A Magnetometer for the Determination of the Vertical Component of the Earth'sMagnetic FieldS. L. Ting and S. T. Lin Citation: Review of Scientific Instruments 15, 171 (1944); doi: 10.1063/1.1770262 View online: http://dx.doi.org/10.1063/1.1770262 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/15/7?ver=pdfcov Published by the AIP Publishing

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DETERMINATION OF THE EARTH'S MAGNETIC FIELD 171

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VOLUME 15. NUMBER 7 JULY. 1944

A Magnetometer for the Determination of the Vertical Component of the Earth's Magnetic Field

S. L. TING AND S. T. LIN National Research Institute of Physics, Academia Sinica, China

(Received February 21, 1944)

The magnetometer consists mainly of two parts: a Helm­holtz-Gaugain coil fixed on a horizontal divided circle for producing a uniform vertical magnetic field to balance the vertical component of the earth's field, and a magnet­balance system for observing the effect of the fields. The vertical component of the earth's field is determined in terms of the coil constants and the strength of the current passing through the coil. The error due to the horizontal component of the earth field and that due to the deviation of the center of gravity of the balance system from th~' rotating axis are eliminated, respectively, by taking the mean value of two observations with the axis of the balance set first in one direction and then turned into the opposite direction by rotating the whole instrument

1. INTRODUCTION

WITH the increased accuracy in the measure­ment of electric current a new method of

determining the earth's magnetic field has been

through 1800 about a vertical axis, and by reversing the polarity of the balance. The special features of the instru­ment are as follows: (1) a pair of small solenoids is fixed inside the balance chamber and the balance can have its polarity reversed in its arrested position without being taken out; and (2) by providing another pair of horizontal coils on the chamber, the magnetometer serves at the same time as a galvanometer with the magnet balance as its moving magnet (the galvanometer is needed for the determination of the current and would otherwise be provided separately). By preliminary tests the instrument is found to give consistent results. Its sensitivity is about ! min. per 'Y.

developed. The underlying principle is to produce a uniform magnetic field in a small region by passing an electric current through a Helmholtz­Gaugain coil so as to balance the effect of the

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172 S. L. TING AND S. T. LIN

earth's magnetic field on some moving magnetic system. The intensity of the earth's magnetic field may thus be determined in terms of the known strength of the current and the coil constants. This principle is theoretically applicable to the determination of both the horizontal and the vertical components of the earth's magnetic field. In practice magnetometers for the determination of the horizontal . component of the earth's magnetic field, which are essentially the same in com;truction as sine galvanometers, have been used both as primary standards in observatories and as secondary instruments in field work. Magnetometers for the determination of the vertical component designed on this same under­lying principle have been constructed and tried, but the attempt has not attained the same suc­cess. The reason is easy to understand. In order to know whether the two horizontal fields balance each other a suspended moving system can be used as a detector, which is not affected by the vertical component of the field. The only trouble, the torsion of the suspension, can be corrected or rendered negligible by immersing in gasoline the whole moving system with a float attached, arid using a very thin quartz fiber for suspension. On the other hand, for detecting the balance of two vertical fields a small horizontal magnetic balance is usually used. In this case the horizontal com­ponent of the field and the weight of the balance will produce effects on the moving system unless the center of gravity of the balance lies exactly below the knife edge, and either the axis of rotation or the magnetic axis of the balance be exactly parallel to the horizontal component of the earth's magnetic field. Both these conditions are difficult to fulfill. The usual method used to overcome these difficulties is to reduce the effects as much as possible and then to eliminate what­ever remains. The effect produced by the hori­zontal component is eliminated by taking the mean value of two observations, the one with the magnetic axis of the balance oriented in one direction and the other with the axis rotated about a vertical axis through 180°. The effect of

FIG.!.

