SOCIETÁ ITALIANA DI FISICA XCVII CONGRESSO NAZIONALE

L’AQUILA, 26-30 SETTEMBRE, 2011

Sergio Severini 1,a, Alessandro Settimi 2,b

1 Centro Interforze Studi Applicazioni Militari (CISAM),Via della Bigattiera lato monte 10, 56122 San Piero a Grado, Pisa, Italy

2 Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Geomagnetismo, Aeronomia e Geofisica Ambientale,via di Vigna Murata 605, 00143 Rome, Italy

a e-mail: sergio.severini@aeronautica.difesa.itb e-mail: alessandro.settimi@ingv.it

11

SEZIONE 6 : Fisica generale, didattica e storia della fisica

SOLENOIDALITY OF A MAGNETIC INDUCTION FIELD AND CONSERVATION OF TOTAL MOMENTUM [1]

22

ABSTRACT

The present talk discusses the physical aspects of a system of non-relativistic massive charge particles moving within an electromagnetic field (e.m.), which propagates through the entire space. The role of total momentum conservation on the solenoidality of a magnetic induction field is demonstrated. After a careful review of all the more widely sustained didactic justifications for the solenoidality of magnetic induction, some properties of the Maxwell stress tensor are defined according to Minkowsky. This talk presents a new framework wherein the necessary condition for the free-divergence of induction derives directly from the total momentum conservation of the system, i.e. particles and field.

33

INDEX

1. Introduction.

2. Magnetic field and dipoles.

2.1. Stationary case.

2.2. Non stationary case.

3. Role of momentum symmetry conservation on a magnetic induction field.

3.1. Momentum of electromagnetic field.

3.2. Implication of symmetry for magnetic induction fields.

4. Conclusions.

Appendix

44

Introduction

The Abraham–Minkoswski controversy regarding the momentum of fields in a dielectric started with Minkowski’s derivation of electromagnetic density momentum pem=D×B [2]. Abraham noted that the unsymmetrical Minkowski tensor did not support conservation of angular momentum [3] and proposed a symmetrical e.m. density momentum pem=E×H/c2, but requiring a new phenomenological force.

This talk adopts the Minkowski’s definition of e.m. density momentum, in agreement with the point of view discussed in ref. [4]. In the authors’ opinion, the macroscopic media (Minkowski) form of the stress tensor is conceptually preferable to the vacuum (Abraham) form in dielectrics, because it indicates stress without any need to attribute to the medium an unrealistic ability to transport a mechanical force field at the velocity of light and sufficient to increase the stress exerted by electromagnetic waves from ε0E2 to ε0εrE2. Clearly, it is electromagnetic wave propagation, not transmission by mechanical means that accounts for the transport of momentum.

55

Magnetic field and dipoles

With respect to an electric field, the magnetic field has substantially different characteristics: while positive electric charges can exist separately from negative charges, it is certainly not possible to isolate magnetic monopoles, which are instead always coupled in magnetic dipoles.

As a consequence magnetic fields have very different characteristics from electrical fields: the lines of force are always closed lines and so the outgoing flow from any closed surface is always zero, while in an electrical field the lines of force arise from positive charges (field sources) and finish on negative charges (field drains).

On the basis of current scientific knowledge of electromagnetic fields, and in particular of the magnetic component B, it is no longer didactically satisfactory to heuristically justify Maxwell second equation:

by asserting first that, for symmetry of the electrical field we should have and then imposing the spatial density of magnetic charge as zero, i.e. ρm=0 [4-9].

0div B B

m B

66

Stationary Case

Oersted Experience

A current-carrying wire reproduces a magnetic field in the surrounding space

Biot-Savart-Laplace law: fundamental law of magnetostatics in vacuum.

0340

dl ΔrdB

ΔrI

The constant µ0,

known as the vacuum magnetic permeability, in SI units is µ0=4π·10-7N/A2.

03( )

40

dl ΔrB

Δrlr I

Filamentary circuits:the surface S of the conductor has negligible linear dimensions

77

Applying the divergence operator ...

0 03 3

03 3

4 4

( ) 04

0

dl Δr ΔrB dl

Δr Δr

Δr Δr dl dl

Δr Δr

l l

l

II

I

This expression is identically null, because both

dl 0

3

1Δr0

ΔrΔr

Maxwell second equation:the field B0 is solenoidal.

ˆ( ) 0S SdS d

0 0 0B B n B The magnetic induction field flow

through any closed surface is null.

88

Non stationary case

The case is now considered in which the electromagnetic field (e.m.) occupies all the space filled by a linear material medium, isotropic and homogeneous, in which the sources are also located.

