Gravity-induced electric field near a conductor

ANNALS

OF PHYSICS: 46, l-11

(1968)

Gravity-Induced

Electric

Field Near

MURRAY Argonne

National

a Conductor*

PESHKIN

Laboratory,

Argonne,

INinois

A new derivation of the gravity-induced electric field is given. The central idea of this treatment is a generalization of the Faraday cage idea. It is assumed here that the force on a test charge surrounded by a conductor is independent of the charge on the conductor and of external electric fields, even when the conductor is differentially compressed by gravity. Under reasonable phenomenological assumptions, it is shown that compressiondependent corrections to the gravity-induced field are negligible. Then, if the test charge has the same e/m as an electron, a shield against electric fields also serves as a shield against gravity. Whether the necessary assumptions apply to real conductors is not certain. I. INTRODUCTION

Free-fall experiments upon electrons (I) have given rise to the speculation (2), (3) that the necessary electrostatic shield may also serve as an effective shield against gravity for electrons. The argument has been put variously. In the version of Ref. (2), it is assumed that the material in the electrostatic shield disposes itself in such a way as to leave zero net force on the conduction electrons. This means that the force exerted on an electron by the induced electric field in the interior of the shield precisely compensates the weight of the electron. It then follows from the laws of electrostatics that the electric field in a region surrounded by the conductor will also compensate the weight of an electron. The qualitative argument has been reinforced by a quantum mechanical calculation (4) based upon a phenomenological model of the conductor. In that calculation, a test charge in the shielded region is taken as the source of a fixed electric field acting on the conductor. In the absence of gravity, it is assumed that the electron density in the conductor gives just the charge density implied by conventional electrostatic theory. The energy in the presence of gravity is then calculated by first-order perturbation theory, which of course uses the zero-order (in g) wavefunctions. The gradient of that energy with respect to the coordinate of the test charge gives the gravity-induced electric force acting on the test charge. When the conductor completely surrounds the test charge, it emerges that the gravity-induced electric field is just equal to (m/e)g, in agreement with the result of the qualitative * Work performed under the auspices of the U.S. Atomic Energy Commission.

1 0 1968 by Academic Press Inc.

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argument. For other geometric arrangements, the gravity-induced field is smaller, but still proportional to g. The fields induced by ion-distortion and lattice-distortion effects caused by the field from the test charge are estimated in Ref. (4) to be 5 0.1 % of the gravity-induced electric field in a typical experiment. On the basis of the perturbative treatment of gravity, it is argued that any gravitational distortion of the lattice results in contributions to the induced electric field which are proportional to higher powers of g, and presumably small. The perturbation-theoretic treatment of gravity raises certain questions. The qualitative argument quoted above from Ref. (2) does not require the assumption of perturbation theory, and it is curious that such an assumption should arise in the quantum mechanical theory. Moreover, there is reason to doubt the reliability of the perturbation theory. One of the effects of gravity is to deform the shape of the conductor by an amount proportional to g. That deformation results in contributions to the electric field which are themselves proportional to g. However, those contributions are not obtained through formal use of perturbation theory. The contradiction arises because the charge density on the conductor drops sharply near the surface, and cannot be expressed as a power series in g. An extreme example is given in the Appendix to illustrate how the perturbation series gives a completely incorrect electric field proportional to g. Although the example is quantitatively unrealistic, it does demonstrate a deformation-dependent correction to the gravity-induced electric field. That correction, which is not covered by the distortion estimate of Ref. (4), could perhaps be appreciable in a realistic case. It is also curious that the quantum mechanical theory should involve itself with an approximation wherein the electron density alone provides the charge density which electrostatic theory postulates. The qualitative argument uses only the assumption that the net force on the conduction electrons vanishes. A closely related question has been raised in the context of a Thomas-Fermi treatment of the conduction electrons. There it has been reported (5) that gravitational deformation of the lattice gives rise to an electric field which is proportional to g, but much larger than (m/e)g. Whether or not the Thomas-Fermi model accurately represents the conduction electrons in a real metal, that result challenges the assumptions of Ref. (4). This report seeks to answer the questions raised above by starting from a somewhat different phenomenological assumption which obviates the perturbative treatment of gravity. It is assumed here that when a charged or neutral conductor is supported in a gravitational field and is simultaneously subjected to applied electric fields, the electric field acting on a test charge macroscopically distant from the conductor is given by 8 = ax) + &3,t(X>, (1.1) where &e(x) is independent

