Electron Inertia and Terrestrial MagnetismAuthor(s): Charles DarwinSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 222, No. 1151 (Mar. 23, 1954), pp. 471-476Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/99198 .

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Electron inertia and terrestrial magnetism

BY SIR CHARLES DARWIN, F.R.S.

(Received 25 September 1953)

The problem is solved of the influence that electron inertia might have on the earth's mag- netism. Owing to the retardation of the earth's rotation electric currents will arise from it. Though the direction is correct, it is shown that even for a sphere of the great size of the earth the magnetic effect is entirely negligible. It is also shown that changes of the accelera- tion rate would have to be extremely slow in order that they should produce any effect at all even on this extremely weak field.

This note gives an account of a failure in the sense that it shows that the effects of electron inertia cannot be invoked to help in the explanation of the earth's magnetic field. It appeared worth while to put it on record so as to save others from wasting time over similar work.

1. In all ordinary experiments electron inertia-that is to say, the lag of the electrons in an accelerating conductor on account of their inertia-is an exceedingly small effect (see, for example, Barnett (1935) for an account of the subject). It was always likely that here again it would be very small, but it seemed worth examining because there are many effects which are negligible on the ordinary scale, but which become important for a body of the size of the earth or a star. Qualitatively the principle has attractive features in relation to terrestrial magnetism. Thus there have been many attempts to relate the earth's magnetism directly to its rotation, but apart from difficulties as to magnitude, they threaten to run into one much more practical difficulty. A study of the residual magnetism in various rocks rather strongly suggests that there have been periods when the earth's field was in the direction opposite to what it is at present. It is impossible to believe that the earth's actual direction of rotation was reversed, and it seems very unlikely indeed that the whole crust should have moved over the core through something like 1 80?, so that it becomes hard to believe that there can be any direct relationship between the earth's field and its angular velocity. This is not so for electron inertia, which would show the effect not of angular velocity, but of angular acceleration. The earth's moment of inertia is not absolutely constant; for example, the onset of an ice age would diminish it through concentrating the ice at the poles, and this would have the effect of shortening the day, so that the normal retardation due to tidal friction would be changed temporarily into an acceleration. It was this considera- tion that led to the present investigation, for if it had been successful it might have enabled us to read the past history of the earth's angular velocity by means of measurements of the residual magnetism in the rocks.

2. It was hardly to be expected that electron inertia would provide the whole earth's field, but it seemed that it might provide an influence which, perhaps strongly amplified by Bullard's (I 949) dynamo effects, would determine the direction of the main field.

[ 471

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472 Sir Charles Darwin

The first qualitative consideration was encouraging, for the field is in the right direction. To see this imagine a telephone line laid round the equator. It is moving with the earth from west to east, but under the present secular retardation it is accelerating westwards. The electrons will lag behind, and so flow eastwards, and owing to their negative charge this means a westerly current in the wire. Such a current gives a northerly magnetic force outside the earth, which is what is observed.

On the other hand, the argument from physical dimensions is entirely dis- couraging. When a sphere is rotating at a variable rate the magnetic field H should depend on the following quantities: (1) the angular acceleration (b, to which it should be proportional; (2) the radius a; (3) the conductivity o, assumed to be uni- form throughout the sphere; (4) the characteristic constant of electron inertia, m/e, the ratio of mass to charge of the electron. In electromagnetic units the dimensions of these are, respectively,

e=-T-2 a=L, =L-2T, mJe=MiL-1, H=ML-iT-1,

from which it follows that

H= oa2. (2.1) e

Here e/mn is 1-8 x 107 and for the earth a 6-4 x 108, while Bullard estimates 22 x 10-22 and o- = 3 x 10-6 for the earth's core; it will make an over-estimate

to apply this for the whole earth. The result is

H _ 1 5 x 10-17 gauss.

Even if, as is very possible, the angular acceleration may at times have been many times greater than it is now, it is not credible that the dynamo amiplification could multiply it more than say a thousand times, and so the dimensional calculation in effect shows that electron inertia can have no influence on terrestrial magnetism.

