LETTERE AL NUOVO CIMENTO VOL. 10, N. 7 15 Giugno 1974

On the Origin of the Magnetic Field.

D. R. ENGLISH

Rhodes U n i v e r s i t y - G r a h a m s t o w n

(ricevuto 1'11 Gennaio 1974)

BEI~GSTR(iM (1) has demonstrated that the magnetic force in a moving system is a Coriolis force resulting from the Thomas rotation caused by the electric force. TOLMAN (3), many years ago, considered the apparent force between two charged par- ticles in vacuo as measured in an inertial frame S in which both particles appear to move at general velocities.

In Tholman's paper it is asserted that in the frame S', in which the particle of charge q'l appears to be at rest and the particle of charge q~ has an instantaneous posi- tion vector r ' and velocity u ' , the force F ' on q~ due to q~ is electrostatic. The trans- forms of special relativity are applied to find the force F on q2, due to ql, in the frame S in which ql appears to have velocity v.

These two papers, separated by a span of sixty years, represent the closest approach to date to an explanation of the magnetic field.

In the earlier paper (2), allowance was not made for the effect of retardation, i.e. for the fact that, in S, the force field due to ql (moving) travels at a finite speed c towards q2. If allowance is made for this (~ retardation effect ,>, the result includes terms of order v/c, (v/c) 2 . . . . independent of u. To avoid this complication it is only neces- sary to regard ql as the charge instantaneously in a small volume ~V~, fixed in S, containing charge density ~ (q~ = Q~ 3V~) and with the charge flowing through it at a velocity v, both ~1 and v remaining sensibly constant during a short period of order r/c and constituting a current density J~. Then the charges in 8V~ are continually being replaced by identical charges following each other in succession. The result of this <( succession effect ~> is that retardation complications may be ignored.

With the ((succession effect ~> in force, for both particles for the sake of symmetry, Tolman's (3) result can be adjusted to give, in MKS units,

(1) F : 7 ~ l ~ V 1 ~ 2 ~ V 2 r 7 J 2 • 2 1 5 + 4 ~ ' o r '3 4~e~ ~'o r'~

where ~ is the permittivity, F - s = 1--v2/e 2 and both charge and permitt ivl ty have !

been regarded as conserved properties, i.e. ~' 8 V ' = ~ 8 V and t 0 = e0.

(1) fl-. BERGSTROM: NUOVO Cimento, 14 B, 235 (1973). (3) R. C. TOLMAN: Phil. 3lag., 25, 150 (1913).

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O N T t I E O R I G I N O F T t t E M A G N E T I C FIELD 2~7

The matter appears to have rested at this stage during the intervening time and equation (1) is quoted in recent (a) monographs.

However there is a difficulty. If it is asserted that the pcrmit t ivi ty of vacuum is invariant and that terms of order {v/c) 4 and higher may be ignored, the second term on the r ight-hand side of equation (1), with /~0e0c2= 1, reduces to the well-known Amp~re-Biot-Savart expression for the magnetic force between the current elements (/*o/4~r')J2x (J, • SV 2, of order (v/e) 2 as compared with the first term. But if terms of this order are retained, the first term does not reduce to the electrostatic force between the two elements.

In the recent paper (4) the system is studied for the case v << c. Magnetic force is shown to be a Coriolis effcct due to Thomas rotation for this case. Since Maxwell's equations arc invariant for all transforms between inertial frames, including those where the velocity difference is large, the derivation is stated to be correct for all v < c.

The disadvantage here lies in the fact that, in order to prove the invariance of Max- well's equations, the existence of the magnetic field must be accepted as an independent phenomenon from the start.

It is now postulated that the permit t ivi ty of vacuum is not invariant. With this idea accepted, the form of its transform may be analysed as follows.

Let the charge q~, as seen in S', instantaneously occupy a small volume 8V~. The /

charge q~ is regarded as a point charge at rest at the origin O' of S'. The permit t ivi ty of the vacuum, at and near q~, is a factor included in the expression for the electro- static force F ' on q2 due to q, and it depends only on the mcan direction 9' of ~V~ from O' and on the solid angle ~o)' subtended at O' by 8V~ : the other factors (magnitude of both charges, scalar distance) appear separately in the expression for F ' . Thus

(2) ~o ~ ~ ' ( ~ ' ) �9

The transform for a small solid angle depends on its direction. In fact, if 8~o is the corresponding small solid angle seen in S in direction r then

(3) yr 3 ~w = r 'a ~,~'.

And so, from equation (2),

(4) eo r'a ~ ~eo ra"

Substituting this identi ty into equation (1) and claiming charge conservation as before, produces the result

(5) F - ~I ~Vl q2~V2r + Ft~ ('ll Xr) ~Vl ~V~ 4~e o r a 4~r a

Thus F, as measured in S, is seen to be the vector sum of the electrostatic force and of the magnetic force, both as measured in S.

Concluding remarks:

a) Maxwell's equations apply to a system where the charge density ~ is continuous at the point of measurement. Under these conditions it can be shown that e 0 is in fact invariant to Lorentz transform.

0) W. G. V. ROSSER: Theory o/ Relativity (1964), p. 311.

2 8 8 D. 1~. ~NGLISH

b) The elementary analysis, whereby the magnetic force between parallel-line currents is shown to be due to electrostatic force as affected by special relativity, can still be curried out with so subject to transform, with the same result.

c) It can be shown that a similar transform applies to e, the permit t ivi ty of a medium.

d) According to a paper by BERGSTR(i•, due to appear shortly (4), (~ fundamental particles ~ such as electrons are shown to be clouds of charge density. When these clouds rotate, (~ succession effect ,~ is applicable and the above analysis together with that in ref. (1) explains the existence of the magnetic moment of the electron and of other (~ fundamental particles ~.

(a) A. BEROSTR(JM: Phys. Rev. D, to be p u b l i s h e d .