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A Treatise on Electricity and Magnetism
by James Clerk Maxwell

Terrestrial Magnetism

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90959A Treatise on Electricity and Magnetism — Terrestrial MagnetismJames Clerk Maxwell

CHAPTER VIII.

ON TERRESTRIAL MAGNETISM.

465.] Our knowledge of Terrestrial Magnetism in derived from the study of the distribution of magnetic force on the earth's surface at any one time, and of the changes in that distribution at different times.

The magnetic force at any one place and time is known when its three coordinates are known. These coordinates may be given in the form of the declination or azimuth of the force, the dip or inclination to the horizon, and the total intensity.

The most convenient method, however, for investigating the general distribution of magnetic force on the earth's surface is to consider the magnitudes of the three components of the force,

$X=H\cos \delta$ , directed due north,	}	(1)
$Y=H\sin \delta$ , directed due west,
$Z=H\tan \theta$ , directed vertically downwards,

where $H$ denotes the horizontal force, $\delta$ the declination, and $\theta$ the dip.

If $V$ is the magnetic potential of the earth's surface, and if we consider the earth a sphere of radius $a$ , then

X={\frac {1}{a}}{\frac {dV}{dl}}

, ⁠

Y={\frac {1}{a\cos l}}{\frac {dV}{d\lambda }}

, ⁠

Z={\frac {dV}{dr}}

(2)

where $l$ is the latitude, and $\lambda$ the longitude, and $r$ the distance from the centre of the earth.

A knowledge of $V$ over the surface of the earth may be obtained from the observations of horizontal force alone as follows.

Let $V_{0}$ be the value of $V$ at the true north pole, then, taking the line-integral along any meridian we find,

V=a\int _{\frac {\pi }{2}}^{l}Xdl+V_{0}

,

(3)

for the value of the potential on that meridian at latitude $l$ .

Thus the potential may be found for any point on the earth's surface provided we know the value of $X$ , the northerly component at every point, and $V_{0}$ , the value of $V$ at the pole.

Since the forces depend not on the absolute value of $V$ but on its derivatives, it is not necessary to fix any particular value for $V_{0}$ .

The value of $V$ at any point may be ascertained if we know the value of $X$ along any given meridian, and also that of $Y$ over the whole surface.

Let

V_{l}=a\int _{\frac {\pi }{2}}^{l}Xdl+V_{0}

(4)

where the integration is performed along the given meridian from the pole to the parallel $l$ , then

V=V_{l}+a\int _{\lambda _{0}}^{\lambda }Y\cos ld\lambda

,

(5)

where the integration is performed along the parallel $l$ from the given meridian to the required point.

These methods imply that a complete magnetic survey of the earth's surface has been made, so that the values of $X$ or of $Y$ or of both are known for every point of the earth's surface at a given epoch. What we actually know are the magnetic components at a certain number of stations. In the civilized parts of the earth these stations are comparatively numerous; in other places there are large tracts of the earth's surface about which we have no data.

Magnetic Survey.

466.] Let us suppose that in a country of moderate size, whose greatest dimensions are a few hundred miles, observations of the declination and the horizontal force have been taken at a considerable number of stations distributed fairly over the country.

Within this district we may suppose the value of $V$ to be represented with sufficient accuracy by the formula

V=V_{0}+a(A_{1}l+A_{2}\lambda +{\frac {1}{2}}B_{1}l^{2}+B_{2}l\lambda +{\frac {1}{2}}B_{3}\lambda ^{2}+\&c.)

,

(6)

whence

X=A_{1}+B_{1}l+B_{2}\lambda

,

(7)

Y\cos l=A_{2}+B_{2}l+B_{3}\lambda

.

(8)

Let there be $n$ stations whose latitudes are $l_{1}$ , $l_{2}$ , ...&c. and longitudes $\lambda _{1}$ , $\lambda _{2}$ , &c., and let $X$ and $Y$ be found for each station.

