=ŷ, noting also that r̂=sinθcosφx̂+sinθsinφŷ+cosθẑ,dI/dΩ=λ4[(p1+p2)×r̂]2/4πc3=λ4[(p12+p22)cos2θ+(p1sinφp2cosφ)2sin2θ]/4πc3. Then, the total intensity, integrated over solid angle is, I=2λ4[p12+p22]/3c3.9 

We can also consider the radiated momentum, dPrad/dt=dΩd2Prad/dtdΩ, where d2Prad/dtdΩ=(cr̂/4π)dI/dΩ.10 Although d2Prad/dtdΩ is nonzero, the total radiated momentum dPrad/dt is zero, recalling the above form of dI/dΩ.

A variant of Narayan's example has been suggested by Onoochin,11 in which the time dependence of the dipoles is pi=p0,itx̂i, and the corresponding charge and current densities are ρi=Iit(/xi)δ3(rri) and Ji=Iidliδ3(rri)x̂i with r1=0 and r2=ax̂. Such a configuration has been called “semistatic,”12 for which the electric and magnetic fields are given by the instantaneous Coulomb and Biot-Savart forms. There is no radiation in this variant. It follows that F12+F21=(I1dl1)(I2dl2)ŷ/a2c2=dPEM/dt, more quickly than in Narayan's example.13 

In a larger historical context, Ampère's insistence that magnetic forces obey Newton's third law earned him the sobriquet by Maxwell14 as the “Newton of electricity.” Ampère's authority held up acceptance of the “Lorentz” force law (stated obliquely by Maxwell in 186115,16) until efforts by Thomson17,18 and Heaviside19 in 1891 clarified that electromagnetic fields carry momentum as well as energy (following the first clear statement of the “Lorentz” force law by Heaviside20 in 1885).

In 1864, Maxwell discussed “electromagnetic momentum,”21 identifying this with Faraday's “electronic state” in Sec. 26, and clarifying in Sec. 57 that the electromagnetic momentum of charge q in an external vector potential A is qA(/c) (in the Coulomb gauge, as favored by Maxwell). This formulation suggests that electromagnetic momentum is a property of the charge, rather than of the electromagnetic field. That Maxwell's electromagnetic momentum is equivalent to the electromagnetic-field momentum of Thomson and Heaviside (the PEM of this Letter) in quasistatic examples was first demonstrated by Thomson.22,23

Applying Maxwell's formulation, PEM=ΣqjA(rj)/c, of electromagnetic momentum to the “semistatic” variant of Onoochin, we see that the (Coulomb-gauge) vector potential of dipole 1 is the same at the position of both charges of dipole 2, and so does not contribute to the momentum. The vector potential of dipole 2 along the x-axis is A2(x)=I2dl2ŷ/c|ax|, so the electromagnetic momentum is PEM=(q1dl1/c)dA2(0)/dx=(I1dl1)(I2dl2)tŷ/a2c2, noting that q1=I1t.

1.
O.
Narayan
,
Am. J. Phys.
89
,
1033
1036
(
2021
).
2.
We exclude the self-momentum of moving charges, and consider only the interaction field momentum.
3.
L.
Page
and
N. I.
Adams
, Jr.
,
Am. J. Phys.
13
,
141
147
(
1945
).
4.
C. G.
Darwin
,
Philos. Mag.
39
,
537
551
(
1920
).
5.
L. D.
Landau
and
E. M.
Lifshitz
,
Classical Theory of Fields
, 4th ed. (
Pergamon
,
New York
,
1975
), Sec. 65.
6.
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
), Sec. 12.6.
7.
P. R.
Berman
,
Am. J. Phys.
76
,
48
54
(
2008
).
9.
The electric-dipole radiation follows from Eq. (67.8) of Landau5 as IE1=2[p¨2]/3c3=2λ4[(p1+p2)2]/3c3=2λ4[p12+p22]/3c3, which is the same as the total intensity.
11.
V.
Onoochin
, private communication.
12.
D. J.
Griffiths
and
M. A.
Heald
,
Am. J. Phys.
59
,
111
117
(
1991
).
13.
For details, see the appendix of kirkmcd.princeton.edu/examples/narayan.pdf.
14.
J. C.
Maxwell
,
A Treatise on Electricity and Magnetism
(
Clarendon
,
Oxford
,
1873
), Vol. 2, Art. 528.
15.
J. C.
Maxwell
,
Philos. Mag.
21
,
338
348
(
1861
), Eq. (77).
16.
See also Appendixes A.28.1.7, A.28.2.6, A.28.3.7, and A.28.4.7 of kirkmcd.princeton.edu/examples/faradaydisk.pdf.
17.
J. J.
Thomson
,
Philos. Mag.
31
,
149
171
(
1891
), Eqs. (2) and (6).
19.
O.
Heaviside
,
Philos. Trans. R. Soc.
183
,
423
480
(
1892
), Sec. 26. This was a clarification of Eq. (7a) of his paper, Electrician 16, 186–188 (1886).
20.
O.
Heaviside
,
Electrician
14
,
178
180
(
1885
).
21.
J. C.
Maxwell
,
Philos. Trans. R. Soc.
155
,
469
512
(
1865
).
22.
J. J.
Thomson
,
Philos. Mag.
8
,
331
348
(
1904
).
23.
For discussion in this journal, see the appendix of
M. G.
Calkin
,
Am. J. Phys.
34
,
921
925
(
1966
); see also kirkmcd.princeton.edu/examples/pem_forms.pdf.