the weight of the balance is eliminated also by taking the mean value of two observations, the one with the balance magnetized in one direction and the other with it magnetized in the opposite direction. The two magnetic moments must be exactly equal and opposite to each other and, meanwhile, the center of gravity of the balance must not be altered. This is a practical difficulty, and it is owing to this that, instead of the balance, other moving systems have been used, such as a small light coil suspended in a vertical plane by horizontal double-wire leads, which, when an alternating current passes through it, will oscil­late if there is any residual vertical field. How­ever, such systems have their own difficulties which are, perhaps, even more difficult to over­come. It is for this reason that, in the design of the present magnetometer, a magnetic balance is preferred to other moving systems. The problem of reversing the polarities of the balance, with fairly strong assurance that equal and opposite magnetic moments will be obtained and that there will be no danger of disturbing the center of gravity of the balance, is solved by fixing two magnetizing coils in the instrument with their common axis parallel to the axis of the balance. The balance is arrested in a definite position while it is magnetized to saturation by sending an electric current through the magnetizing coils in one direction or the other. As the balance is not disturbed and the interval between the two ob­servations is very short, it can be safely assumed that both the center of gravity of the balance and its· temperature remain unchanged during th.is interval. This assumption is experimentally con­firmed by the results obtained with the new instrument, which are given in one of the follow­ing sections of the present paper.

2. THEORY

(a) The Balance

Figure 1 shows diagrammatically the essential parts of the magnet balance: the magnetic needle B, the knife edge K, and the reflecting mirror R. The reflecting surface of the mirror is parallel to the knife edge, and the knife edge and the axis of the needle are perpendicular to each other. Be­fore magnetization of the needle the center of gravity of the balance is adjusted to lie just below the knife edge so as to give the desired

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DETERMINATION OF THE EARTH'S MAGNETIC FIELD 173

sensitivity and to allow the needle, to rest in equilibrium in a horizontal position. The needle is then magnetized and the rotating system to­gether with an auto-collimating telescope point­ing at the mirror is rotated about a vertical axis until the knife edge of the balance is brought into the earth's magnetic meridian plane in which the knife edge is to be kept throughout the experiment.

(b) The Reference Scale for the Image

The equilibrium of the balance is determined by observing the position of the image of the telescope cross wire reflected back from the mirror onto a scale inside the telescope. A line on the scale must first be chosen as a reference posi­tion for the image. Theoretically, if the needle can be completely demagnetized, the image posi­tion of the balance with its needle in that state would be the ideal reference position; and then the weight of the balance would have no effect on the balance. In practice, as it is not easy to demagnetize the needle completely, a reference position must be chosen to the best advantage for the purpose of the experiment. It is obtained by the following method. The needle is first magnet­ized by the magnetizing coil. The vertical com­ponent of the earth's magnetic field Z is then balanced by-the artificial field produced by the Helmholtz-Gaugain coil, a certain line, for in­stance, the center of the scale, being used as a tentative reference position. Suppose the field required is ZI. The polarities of the needle are then reversed and the vertical component of the earth's field is again b~lanced. Suppose this time the required field is Z /. Owing to the effect of the weight of the balance, ZI' will generally differ from Zl. The value of the difference ZI-ZI' will mainly depend upon the line on the scale used as the reference position. The best line is that which makes the difference ZI-ZI' a minimum. In practice, it is only necessary to reduce this differ­ence to a limit depending on the percentage difference between the original and the reversed magnetic moments of the needle.

(c) Determination of Z

Let A, in Fig. 2, be a unit vector in the direc­tion of the axis of rotation; M the magnetic moment of the needle projected in a plane per- .

t z. I I * -A / 00\ ~--------,--,",H

M I I

I I

1 Z,W

FIG. 2.

pendicular to A; L the perpendicular distance from the axis of rotation to the center of gravity of the balance; W the weight of the balance; H and Z the horizontal and the vertical components of the earth's magnetic field, respectively; and ZI the artificial vertical field required for balancing Z. Then, for equilibrium, we have

[MHJA+[M(Z-ZI)JA+[LWJA=O. (1)

After rotating the whole instrument 1800 about a vertical axis and again balancing the vertical component Z by adjusting the artificial field to Z2, using the same reference position for the reflected image, we have

-[MHJA+[M(Z-Z2)JA+[LWJA=0, (2)

which is the same as Eq. (1) except the replace­ment of ZI by Z2, and the change in sign of H. Adding Eqs. (1) and (2) and dividing the sum by 2, we obtain

[ ( Zl+Z2)] M Z--2- A+[LWJA=O. (3)

In Eq. (3) the effect due to H is eliminated. To eliminate the effect due to the weight of the balance, the polarity of the needle is reversed and two similar observations made. Suppose the new values of the required artificial field are Zl' and Z2'; then, changing M in Eqs. (1) and (2) into -M, we have

-[MHJA-[M(Z-Zl')JA+[LWJA=O, (4)

[MHJA-[M(Z-Z2')JA+[LWJA=0. (5)

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174 S. L. TING AND S. T. LIN

By adding Eqs. (4) and (5) and dividing the sum by 2, we obtain

[ ( Z '+Z'

- M Z-~~) ]A+[LW]A=O. (6)

Subtracting Eq. (6) from Eq. (3), and dividing the difference by 2, we obtain

Z=HZIfZ2+ZI'+Z2'), (7)

or, more simply we can obtain, from Eqs. (1), (5) and (2), (4),

Z=!CZI+Z2'),

Z = !CZ2+Zr').