of the charges on the conductor and test body and of

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applied electric fields. The electrostatic terms 8&x) is defined as the electric field acting on the test charge according to the laws of electrostatics, with the given charge on the conductor and the given applied electric field, under the condition that the deformed surface of the conductor be equipotential. The present phenomenological assumption is suggested by common experience with electric fields in terrestrial experiments involving much stronger electric fields than those to be considered here. In particular, for a test charge surrounded by a conductor, this assumption asserts that the motion of the test charge is not influenced by the net charge on the conductor, nor by externally applied electric fields. To that extent, this assumption appears to underlie the use of an electrostatic shield in any free-fall experiment upon electrons. The term as(x) will be called the gravity-induced electric field. Actually, c&t(x) also depends upon gravity (through the deformation of the surface), but it is assumed that the implied electrostatic calculation automatically includes that effect. Under the phenomenological assumption of this treatment, it is possible to find an approximation (Section III) in which the gravity-induced electric field in any geometric arrangement is obtainable without a perturbative treatment of gravity. The result of that approximation agrees with the result of Refs. (2) and (4). A test electron that is shielded against electric fields is also shielded against gravity. The present treatment leads to model-dependent corrections, proportional to g. These are estimated (Section IV) by using a free-electron model, and are found to be negligible under certain reasonable assumptions. However, those corrections are not negligible in the strict Thomas-Fermi model of Ref. (5).

II.

ADIAFSATIC

APPROXIMATION

The full Hamiltonian J’& for a test particle near a material body, all in a gravitational field g and a uniform external electric field E, is given by %(4, g, E, X, P, Xi 3Xi) = %(Xi 3 Xi) f 8(X, P> + 4%(X, Xi 3 XJ -gg~X+m~Xi+M~Xil - E [p - e C Xi + Ze 1 Xi]. l

q, ~1,x, p = charge, mass, coordinate, momentum

of the test particle. -e, m, xi = charge, mass, coordinates of electrons in the body. Ze, M, Xi = charge, mass, coordinates of nuclei in the body. & = Hamiltonian for the body, including any structure that supports it, for g = E = 0.

(2.1)

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tit = Hamiltonian forces other qZc = Hamiltonian particle and

for the test particle, including any external than those represented by g and E. for Coulomb interaction between the test the body.

The momentum variables of the electrons and nuclei in the body are suppressed in Eq. (2.1) and hereafter. The Coulomb energy operator is 4%(x,

xi , Xi> = -qe 1 (I x - xi 1-l) + qZe C (I x - Xi 1-l).

(2.2)

It is assumed that for relevant values of x, at macroscopic distances from the body, the motion of the test particle is given accurately by the adiabatic approximation which follows. First, the state of the body is determined, for a fixed position x of the test particle, by the approximate Hamiltonian ~b,

g, E,

X 1 Xi

, xi>

=

Hb(xi

, xi>

+

q&(x

I Xi

, xi>

- (mg - eE) * c xi - (Mg + ZeE) * c Xi . (2.3)

This approximate Hamiltonian determines a density operator p(q, g, E, x 1Xi , Xi) which, like 2, depends upon the external parameters q, g, E, x, and operates upon the dynamical variables xi , Xi and upon their (suppressed) corresponding momenta. In reality, p also depends upon the temperature, but for present purposes it could just as well be regarded as representing the square of the wave function for the ground state of .%. Either way, the energy of the body is equal to

Wq, g, E I X>= Wpk, g, E, x I Xi yXi> s(q, tit,JGX I Xi 3X2>.

