Hidden momentum, field momentum, and electromagnetic impulse

The relation between momentum density and energy flux is not peculiar to electrodynamics. See

R. P.

 

Feynman

,

R. B.

 

Leighton

, and

M.

 

Sands

, (, , ), Vol. , Eq. (27.21).

F. S.

 

Johnson

,

B. L.

 

Cragin

, and

R. R.

 

Hodges

, “,”  , – ().

K. T.

 

McDonald

, “” <www.hep.princeton.edu/~mcdonald/examples/cap_momentum.pdf>.

As we shall see, this turns out to be incorrect.

W. H.

 

Furry

, “,”  , – ().

R. H.

 

Romer

, “,”  , – (). Romer did not use a polarized/magnetized sphere but a sphere carrying the same surface charge and current distributions as they would produce. This avoids the awkward question of whether we should use $D×B$ in place of Eq. (1) and preempts conceptual problems about hidden momentum in bound currents. If such issues arise, this example is to be interpreted in Romer’s way, as a configuration of free charges and free currents.

Hidden momentum occurs in moving systems as well. See, for instance,

E.

 

Comay

, “,”  , – (), but it is most striking in static configurations, and we shall concentrate on such cases.

In this paper all such processes will be carried out quasistatically to avoid electromagnetic radiation (which would remove momentum from the system).

This, incidentally, is the answer provided in the Solution Manual to

D. J.

 

Griffiths

, , 3rd ed. (, , ), Problem 8.6. The error was caught by David Babson.

See

S.

 

Coleman

and

J. H.

 

Van Vleck

, “,”  , – () and Refs. 5 and 11.

M. G.

 

Calkin

, “,”  , – ().

Center-of-energy is the natural relativistic generalization of center-of-mass: $∫rudτ/∫udτ$⁠, where $u$ is the energy density.

It is notoriously dangerous to speak of the momentum (or energy) of a configuration that is not localized in space. In this context a uniform field should always be interpreted to mean locally uniform, but going to zero at infinity.

See Refs. 11, 15, and 16.

L.

 

Vaidman

, “,”  , – ().

V.

 

Hnizdo

, “,”  , – ().

D. J.

 

Griffiths

, “,”  , – ().

Because electromagnetic momentum [see Eq. (1)] is linear in $B$⁠, this result—expressed in terms of the total magnetic dipole moment—holds for any collection of dipoles, and hence in particular for the rectangular loop (which could be built up as a tessellation of tiny squares).

See Refs. 11, 15, 16, and 3.

See Ref. 15 for the nonrelativistic argument and Ref. 16 for the relativistic version.

See, for example,

R. J.

 

Adler

,

M. J.

 

Bazin

, and

M.

 

Schiffer

, , 2nd ed. (, , ), Sec. 9.2.

Why should a moving fluid under pressure carry extra momentum? This is a surprisingly subtle relativistic effect. For a lovely explanation, see

K.

 

Jagannathan

, “,”  , – ().

The essence of the argument appears also in Ref. 7, Sec. IIB. A variation is suggested by

K.

 

Szymanski

, “,”  , – (), Sec. V;

see also

R.

 

Medina

, “,”  , – ().

A more general version of the following argument is found in Ref. 16 and in a slightly different form in

V.

 

Hnizdo

, “,”  , – ().

This realization casts doubt on our original naive expression for the field momentum in Eq. (2), which took the electric field to be uniform inside the capacitor and zero outside, and as we have seen, that equation is in error (Ref. 3).

Because electromagnetic momentum [see Eq. (1)] is linear in $E$⁠, our result—expressed in terms of the total electric dipole moment—will hold for any collection of dipoles, and hence in particular for the original capacitor model. This is how McDonald (Ref. 3) fixed the error in Eq. (2).

More precisely, we want an electrically neutral spherical shell that carries a surface current density $K=σ(ω×r)$⁠. In the presence of an electric field, the charges constituting this current will speed up and slow down (in the first model of Sec. III), spoiling the simple picture of a rigid spinning sphere. The incompressible fluid model is, in this sense, closer in spirit to the spinning sphere.

More simply, we can use the general expression for the momentum of an electric dipole in a magnetic field (Ref. 17): $pem=(p⋅∇)A$⁠, where in this case $A=−12(r×B)$⁠.

This is the “hidden momentum force” that began the whole story.

W.

 

Shockley

and

R. P.

 

James

, “ ‘,”  , – ().

One might worry, of course, about assuming that the solenoid is infinitely long, or—if finite—the approximations involved in treating its field as uniform inside and zero outside.

The rotational analog is strikingly different: Angular momentum in static fields is not (typically) compensated by hidden angular momentum (there is no rotational analog to the center-of-energy theorem), and when the fields are removed, the system starts to rotate, with angular momentum equal to that originally stored in the fields. The classic case is the “Feynman disk paradox” (Ref. 1, Sec. 17-4). It is easy to construct configurations with hidden angular momentum, but because there is no analog to the center of energy theorem, hidden angular momentum is not forced on us as dramatically as hidden linear momentum.

A possible counterexample is given in Ref. 7, Sec. III, where the role of hidden momentum is played by standing electromagnetic waves.

For example, if the magnetic field is produced by stationary magnetic monopoles, instead of electric currents, then there is no hidden momentum, but in that case the electromagnetic momentum also vanishes (Ref. 17).

Although hidden momentum was first discovered 40 years ago (Ref. 28), it continues to carry a mysterious aura and has been misunderstood and misused by a number of authors (including one of us, D.J.G.). Just last year

Jon

 

Thaler

(personal communication, August 26, 2007) and

Timothy

 

Boyer

, “,”  , – () pointed out that the coaxial cable is not an example of hidden momentum—there is momentum in the fields, but the center of energy is not at rest. The battery at one end is losing energy, and the resistor at the other end is gaining energy, and the momentum in the fields is precisely the momentum associated with the motion of the center of energy.