(8)

(9)

The derivation of the above equations is based upon the assumption that the two magnetic moments of the needle before and after the re­versing of its polarity are exactly equal and opposi teo As this assum ption has not been justified we must consider the more general case, in which the vector -M in Eqs. (4) and (5) must be replaced by a new vector -M' representing the reversed magnetic moment of the needle, and instead of Eq. (6), we have

[ ( Z'+Z'

-: M' Z 1 2 2) ]A+[LW]A=O. (10)

Subtracting Eq. (10) from Eq. (3), and dividing by 2, we obtain

[ ( ZI+Z2)] M Z--- A

2

[ Z '+Z'

+ M'( Z- 1 2 2) ]A=O,

( ZI+Z2)

M Z --2- sin (M, Z)

( ZI'+Z/)

+M' Z 2 sin (M', Z) =0,

Z = HZI +Z2+Zr' +Z2')

+~ Msin (M, Z)-M' sin (M', Z')

4 M sin (M, Z) + M' sin (M', Z')

X (Zl+Z2-Z1' -Z2') =0.

(11)

In the above equation the angles (M, Z) and

(M','Z') are both nearly equal to 90°; the nu­merical values of M and M' will not differ much from each other since the needle is always magnetized to saturation; and ZI and Z2 are nearly equal, respectively, to ZI' and Z2' owing to the use of the chosen reference scale. Hence the second term in Eq. (11) is only a small quantity. Putting sin (M,Z)=sin (M',Z')=90°, M'=M ±~M, and M+M+~M=2M, we obtain

Z=HZI+Z2+Zr'+Z2')

l~M +--(ZI-ZI'+Z2-Z2'). (12)

42M

If, in Eq. (12), ~M/ M is less than 2 percent and (ZI+Z2)/2 and (ZI'+Z2')/2 differ by not more than 200,)" the value of the second term will be less than 1,),. Hence the second term in Eq. (12) can be neglected; and the equation is reduced to the same form as Eq. (7). In the actual experiment to be described the value of ~M/ M is less than 1 percent. The mean value of the difference (ZI +Z2)/2 - (ZI' +Z2')/2 is about 70,),. The latter value could be'reduced further if a better reference position for the 'reflected image were chosen.

(d) Sensitivity and Stability

Taking the variation of the quantities in Eqs. (1) and (2) and regarding H, Z, W, and A as constants, we obtain the following equation:

±[oMH]A+[oM(Z-Zo)]A

-[MoZo]~+[oLW]A=O, (13)

where Zo is put for ZI or Z2 and oZo for OZI or OZ2. Let a denote the small angle through which the balance is rotated to regain equilibrium when the artificial field is given the small increment oZo. Then, remembering that both M and L are perpendicular to A, we can put

oM/M=oL/L=a.

By this relation Eq. (13) becomes

±aG:H ]A+a[::(Z-Zo) JA

. [M OZO] LliV:[OL W] -oZo M 5Z

o A+a M oL W A=O,

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DETERMINATION OF THE EARTH'S MAGNETIC FI~LD 175

5Zo (14)

a

Assuming that both M and A are nearly hori­zontal, L is nearly vertical; putting

[M 5Z0]A=[5L W]A =1, M5Zo 5L W

and neglecting small terms of the second order, we obtain

1 5Zo LW -= ±Iisin (H, A)+-. (15)

Sensitivity ,a M

As evident from the above equation, if Ii sin (H, A) >LWjM, the system will be unstable. Hence to insure the stability of the system the term H sin (H, A) must be made as small as possible, i.e., the axis of rotation must point in the mag­netic north-south direction. The sensitivity of the balance then will depend mainly upon the nu­merical value of L W j M and can be varied by adjusting the distance L between the center of gravity of the balance and the knife edge. The sensitivity of the actual instrument is about ! min. per 'Y. The weight of the balance is 2.6 g,

(a)

(c)

FIG. 3.