D E

H B

Dielectric permittivity and magnetic permeability of material medium:

0 r

0 r

99

The Maxwell equations in SI units:

D

0 B

t

B

E

t

D

H J

Current continuity equation:

0t

J

1010

The second Maxwell equation can be deduced from the third Maxwell equation as follows. The divergence operator is applied to the latter. Considering that the divergence of a rotor is identically null, this gives:

0 ( )t t

B

E B

Given that the temporal derivative of the divergence of field B is null everywhere, it follows that, in every spatial position, it is constant through time:

const B

On the other hand, at the initial instant (before the sources were realized) it was B=0 everywhere and so also ;and since the equation must be valid, independently of the modifications to which the sources might be subjected, the second Maxwell equation follows.

0 B

1111

Role of momentum symmetry conservation on a magnetic induction field

In the following sections:the concept of momentum associated with a radiating field will be introduced, from a general point of view; this definition will be used to discuss the central idea of the present talk, i.e. the demonstration of a link between the invariance of the general momentum of a systemand the solenoidality of B.

Momentum of electromagnetic field

The system considered comprises a “continuum” of material particles, with electrical charges, moving in a region τ of space, of constant form through time, assuming that these interact only electromagnetically

1212

Lorentz force law

f E J B

ˆ

F f

Π T nS

d

dd dS

dt

Poynting vector

Π E H E B

Maxwell electromagnetic stress tensor

221 1 1

, [ , , , ]2 2ij i j i j ij

BT E E B B E i j x y z

The e.m. stress tensor has the dimensions of a pressure [in units of Pa = N/m2]. However, it represents the momentumtransmitted by the electromagnetic field through the surface element , and not (in general) the pressure exerted by the e.m. field on the surface . This happens only if the surface is physically realized with a totally absorbent surface material.

The electromagnetic stress tensor T acts analogously to the (mechanical) stress tensor in a material body, causing, by its flow through the surface of the system, the variation of total momentum of the latter(equal to that of the material plus that of the e.m. field).

ˆdSn

1313

Due to the propagation of electromagnetic waves, the e.m. fields do not tend towards zero, for a large r (r→∞), "at least as r-2 '' (as happens in static cases); however, taking into account the finite propagation velocity of the waves, it is always possible to choose a surface S∞ so large that the field has not yet reached it.

dd

dt

F Π

First cardinal equation of the dynamics of systems

d

dt

PF mat d

P v

d dd

dt dt

P

Π

Although the physical system is isolated (in the sense that it is not subject to influence from any other systems), the momentum of the material contained within it is not an invariant through time, in other words it is not conserved.

Electromagnetic interaction does not respect the principle of equality between action and reaction.

1414

Momentum of the electromagnetic field (Minkowski)

d

emP Π

Extended principle of momentum conservation

emP P const An e.m. field, striking a body, exercises a radiation pressure on the bodyin the same way as a beam of material particles.

Local form

emp p const d

d

Pp

2( )

emem

Pp D B

Π E H Π

d

d

v

emp Πv v

Momentum contained in a cylinder of unitary base and height equal to the velocity v of the wave.

Radiation pressure (impulse per unit of surface and per unit of time)

1515

Implication of symmetry for magnetic induction fields

Every particle of mass m, charge q, with a position rα(t) and a velocity (non-relativistic) vα(t)=drα/dt is subject to the following Newton-Lorentz equation:

2

2

( )( ( ), ) ( ) ( ( ), )

d tm q t t t t t

dt αα α α

rE r v B r

when E is the electrical field vector and B is the magnetic induction field. The equation is of evolution for the system under investigation (charged particle + field). The total momentum P(t) (Minkowski) is given by:

( )( ) ( , ) ( , )

d tt m t t d

dt

αrP D r B r

where D is the electrical displacement vector, i.e. D=εE. If the integral is extended to any closed region (indicated with τ∞) so large that τ∞ is an isolated system, then the momentum becomes a constant of motion; in equivalent terms, the magnetic induction B must be solenoidal, and so Maxwell’s second equation is valid, i.e.

0 BP 0d dt

1616

Demonstration

Expressions of the spatial density of electrical charge and current, with reference to a discrete set of point sources:

( , ) ( ( ))t q t

αr r r

( , ) ( ) ( ( ))t q t t

α αJ r v r r

( ) 1d

r

Dirac delta distribution

( , ) ( , ) ( ) ( ( )) ( , )

( ) ( ( )) ( , ) ( ) ( , )

t t d q t t t d

q t t t d q t t

α α

α α α α

J r B r v r r B r

v r r B r v B r

( , ) ( ) ( , )

( , ) ( , ) ( , )

1( , ) ( , ) ( , )

α α α

α

PE r v B r

H J r B r D r E

E r B r B E r E

dq t t t

dt

t t t d

q t t t d

1717

For the condition of evanescence of the electromagnetic field (E, B), both the volume integrals of the second member of the equation are reduced to:

( , ) ( ) ( , )( )t d t d

E r E E r E

( , ) ( ) ( , )( )t d t d

B r B B r B

1( , ) ( , )( ) ( , )( )

dq t t t d

dt

α

PE r B r B E r E

( , )( , )( ) ( , ) ( , ) ( , )

( , ) ( ( )) ( , ) ( ( )) ( , )

tt d t d t t d

E t q t d t q t d q t

α α α

rE r E E r E r r

r r r E r r r E r

1( , )( )

dt d

dt

P

B r B

Considering last equation, if the momentum P is conserved, then the stationary condition is applied: dP/dt=0. Therefore, the second member of the equation must also be null.