(2.4)

The trace operation, which represents a sum over quantum states of the body, integrates out the dynamical variables. The adiabatic approximation is completed by using W(q, g, E I x) as an effective potential energy in the approximate Hamiltonian %d(%

for the motion

g, E

1 x,

P) = 8(x,

P) - (pg + qE) * x + w(q, g, E 1x)

(2.5)

of the test particle. Then the force on the test particle is equal to

W, g, E I x, P)

= -V&(x,

P) + 018 + qE) - VWq,

g, E I x).

(2.6)

The first term represents external forces such as that of the magnetic guide field in the Stanford experiment (I). The net electric field acting on the test particle is given by

&q, g, E I x> = E - $ VW, g, E I 4 = E - Vn W(q, g, E, x I xi , Xi> %Xx I xi , &>I

= E - Wdq, g, E, x I xi , Xi> V&(x

I xt , W.

(2.7) (2.8) (2.9)

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A CONDUCTOR

No gradient of p appears in Eq. (2.9) because the variation principle guarantees the stability of energy averages against small variations in wavefunctions or density operators. In terms of the densities pc and pM of charge and of electric dipole moment in the body, the energy is given by

q-l Wq, g, E I x> = j- p&, g, E, x I ~‘1I x - x’ 1-l dx’

-s

pM(q, g, E, x I x’)

l

V, I x - x’ 1-l dx’,

(2.10)

where the integrals are carried over points x’ in the deformed body.

III.

GEOMETRIC

DEFORMATION

APPROXIMATlON

Under the phenomenological assumption discussed in Section I, it is necessary to consider only the case of a neutral conductor, since b,t can always be modified to include the electrostatically determined field due to a charge on the conductor. Then the electric field on the test charge is given by g(q, g, E I x> = 4dx)

+ tp,t(q, g, E I x>,

(3.1)

where gst vanishes for q = E = 0, and is otherwise calculated by conventional electrostatics. The gravity-induced electric field 4m

is most easily approximated

= 60, g, 0 I XI

0.2)

by using

&T(X) = JW, g, Eg I x1 - &40, g, E, I x1

(3.3)

with Eg = (m/e)g -

For that choice of the arbitrary reduces to %(g

I xi 3 &I = W4 =

%(x$

IO-lOV/rn.

(3.4)

external field, the Hamiltonian

for the body

g, Eg , x I xi , Xi) ,

xi)

-

(hf

+

zm)g

l

c

xi

.

(3.5)

The density operator derived from X0 , namely, po(g I xi 3 Xi) = ~(0, g, J&z, x I xi , Xi),

cannot depend upon x, since so does not.

(3.6)

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The external field Es was chosen to make the net external force on electrons vanish. Then the Hamiltonian .?i$ represents a problem in which a gravitylike force acts on the nuclei, but not on the much more mobile electrons. This circumstance suggests an approximation wherein it is assumed that the electrical structure of the body is deformed by gravity in a purely geometrical way, so that charge densities are shifted to corresponding points. The accuracy of that approximation is discussed in Section IV. For the present, it is assumed. Then p,, represents a system that is everywhere neutral, when viewed macroscopically, and l

b(0, g, Es 1x) = Es - VW, = Es, 4T’e(x>= Eg - &t(O, g, Eg I 4. In the complete-shielding vanishes for q = 0. Then

situation, 440,

(3.7)

according to electrostatics, the field at x

g, EB , x> = 0,

&g(x) = Eg = (m/e) g, and the force on the test charge for nonvanishing W, g, 0, x, P) = -VSt(x, The last term represents in the usual way, using surface. If JLJ~ = -m/e, For other geometric with the external field surfaces.

IV.

(3.8)

q is

P) + (P + T)

g + 44&,

g, 0,x).