Let

l_{0}={\frac {1}{n}}\sum (l)

, ⁠ and ⁠

\lambda _{0}={\frac {1}{n}}\sum (\lambda )

,

(9)

l_{0}

and

\lambda _{0}

may be called the latitude and longitude of the central station. Let

X_{0}={\frac {1}{n}}\sum (X)

, ⁠ and ⁠

Y_{o}\cos l_{o}={\frac {1}{n}}\sum (Y\cos l)

,

(10)

then $X_{0}$ and $Y_{0}$ are the values of $X$ and $Y$ at the imaginary central station, then

X=X_{0}+B_{1}(l-l_{0})+B_{2}(\lambda -\lambda _{0})

,

(11)

Y\cos l=Y_{0}\cos l_{0}+B_{2}(l-l_{0})+B_{3}(\lambda -\lambda _{0})

.

(12)

We have $n$ equations of the form of (11) and $n$ of the form (12). If we denote the probable error in the determination of $X$ by $\xi$ , and that of $Y\cos l$ by $\eta$ , then we may calculate $\xi$ and $\eta$ on the supposition that they arise from errors of observation of $H$ and $\delta$ .

Let the probable error of $H$ be $h$ , and that of $\delta$ , $d$ , then since

dX=\cos \delta \centerdot dH-H\sin \delta \centerdot d\delta

,

\xi ^{2}=h^{2}\cos ^{2}\delta +d^{2}H^{2}\sin ^{2}\delta

.

Similarly

\eta ^{2}=h^{2}\sin ^{2}\delta +d^{2}H^{2}\cos ^{2}\delta

.

If the variations of $X$ and $Y$ from their values as given by equations of the form (11) and (12) considerably exceed the probable errors of observation, we may conclude that they are due to local attractions, and then we have no reason to give the ratio of $\xi$ to $\eta$ any other value than unity.

According to the method of least squares we multiply the equations of the form (11) by $\eta$ , and those of the form (12) by $\xi$ to make their probable error the same. We then multiply each equation by the coefficient of one of the unknown quantities $B_{1}$ , $B_{2}$ , or $B_{3}$ and add the results, thus obtaining three equations from which to find $B_{1}$ , $B_{2}$ , and $B_{3}$ .

P_{1}=B_{1}b_{1}+B_{2}b_{2}

,

(\eta ^{2}P_{2}+\xi ^{2}Q_{1})=B_{1}\eta ^{2}b_{2}+B_{2}(\xi ^{2}b_{1}+\eta ^{2}b_{3})+B_{3}\xi ^{2}b_{2}

,

Q_{2}=B_{2}b_{2}+B_{3}b_{3}

;

in which we write for conciseness,

b_{1}=\sum (l^{2})-nl_{0}^{2}

, ⁠

b_{2}=\sum (l\lambda )-nl_{0}\lambda _{0}

, ⁠

b_{3}=\sum (\lambda ^{2})-n\lambda _{0}^{2}

,

P_{1}=\sum (lX)-nl_{0}X_{0}

, ⁠

Q_{1}=\sum (lY\cos l)-nl_{0}Y_{0}\cos l_{0}

,

P_{2}=\sum (\lambda X)-n\lambda _{0}X_{0}

, ⁠

Q_{2}=\sum (\lambda Y\cos l)-n\lambda _{0}Y_{0}\cos l_{0}

.

By calculating $B_{1}$ , $B_{2}$ , and $B_{3}$ , and substituting in equations (11) and (12), we can obtain the values of $X$ and $Y$ at any point within the limits of the survey free from the local disturbances which are found to exist where the rock near the station is magnetic, as most igneous rocks are.

Surveys of this kind can be made only in countries where magnetic instruments can be carried about and set up in a great many stations. For other parts of the world we must be content to find the distribution of the magnetic elements by interpolation between their values at a few stations at great distances from each other.

467.] Let us now suppose that by processes of this kind, or by the equivalent graphical process of constructing charts of the lines of equal values of the magnetic elements, the values of $X$ and $Y$ , and thence of the potential $V$ , are known over the whole surface of the globe. The next step is to expand $V$ in the form of a series of spherical surface harmonics.

If the earth were magnetized uniformly and in the same direction throughout its interior, $V$ would be an harmonic of the first degree, the magnetic meridians would be great circles passing through two magnetic poles diametrically opposite, the magnetic equator would be a great circle, the horizontal force would be equal at all points of the magnetic equator, and if $H_{0}$ is this constant value, the value at any other point would be $H=H_{0}\cos l^{\prime }$ , where $l^{\prime }$ is the magnetic latitude. The vertical force at any point would be $Z=2H_{0}\sin l^{\prime }$ , and if $\theta$ is the dip, $\tan \theta =2\tan l^{\prime }$ .