The condition that satisfies the integral equation achieves the predicted result.

Finally, this condition holds even in the following cases, to be excluded:

I) The magnetic induction field is null, i.e. B=0; the case I) is the trivial case.

The integral must be extended to any closed region (indicated with τ∞), so generally not symmetric, but large enough that τ∞ is an isolated system.Independently from the volume τ∞, the law of product annulment can be applied to the integrating function . For as long as the total momentum is conserved, if the magnetic induction field B(r,t) is generally different from zero, then this field must necessarily be solenoidal, i.e. .

( , )( )t B r B

0B

1818

II) The integral is extended to a region τ∞ symmetric with respect to space origin O and the field B(r,t) is an even [B(-r,t)=B(r,t)] or odd [B(-r,t)=-B(r,t)] function of position vector r, so that its divergence is generally a non vanishing function and respectively odd or even. Even though the case II) is mathematically true, indeed the integral can be null for an even or odd field defined in a symmetric volume τ∞; anyway, it is physically absurd, indeed the magnetic induction cannot be defined by a divergence different than zero, since the existence of magnetic monopole has not yet been proven [4-9]: in other terms, the magnetic induction field cannot subsist in some symmetries, including the radial one.

0 B

1919

Appendix

ˆi

i

A iA e

21( ) ( ) ( )

2A A A A

DIM

A

ˆ ˆ ˆ ˆ( ) ( ) ( )

ˆ ˆ( ) ( ) ( ) ( )

i i i i

i i

e A e A A A e e A

A e e A A A A A

i i i i ii i i i

i ii i

A A A A A

A A

2 21 1ˆ( ) ( ) ( ) ( ) ( ) ( )

2 2 ii

A A A iA A A A e A A A

2

2

1ˆ( ) ( ) ( ) ( )

2

1ˆ ˆ ˆ ( )

2

i

i

A A e A A A

n e A n A A

ii

divergence and gradient theorems

iS Si

d A d A d d

A dS A dS d

2

0 0

( ) lim ( )

1ˆ ˆ ˆlim lim lim ( )

2

i

A A A A

n e A n A A

condition of evanescent field

iS SS S S Si

d d

A dS A dS d

lim ( ) ( ) A A A Ad d

2020

CONCLUSIONS

The present talk discussed the physical problem of a system of non-relativistic massive charge particles moving in the presence of an electromagnetic (e.m.) field, which propagates within the entire space, embedded in a linear, isotropic and homogeneous material medium, where the sources are located. In order to explain the solenoidality of the magnetic induction field B, the authors have explored a new perspective on the divergence-free property of the B-field, based directly on the conservation of total momentum. The more popular didactical justifications for the solenoidality of magnetic induction B were suitably reviewed and compared with the newly proposed perspective, which, to the best of the authors’ knowledge, has never been previously discussed in literature and represents a novelty of potential didactic value to graduated, Ph-D students and eventually general physicists interested in symmetry in physics.

2121

ACKNOWLEDGMENTS

The authors, Drs. A. Settimi and S. Severini, are extremely grateful to Prof. Luciano Mistura and Dr. Omar El Gawhary for their stimulating suggestions and discussionson the solenoidality of magnetic induction fields and the conservation of total momentum.

2222

REFERENCES

[1] S. Severini, A. Settimi, "Solenoidality of a magnetic induction field and conservation of total momentum", accepted for publication on Nuovo Cimento C (30/09/2011); ibid., arXiv:1103.6253v2 [physics.class-ph] (2011).

[2] H. Minkowski, Nachr. Ges. Wiss. Gottingen, 53 (1908); ibid., Math. Ann., 68, 472 (1910). [3] M. Abraham, Rend. Circ. Matem. Palermo, 28, 1 (1909); ibid. 30, 5 (1910).

[4] R. N. C. Pfeifer et al., Phys. Rev. A 79 (2), 023813 [7 pp.] (2009).

[5] G. T. Hooft, Nuclear Physics B, 79 (2), 276-284 (1974). [6] P. B. Price et al., Phys. Rev. Lett. 35 (8), 487–490 (1975). [7] P. T. Leung, Eur. J. Phys., 16 (1), 43 (1995). [8] Jun S. Song, J. Undergrad. Sci. 3, 47-55 (1996). [9] Z. Fang, et al., Science, 302 (5642), 92-95 (2003).