(3.9)

the image force, which is proportional to 4%.It is calculated the deformed surface of the conductor as an equipotential gravity is precisely compensated. arrangements, one must solve the electrostatic problem equal to (m/e)g, and with the appropriate equipotential

DYNAMIC

EFFECTS

OF

DEFORMATION

The Hamiltonian X0 involves no external force on electrons. In the geometricdeformation approximation, the conductor is deformed by gravity, but no macroscopic density of charge or electric moment is induced by the deformation. In reality, an electric dipole moment is induced in each atom or ion by the weight of the nucleus. More importantly, the lattice is differentially compressed, more near the bottom than near the top. In the geometric-deformation approximation, the conduction electrons respond by preserving macroscopic electrical neutrality everywhere. That approximation therefore corresponds to the condition of mini-

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mum electrostatic energy. In a real conductor, the kinetic energy of the conduction electrons causes a small shift in the direction of uniform distribution of the condw tion electrons. Consequently, there is a net positive charge near the bottom and a net negative charge near the top. Both of these phenomena produce an electric field near the conductor. That field cannot be calculated without a theory of the structure of the conductor and especially of its surface. However, it is possible to obtain a crude estimate from a natural extension of the phenomenological assumption of this treatment. It will now be assumed that charges which are buried deeper inside the conductor than some depth t are effectively shielded by the conductor; they leave the surface equipotential. Consider first the polarization of the atoms and ions. The displacement d of the nucleus from the center of the distribution of negative charge is given by &r/3)

p,ZGd

= Mg,

where (-e) pn is the central density of the negative charge distribution. induced dipole moment per atom is Zed = 3Mg/hpne.

For (M/p*)

-

(4.1)

The (4.2)

10-48g cm-3, Zed -

10-asesu cm.

(4.3)

The contribution to the gravity-induced electric field outside of the conductor depends upon the geometric arrangement. For a long hollow cylinder of radius r, it is given to order of magnitude by d -

(Zed) pot/r.

(4.4)

For t -=z10-3 cm, which seems very conservative, and r = 10 cm, 8 2 IO-13V/m < (m/e)g.

(4.5)

This reasoning is in essence the same as that of Ref. (4). The charge separation of the conduction electrons would seem to be safely overestimated by a simplified example of the Thomas-Fermi calculations of Ref. (5). Otherwise free electrons in a box are supposed to move through a smooth distribution of positive charge whose density is given by ep+(d = 41 where pa and y are phenomenological

- ~8,

(4.6)

constants, and z is the height. For a neutral

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conductor, infinitely extended in the horizontal directions and having height L, the electron density p-(z) must minimize the energy U per unit horizontal area, where

lJ= i::,, (&

hying

+ ~[P-(z)]~I~) dz.

The constant coefficient cxis provided by the Thomas-Fermi internal electric field in the conductor is &in(Z)

= 47rC?Jr,,

[p+(“> - P-(z’)l dz”

(4.7) approximation.

The

(4.8)

If the kinetic energy term is neglected, it follows that (II = 0. Then the energy is minimized by maintaining p-(z) = p+(z), in agreement with the geometric-deformation approximation. The minimum condition on U for zero net charge is equivalent to f”(z) + y = s2f(z)[l - yz -f’(z)]“3,

f(-w>

=f(+m

= 0,

w9

where the newly introduced symbols are defined by f(z) = (47=&-l s2 = iEEf 5cL

4&), #3.

(4.10) (4.11)

Reasonable numerical values for a metal are y - lo-* cm-l, p,, - 10z3cm-3, CY- lO-27 erg cm2. Then s2 - lOI cm-2. For L < IO3 cm, the nonlinear factor in the differential equation may be neglected, and the solution of Eqs. (4.9) is given very accurately by (4.12) The internal electric field near the middle, c&(O) = 47rep,yr2 -

10-sV/m,

(4.13)

is much greater than (m/e)g. Under the shielding assumption discussed above, a rough estimate of the external field in a hollow cylinder is given by assuming a

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charge density (h)-lMi,/aZ on a cylindrical shell of thickness t. To order of magnitude, 6’ is then given by (4.14) d - (rt/L2) &in . For t < 1O-3 cm, I = 10 cm, and L = 1000cm, C 5 lo-14V/m < (m/e)g.