In the case of the earth, the magnetic equator is defined to be the line of no dip. It is not a great circle of the sphere.

The magnetic poles are defined to be the points where there is no horizontal force or where the dip is 90°. There are two such points, one in the northern and one in the southern regions, but they are not diametrically opposite, and the line joining them is not parallel to the magnetic axis of the earth.

468.] The magnetic poles are the points where the value of $V$ on the surface of the earth is a maximum or minimum, or is stationary.

At any point where the potential is a minimum the north end of the dip-needle points vertically downwards, and if a compass-needle be placed anywhere near such a point, the north end will point towards that point.

At points where the potential is a maximum the south end of the dip-needle points downwards, and the south end of the compass-needle points towards the point.

If there are $p$ minima of $V$ on the earth s surface there must be $p-1$ other points, where the north end of the dip-needle points downwards, but where the compass-needle, when carried in a circle round the point, instead of revolving so that its north end points constantly to the centre, revolves in the opposite direction, so as to turn sometimes its north end and sometimes its south end towards the point.

If we call the points where the potential is a minimum true north poles, then these other points may be called false north poles, because the compass-needle is not true to them. If there are $p$ true north poles, there must be $p-1$ false north poles, and in like manner, if there are $q$ true south poles, there must be $q-1$ false south poles. The number of poles of the same name must be odd, so that the opinion at one time prevalent, that there are two north poles and two south poles, is erroneous. According to Gauss there is in fact only one true north pole and one true south pole on the earth s surface, and therefore there are no false poles. The line joining these poles is not a diameter of the earth, and it is not parallel to the earth s magnetic axis.

469.] Most of the early investigators into the nature of the earth s magnetism endeavoured to express it as the result of the action of one or more bar magnets, the position of the poles of which were to be determined. Gauss was the first to express the distribution of the earth s magnetism in a perfectly general way by expanding its potential in a series of solid harmonics, the coefficients of which he determined for the first four degrees. These coefficients are 24 in number, 3 for the first degree, 5 for the second, 7 for the third, and 9 for the fourth. All these terms are found necessary in order to give a tolerably accurate representation of the actual state of the earth's magnetism.

To find what Part of the Observed Magnetic Force is due to External and what to Internal Causes.

470.] Let us now suppose that we have obtained an expansion of the magnetic potential of the earth in spherical harmonics, consistent with the actual direction and magnitude of the horizontal force at every point on the earth' surface, then Gauss has shewn how to determine, from the observed vertical force, whether the magnetic forces are due to causes, such as magnetization or electric currents, within the earth s surface, or whether any part is directly due to causes exterior to the earth's surface.

Let

V

be the actual potential expanded in a double series of spherical harmonics,

V=A_{1}{\frac {r}{a}}+\mathrm {\&c.} +A_{i}\left({\frac {r}{a}}\right)^{i},{}+B_{1}\left({\frac {r}{a}}\right)^{-2}+\mathrm {\&c.} +B_{i}\left({\frac {r}{a}}\right)^{-(i+1)}

.

The first series represents the part of the potential due to causes exterior to the earth, and the second series represents the part due to causes within the earth.

The observations of horizontal force give us the sum of these series when $r=a$ , the radius of the earth. The term of the order $i$ is

V_{i}=A_{i}+B_{i}

.

The observations of vertical force give us

Z={\frac {dV}{dr}}

,

and the term of the order $i$ in $aZ$ is

aZ_{i}=iA_{i}-(i+1)B_{i}

.

Hence the part due to external causes is

A_{i}={\frac {(i+1)V_{i}+aZ_{i}}{2i+1}}

,

and the part due to causes within the earth is

B_{i}={\frac {iV_{i}-aZ_{i}}{2i+1}}

.

The expansion of $V$ has hitherto been calculated only for the mean value of $V$ at or near certain epochs. No appreciable part of this mean value appears to be due to causes external to the earth.