3. In spite of this negative result it seemed worth giving a more detailed calcula- tion of the effect, and even more discouraging consequences emerge from the work!

The theory of electric conductivity has been much discussed with the use of the quantum theory and it proves to be very elaborate. However, these complications do not affect the question of electron inertia, as has been shown by the present writer (I936). For this it suffices to use classical theory. An electron (charge - e) is in a conductor that is moving with variable velocity v and has co-ordinate x relative to the medium. Then in electromagnetic units its motion is given by

m(k+?)- = (-e) {E [+v, B]j}-Kk. (3.1)

Here E and B are the electric force and magnetic induction at the place x, and the last term represents the ohmic resistance. This equation is to be averaged for all the electrons. Now if N is the electron density and J the current density,

J = N(-e).

For a stationary conductor J - oE, and it follows that

K = Ne2/0. (3.2)

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Electron inerti'aa and terrestrial magnetism 473

In averaging the other terms of (3.1) x will vanish, and the term - eL[, B] will represent the Hall effect, of which the full theory is troublesome, but the term will be omitted here as of no present interest. So (3.1) becomes

mr-- e{E + [v, B]} + Je/o-,

or J o-(E + [v, B]?+-v;. (3*3)

It is the last term in this that introduces the effect of electron inertia. This equation is to be combined with the usual electromagnetic equations

divB = O, (3'4) B =-,H, (3.5)

47TJ curl H, (3.6)

t curlE. (3 7)

The dielectric displacement terms have been omitted as of no interest in the present problem.

4. These equations are to be applied to a sphere. The boundary conditions at the surface are as usual that B norm, and Htgtl should be continuous, and also Etgti is continuous. Enorm is not continuous, because a surface charge is induced. This charge must be of such magnitude that the external field can be derived from a potential. Thus after the rest of the problem has been solved the surface charge and external electric field can be determined so as to satisfy the continuity of Etgtl, but the result has no direct interest and so it is unnecessary to consider it further.

It is easy to solve the problem with any permeability for the sphere, but the results become slightly complicated, and for the present purpose it will suffice to take ,u-1, which makes them a good deal simpler, because then the boundary conditions are simply that all components of H should be continuous.

Arbitrary electric currents in a sphere are best described in polar co-ordinates and spherical harmonics. The formulae are often written in terms of half-order Bessel functions, and this notation is to say the least clumsy, because these func- tions, being in fact sines and cosines, are far simpler than Bessel functions. For such a simple problem as the present one it seemed easier to use rectangular co-ordinates.

5. Taking axes fixed in space the motion of the sphere is given by

v = -y-W, ox, (O,-}(5.01 V- = -6Y-Ct2X' C)X - 0)2y, 0.

The terms in )2 are actually much greater than those in d but, as will appear, these centrifugal terms only affect the electric field and not the magnetic.

Inside the sphere assume the solution

HxI=3xzf(r,t), H, = 3yzf, Hz = (3Z2-r2)f+g(r,t), (5-2)

where f and g are functions of time and radius. To match the first two the external field must be

Hz f3xz 3yz a 3z2-r Hx a5f(a, t), Hy~= a5af(a, t), 14 = r5 a5f(a, t), (5*3)

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474 Sir Charles Darwin

and then, to satisfy the third equation,

g(a, t) = 0. (5.4)

The total magnetic moment of the sphere can, of course, be found from the solution, but physically a more convenient quantity is the northward magnetic field at the equator. This will be

Heq. -a2f(a, t). (5 5)

Equation (5.2) is now to be substituted in the equations of ? 3. Then (3 4) gives lOf + 2rf ' + g'/r = 0, which may be written as

a-r (r5f) =--9r3g'. (56)

3y = 3x 4nJ70 From (3.6) 47TJ=J, = . (57)

From (3.3) the first equation gives

x = 1-_g 2 Y x{(3z2-r2) f+ q} + m ()y + ot2x). 47To 2 r ge

From this and the other two, (3.7) gives, after some reduction,

aHlx 1 3xz a 3' At 4ioi2rarrr

aH,_ 1 3yzag' At 4rrc 2r ar r'

aH' H 1 3 X2+y2 ag m .- ,> = _ _2_ + _-_ - 2 t).

at 47T4r 2 r r ar r e

It will be noticed that the terms in (o and in o2 have disappeared. The first two equations are satisfied by

I I a 8 ' (5.8)

and the third then requires , 2 2m

or (rg) = 47o- t (rg) - 2r -mA7o (5.9) ar2 at

which is the equation yielding the whole solution in conjunction with (5.6) and the boundary condition (5.4).