(b)

the magnetic moment of the needle is about 24.4, and the distance between the center of gravity and the knife edge is therefore about 6.6 X 10-4 cm.

3. THE INSTRUMENT

(a) The Balance System

The construction and the important parts of the balance system are shown in Figs. 3 (b) and 3(c). B is the magnetic needle, which is a cobalt­steel rod about 35 mm long and 2 mm in diameter; G the agate piece, of which the lower part K is a knife edge, the front face F serves as the re­flecting mirror, and the two triangular side arms R serve as the supporting beam when the balance is arrested; P the resting plate for the knife edge which is ~xed on a metal base E, placed in a circular hole and screwed on the bottom of the balance chamber; Q is the needle holder which is screwed tightly to the agate piece with the axis of the needle normal to the mirror. The con tact area between the knife edge and its resting plate is made small by removing the two upper edges of the plate, as shown in Fig. 3(a).

(b) The Magnetizing Solenoids

Two solenoids, 4 cm long, 2.8 cm in outer di­ameter, and 1.1 cm in inner diameter, are used as the magnetizing coils. They are placed as near to each other as possible in the two side holes of the balance chamber. The two outer ends of the solenoids are closed by glass windows to allow light to pass through from the telescope to the reflecting mirror. The magnetizing field required for producing saturate magnetization of the needle is about 1sor. The current used for pro­ducing this field is about 1 ampere. The resistance of the coils is about 4 ohms.

(c) The Balance Chamber

The balance chamber, which is made of brass, consists of a cubic block at the middle and two circular disks, one at the top ·and the other at the bottom (Fig. 4). A large vertical circular hole is made through the center of the block, closed at the bottom by the base of the resting plate of the knife edge and at the top by a cover, to which the device for arresting the balance is attached. To arrest the balance the two arms [R in Fig. 3 (a)] of the agate piece are raised and pressed against two

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176 S. L. TING AND S. T. LIN

FIG. 4.

straight edges parallel to the needle. The direc­tion of the needle is therefore always the same when the balance is in its arrested position, and so is the direction of its magnetic axis. Two other circular holes for inserting the solenoids are sunk into the block from its front and back surfaces leaving a partial partition at the center just ~ little thicker than the thickness of the agate piece. On the upper and lower disks of the block two deep grooves are made for the winding of two other coils, the purpose of which will be explained below.

(d) The Helmholtz-Gaugain Coil

The coil is wound on a metal frame rigidly fixed on a horizontal divided circle, with its axis exactly vertical. On the frame is fixed the auto­collimating telescope for observing the equilib­rium of the balance. The optical axis of the tele­scope is horizontal and hence normal to the reflecting mirror when the magnetic axis of the needle is horizon tal. The diameter of the coil is 30 cm and the mean distance between the two halves of the coil is, therefore, 15 cm. The total number of turns is 104; and the constant of the coil obtained by rough calculation from the num­ber of turns and the dimensions of the coil is about 311,660'Y/amp. The exact value of the constant must be calibrated by a standard instru­ment, as the present instrument is intended for use in field work as a secondary standard. But this type of balance can just as well be used as a primary standard instrument.

( e) The Potentiometer

The current passing through the Helmholtz­Gaugain coil is determined by a specially designed

potentiometer and a standard cell. The potenti­ometer is connected in series with the coil. The poten tial drop along a known resistance in the circuit is balanced by a standard cell. Thus the value of Z is actually determined in terms of the coil constant, the resistance, and the e.mJ. of the cell. The potentiometer circuit consists of a series of seven resistance coils of 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, and 6.4 ohms, respectively, and a slide wire of 0.12 ohm. Any resistance below the total can be obtained by the combination of these coils and the slide wire. From the approximate value of the earth's field the coils needed to give the required resistance can be predetermined. The rest of the resistances are then short-circuited. This par­ticular arrangement of the potentiometer resist­ance coils has two advantages. The resistance of the slide wire can be very low. Hence a short wire can be used. The number of the resistance coils needed is very much reduced in comparison with that in the ordinary potentiometer. To supply the current dry cells can be used, as the current required is very low. This gives great convenience to the field worker. The same current source is used for the magnetization of the balance. A double-throw switch is provided on the potenti­ometer, by which the battery leads can be switched from the potentiometer circuit on to the solenoid circuit, orvice versa. The circuit diagram is shown in Fig. 5.