471.] We do not yet know enough of the form of the expansion of the solar and lunar parts of the variations of $V$ to determine by this method whether any part of these variations arises from magnetic force acting from without. It is certain, however, as the calculations of MM. Stoney and Chambers have shewn, that the principal part of these variations cannot arise from any direct magnetic action of the sun or moon, supposing these bodies to be magnetic^[1].

472.] The principal changes in the magnetic force to which attention has been directed are as follows.

I. The more Regular Variations.

(1) The Solar variations, depending on the hour of the day and the time of the year.

(2) The Lunar variations, depending on the moon's hour angle and on her other elements of position.

(3) These variations do not repeat themselves in different years, but seem to be subject to a variation of longer period of about eleven years.

(4) Besides this, there is a secular alteration in the state of the earth' magnetism, which has been going on ever since magnetic observations have been made, and is producing changes of the magnetic elements of far greater magnitude than any of the variations of small period.

II. The Disturbances.

473.] Besides the more regular changes, the magnetic elements are subject to sudden disturbances of greater or less amount. It is found that these disturbances are more powerful and frequent at one time than at another, and that at times of great disturbance the laws of the regular variations are masked, though they are very distinct at times of small disturbance. Hence great attention has been paid to these disturbances, and it has been found that disturbances of a particular kind are more likely to occur at certain times of the day, and at certain seasons and intervals of time, though each individual disturbance appears quite irregular. Besides these more ordinary disturbances, there are occasionally times of excessive disturbance, in which the magnetism is strongly disturbed for a day or two. These are called Magnetic Storms. Individual disturbances have been sometimes observed at the same instant in stations widely distant.

Mr. Airy has found that a large proportion of the disturbances at Greenwich correspond with the electric currents collected by electrodes placed in the earth in the neighbourhood, and are such as would be directly produced in the magnet if the earth-current, retaining its actual direction, were conducted through a wire placed underneath the magnet.

It has been found that there is an epoch of maximum disturbance every eleven years, and that this appears to coincide with the epoch of maximum number of spots in the sun.

474.] The field of investigation into which we are introduced by the study of terrestrial magnetism is as profound as it is extensive.

We know that the sun and moon act on the earth's magnetism. It has been proved that this action cannot be explained by supposing these bodies magnets. The action is therefore indirect. In the case of the sun part of it may be thermal action, but in the case of the moon we cannot attribute it to this cause. Is it possible that the attraction of these bodies, by causing strains in the interior of the earth, produces (Art. 447) changes in the magnetism already existing in the earth, and so by a kind of tidal action causes the semidiurnal variations?

But the amount of all these changes is very small compared with the great secular changes of the earth's magnetism.

What cause, whether exterior to the earth or in its inner depths, produces such enormous changes in the earth's magnetism, that its magnetic poles move slowly from one part of the globe to another? When we consider that the intensity of the magnetization of the great globe of the earth is quite comparable with that which we produce with much difficulty in our steel magnets, these immense changes in so large a body force us to conclude that we are not yet acquainted with one of the most powerful agents in nature, the scene of whose activity lies in those inner depths of the earth, to the knowledge of which we have so few means of access.

↑ Professor Hornstein of Prague has discovered a periodic change in the magnetic elements, the period of which is 26.33 days, almost exactly equal to that of the synodic revolution of the sun, as deduced from the observation of sun-spots near his equator. This method of discovering the time of rotation of the unseen solid body of the sun by its effects on the magnetic needle is the first instalment of the repayment by Magnetism of its debt to Astronomy. Akad., Wien, June 15, 1871. See Proc. R. S., Nov. 16, 1871.

[1] Professor Hornstein of Prague has discovered a periodic change in the magnetic elements, the period of which is 26.33 days, almost exactly equal to that of the synodic revolution of the sun, as deduced from the observation of sun-spots near his equator. This method of discovering the time of rotation of the unseen solid body of the sun by its effects on the magnetic needle is the first instalment of the repayment by Magnetism of its debt to Astronomy. Akad., Wien, June 15, 1871. See Proc. R. S., Nov. 16, 1871.

[1]

A Treatise on Electricity and Magnetism/Part III/Chapter VIII

CHAPTER VIII.

ON TERRESTRIAL MAGNETISM.