(4.15)

V. DISCUSSION

The substance of this paper lies in its phenomenological assumptions, which have been designed to give the same principal conclusion as do the assumptions of Refs. (2) and (4). Under those assumptions, the gravity-induced electric field near a conductor can be calculated by solving the appropriate electrostatic problem with the external field equal to (m/e)g. If the conductor is charged and the shielding incomplete, there is an additional force proportional to 4, and there is always an image force proportional to q2. Both of these involve terms proportional to g, because the shape of the equipotential surface depends upon g, but both are accounted for by the electrostatic calculations. The present assumptions provide an estimate of corrections to the geometricdeformation approximation. Although proportional to g, the corrections are negligible in practical cases. The main uncertainty of the analysis given here lies in the extrapolation of the known behavior of metals under the influence of strong electric fields to the case of much weaker fields. The extrapolation discussed in Section I requires that the induced field depends linearly upon the applied field, but it does not require that the linear dependence should be given correctly by perturbation theory. A similar assumption was made tacitly in Section IV. There it was assumed that shielding reduces the induced electric field by a factor ~10~. That reduction was made plausible by choosing a thickness t N 1O-3 cm, which is more than 100 times any reasonable penetration depth for electric fields. It does not appear to be possible to extract a reliable estimate of the error induced by these phenomenological assumptions from existing information. The relation of the Thomas-Fermi model to the present assumptions is clear. That model does not possess the shielding property which was assumed in Sec. IV. That disagreement does not weaken the conclusion of Sec. IV under the latter’s own assumptions, since there the Thomas-Fermi model was used only to obtain a presumed overestimate of the charge separation in a real conductor. It was pointed out in Ref. (5) that the Thomas-Fermi model illustrates how the gravity-induced electric field depends upon the dynamics of the conductor in an important way. From the present point of view, it appears that the gravity-induced field is particularly dependent upon the shielding property of the surface region.

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APPENDIX

Under the assumptions of Ref. (4, the perturbative

treatment of gravity yields

q&d4= -mv, j (g - x’)f.& Ix’) - pald3x’,

(A-1)

where the integration is carried over points in the undeformed body. The electron density P&X ( x’) is calculated with no uniform external field, but with the test charge serving as the source of an applied field, and with g = 0.

1

h 0

A

0

.

B

FIG. 1. Hollow conducting cylinder hanging from a spring in the absence (left) and presence (right) of gravity. The weight of a test electron is compensated at point B, but not at point A.

An extreme example which illustrates the difficulty with a perturbative treatment of gravitational deformation is indicated in Fig. 1. There, a hollow conducting cylinder is hung from a soft spring. Under the influence of gravity, the spring is stretched. The perturbation-theoretic formula makes use of the charge density induced on the cylinder in the upper position. It gives approximately GW

= (mlels,

4m

= 0,

64.2)

while the correct result is approximately &(A) = 0, The mathematical

4W

= Wek.

(A.3)

source of the error is illustrated by assuming a charge density

f(h 4 = [ew(+--)

+ I]-‘,

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which falls rapidly to zero near the surface x = b, if c is small. Under the influence of gravity, the surface may be shifted so that b is proportional to g. The series expansion p(b, 4 = t P&> b” n--O

(A. 5)

diverges for 1b 1 > TC exp(-x/c). The induced electric field may nevertheless have a series expansion in g, but it cannot be found from a series expansion of the charge density. RECEIVED:

June 26, 1967 REFERENCES

1. 2. 3. 4. 5.

F. C. W~BORN, Unpublished doctoral dissertation, Stanford University, 1965. Reference (I), p. 66. W. M. FAIRBANK AND F. C. Wrrrmo~~ (private communication). L. I. Scxim AND M. V. BARNHILL, Phys. Rev. 151, 1067 (1966). Bull. Am. Phys. A. J. DESSLER, F. C. MICHEL, H. E. RORSCHACH,JR., AND G. T. TRAMMELL, Sot. 12,183 (1967).