6. There is the usual type of solution composed of a particular integral and a complementary function. This last is, of course, well known. It is

rg A sin kr et2/4 c, (6.1)

and to satisfy (5.4) k = n7r/a.

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Electron inertia and terrestrial magnetism 475

Then (5.6) gives I Isinkr 3 coskr 3sin kr2 (6.2) f r3+ kr4 k 2r5'

and hence Heq 3A(-21) e-tn27i4raa (6.3)

A series of terms of this type will represent any decaying field, but it must be noted that the distribution of magnetization in the sphere is not fully described by giving IT~ Hleq.-

For the particular integral, first take the case of constant d). Then g is independent of the time, and is

g = (a2_r2) --m P4To (6.4) 3a e so as to satisfy (5.9) and (5.4). Then

f= =.7w47Tro (6.5)

and Heq. =-47arm . (6.6)

This confirms the dimensional calculation of ? 2, and since 6 is negative it also confirms the field's direction.

7. For the case where b is not constant, take it first as harmonically varying. Let

= o +0/cos vt. (7.1)

Then g is the real part of a quantity proportional to elPt in the equation

d2 2rfi4r d (rg) = iv 47To-rg -2r -,8iv 47To-. (7 2) 2~~~~~~~~~~~~~~~~72 dr2(Y viq el

If p2 = -iv4nro, (7.3)

2m sin pr a 1 this gives g m-i8pr a 1

ec r sin pa-

m a sinpr 3 cospr 3 sinprl e sin pa L r3 pr4 p2r5 j'

from which H = --- i1+?3 c7a(pa)2}4 (7)

For acceleration varying with time in an arbitrary manner, a formal solution can be set down from (7.4) by means of a Fourier integral.

The character of the field will evidently depend mainly on the magnitude of pa. If this is small the cotangent can be expanded, and the selection of the real part is easily seen to lead back to (6.6). If pa is large the expression reduces to

Heq. -- (-c)O), (7 5)

and the interesting point is that this is independent of the size of the sphere. To see the meaning of this take as time scale

T-4Toa2. (7.6)

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476 Sir Charles Darwin

Then (pa)2= _-ivT, and from (6 3) the rate of decay of a field is proportional to e-72tIT. If then w is changing rapidly when measured on the time scale T, the effects will have no time to diffuse through the sphere before the acceleration has been reversed, and so the size of the sphere will be irrelevant.

For the case of the earth, with the constants adopted above, T is 150 000 years. It has already been seen that the field under constant acceleration is much too small to be detected, but now it can also be seen that even if it could be observed, only changes of acceleration so slow as to be measured by geological epochs would give any effect. The self-induction of the earth would suppress all changes at a rate more rapid than this.

A star can hardly be treated as a rigid body like the earth, but, overlooking this point, it is even less possible to attribute to electron inertia any effects in it. Thus there may well be very considerable angular accelerations in a variable star if its radius is expanding and contracting, but for a star T will be at least of the order of a million years, and so the pa of the star will be large number, and the size of the star will be without effect.

The only practical way of studying electron inertia is by means of the great accelerations of a small body that can be achieved in a laboratory, and there is no profit in thinking about the effects of it in a large body such as the earth.

REFERENCES

Barnett, S. J. 1935 Rev. Mod. Phys. 7, 159. Bullard, E. C. 1949 Proc. Roy. Soc. A, 197, 433. Darwin, C. I936 Proc. Roy. Soc. A, 154, 61.

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