(f) The Galvanometer

To balance the potential drop in the potenti­ometer circuit with the standard cell a sensitive galvanometer is needed. As the magnetic balance under the balance condition of the earth's field is

6Ar 3.2r 1.6r .ar .4r .2r .Ir

·To1

Sol~noid

llill B

s

FIG. 5. B=battery, M=magnetometer R=rheostat of the potentiometer, S=slide wire, E=s~ndard cell, and G=galvanometer.

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DETERMINATION OF THE EARTH'S MAGNETIC FIELD 177

the ideal moving system for a galvanometer of the moving-magnet type, the magnetometer itself is used as a galv~nometer. A pair of coils is wound in the grooves of the upper and lower disks of the balance chamber and serves as the galvanometer coils. This contrivance saves .not only the pro­vision of an extra instrument but also reduces the operating personnel. For, usually, while one person is working at the magnetometer, adjusting the rotating system to balance the magnetic fields, another person is needed to balance the potentiometer. When the same moving magnetic system is used for both instruments the two separate observations are combined into one. As soon as the image of the cross wire ·is brought to the reference position (this completes the mag­netometer adjustment) this same position is used as the zero line of the galvanometer scale, and the deflection of the image is observed for the potentiometer adjustment.

4. THE EXPERIMENT AND THE RESULTS

The instrument was tested at a field station. Altogether six experiments were made during four consecutive days. Each experiment con­sisted of two or three pairs of observations, the first with the needle magnetized in one direction, the second with the direction of magnetization of

..I

the needle reversed, and the third with the direc-

TABLE I.

Artificial field in gauss X 10' HZ.+Z, Ex- Zin -Z.'-Z,')

peri- gauss in gauss X 10' ments Telescope E Telescope W X 10' (factor in cor-

Z. Z.' Z, Z,' rection terms)

36951.7 36949.5 36866.6 83.9

1 36783.5 36782.2 36864.0 61.1

36945.1 36945.1

36905.9 36904.6 36874.0 31.3

2 36841.4 36843.9 36872.1 29.4

36899.7 36903.2

36954.1 36951.2 3 36867.6 85.5

36784.4 36780.8

36910.6 36915.0 4 36861.9 55.9

36812.4 36809.5

36947.4 36948.4 5 36865.1 82.7

36782.3 36782.4

36997.6 36998.5 6 36866.8 131.3

36735.6 36735.5

TABLE II. Strength of magnetizing current=O.99 amp. Intensity of magnetizing field = 150r.

Direction of Number of magnetization reversals of Time for 50

of needle current oscillations

Marked end S 5 98.0 sec. Marked end S 5 97.8 sec. Marked end N 5 98 sec. Marked end N 5 98 sec. Marked end S 5 98.4 sec. Marked end S 5 98.3 sec. Marked end N 5 98.5 sec. Marked end N 5 98.6 sec. Marked end N 2 98.4 sec. Marked end N 1 97.8 sec. Marked end N 1 98.0 sec. Marked end S 1 98.7 sec. Marked end S 1 97.8 sec. Marked end S 1 98.1 sec.

Mean value of period of oscillation T = 1.97 sec. Maximum value of percent variation t1M/M =0.88 percent.

tion of the magnetization. reversed back. In each pair of observations the telescope was first set at the east or west and. then turned to the opposite side by rotating the . whole instrument through 1800 about a vertical axis. As the purpose of these experiments was to test the consistency in the value of the vertical component of the earth's magnetic field to be obtained by the instrument rather than to determine its actual value, the instrument was not calibrated before its use. The coil constant used for the calculation of the artificial field was obtained by rough calculation from the number of turns and the dimensions of the coil. The constant must, however, be cor­rected for temperature variation for each experi­ment, and its temperature coefficient must there­fore be known. This was obtained indirectly by simple calculation from the coefficient of the thermal expansion of the frame metal. The results of these experiments are given in Table I, from which the consistency in the value Z can clearly be seen.

A preliminary test for the constancy of the magnetic momen t of the needle was also made by observing its period of oscillation in the earth's field. The polarity of the needle was reversed for a number of times, and after each reversal the needle was suspended by a thin thread and set in oscillation in a horizontal plane, and its period determined. The results of the test given in Table II show that with the magnetizing field used the maximum value of the percent variation t:..M/ M is less than 1 percent.

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