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an opening through which passed the lower projecting end of the tube in the other coil, and to this was attached a hook and scale-pan.
The current, obtained from a battery of accumulators, was measured by a Siemens and Halske's torsion-galvanometer, standardized by electrolysis.
The magnetization was measured in independent experiments by the magnetometer method.
The ellipsoid and coils were placed in the magnetic east and west line east of a delicate magnetometer read by telescope and scale.
The intensity of the earth's horizontal field was measured by the method of Gauss and by the tangent-galvanometer method, a Kohlrausch's local-variometer being found useful for measuring its variations.
The magnetization-curve was first obtained, showing the relation between I and H, allowance being made, as usual, for the demagnetizing-force of the ellipsoid calculated from the expression given in Maxwell's 'Electricity and Magnetism' (Sect. 438). The intensity of magnetization I was calculated in absolute units from the formula*
where hearth's horizontal field, 0= deflexion of needle, v=volume and e=eccentricity of ellipsoid, d= distance of needle from centre of ellipsoid (the former being in the pro
longation of the axis of the latter), and n=semiaxis of the ellipsoid.
c being the
This curve being determined once for all, the induction B corresponding to any current could be calculated, since B=Н+4πI†. All precautions were taken to determine the curve as accurately as possible, the torsion of the magnetometer-fibre, the exact position of the needle with respect to the axis of the ellipsoid, the length of a division on the scale, the effect of the compensating-coil at all parts of the scale (by deflecting with permanent magnets), and the temperature of the room during the local-variometer experiments all being carefully examined. The residual magnetism was very small, being less than a sixtieth of the total magnetization when the ellipsoid was strongly magnetized. Each current was
*This was obtained from Thomson and Tait's expression for the potential at an external point due to an ellipsoid of attracting matter by differentiating twice with respect to the distance of the point. (Cf. Roessler, Dissertation, Zürich, 1892).
+ Where H=Field due to coils-demagnetizing force of ellipsoid.
reversed, and half the double deflexion of the needle taken as the deflexion corresponding to the mean of the two currents, which were generally the same*.
The "lifting" experiments were then proceeded with :
The upper coil was first suspended vertically and symmetrically above the opening H, in the platform P (fig. 1), the lower bar and coil being thus supported (when the current was made) so that its tube T passed freely through the opening H.
Enough weights were placed in the scale-pan to nearly overcome the stress, the remainder being slowly poured in in the form of fine shot until the lower coil fell. The current was then read off and the shot weighed.
This was repeated for a number of currents ranging from 1 to 10 amperes (the corresponding inductions ranging from 6000 to 20,000 C.G.S.); the adjustment of the upper coil being made before every reading, and the surfaces carefully cleaned with a soft dry brush.
The "reversed" readings were also taken, as in the magnetometer experiments.
The inductions corresponding to the currents used were calculated from the magnetization-curve. The inductions and the square roots of the observed weights were plotted in a diagram along with the straight line representing Maxwell's law.
The first experiments were made with the contact-surfaces of the ellipsoid turned truly plane but not polished.
The curve representing the observed results was at low inductions considerably above the straight line (the observed weights being greater than those given by Maxwell's law), soon crossing it and remaining below it for high inductions, the difference increasing with the induction.
The correction due to the excess of area of the coils over that of the core was found by removing the cores and supporting the lower coil in the same position on a balance.
* (1) It may be objected that the magnetization in the "lifting' experiments was not the same as in the magnetometric experiments, on account of the longitudinal pull existing in the former; but a small calculation shows that, with the weights used, this effect was in general very small, and could, except perhaps at the highest inductions, be neglected.
(2) The magnetization-curve was determined both before and after the ellipsoid was cut, and the demagnetizing-force (which was affected by the shortening of the ellipsoid after the cutting) calculated in both cases. The curve for the cut ellipsoid was lower than the other; but the difference was very slight, especially at high fields.
Phil. Mag. S. 5. Vol. 39. No. 238. March 1895.
The current was made, and weights added until the coil returned to its original position. These added weights measured the total attraction of the coils. From this must be subtracted the part corresponding to the area occupied by the core, since this is already included in B.
The correction was found to be negligible, never being more than a sixth per cent. of the attraction of the electromagnets, and for small currents less than a twentieth per cent.
Another correction is due to the effect of the surfacedistribution of magnetism on the ellipsoid in so far as the corresponding tubes of force pass through the air. This correction, which vanishes with ring-magnets and infinitely long cylinders, was found on calculation to amount to less than per cent.
The interfaces of the ellipsoid were then polished by the firm Hartmann and Braun of Frankfurt. In this process the surfaces were surrounded with wide "guard-rings" to keep the edges as sharp as possible. The mirrors showed a so-called "black polish," and gave a perfectly clear image with a 32-magnifying-power telescope and scale at a distance of 5 metres. It would be hardly possible to obtain on metal a nearer approach to an absolutely geometrical plane.
Also in order better to guide the lower coil, the opening in the platform P (fig. 1) was made smaller, just large enough, in fact, to allow the tube to pass freely through it. A series of readings was taken, as before, and these agreed considerably better with Maxwell's theory both for low and for high, but especially for low inductions.
This I attributed chiefly to the improved guiding of the lower coil; and it was found that at low inductions, by very slightly altering the position of the upper coil, a position could be found in which the tractive force was a minimum, and that if the applied weights were rather less than this minimum value the upper coil (never being absolutely rigidly fixed), on being slowly moved towards the "minimum position, suddenly jumped across it, thus showing, so to speak, ference for positions in which it could support heavier weights or the actual weight more easily.
The measurements were therefore made as follows:-The current being kept constant, shot was poured into the scalepan in small instalments, the upper coil after each addition being carefully moved by hand until the "minimum" position (easily observed by the jerk) was found. This was repeated until the coil fell in this position.
The explanation of this is given in Threlfall's paper (l. c.), in which it is shown that at small inductions the tractive force is less when the two pole-faces are everywhere in contact
than when they are separated at one side, thus enclosing a wedge-shaped gap.
This was the case in the present experiments up to inductions of about 14,000 C.G.S.
For higher inductions the tractive force is greatest when the surfaces are everywhere in contact, the upper core and coil therefore assuming naturally the proper position, provided, of course, the screw-adjustment is first sufficiently good. For inductions up to about 14,000 C.G.S., therefore, the lower bar was in unstable equilibrium when in good contact with the upper, for higher inductions in stable equilibrium; at about 14,000 the equilibrium was indifferent, and the nature of the contact was found within wide limits to have no influence on the Tractive Force.
As regards the results showing better agreement with the theory than before at higher inductions, I attribute this to a better method of testing the screw-adjustment of the upper coil, and partly, perhaps, to the improved state of the contactsurfaces after polishing.
At this stage the weights were all smaller than those calculated from Maxwell's expression, but their square roots were approximately proportional to the induction for inductions up to about 14,000 units, being still smaller at higher inductions, the deviation from the theoretical values increasing to about 3 per cent. at B=20,000.
The uniformity of these results led me to believe that the errors, if any existed, were not accidental but due to some cause which acted always in the same way. Accordingly, I tried the effect of increasing the distance between the coils, thus leaving a greater part of the core near the plane of contact unsurrounded with coil-windings. I found that the effect of increasing this distance by about 2 millim. was very slight for inductions up to about 14,000 C.G.S., and for higher inductions was a diminution of Tractive Force of a few hundred grammes, varying with the induction-differences of the same order as the differences between the above observed and calculated values. I could also increase the Tractive Force by an amount of the same order by putting between the coils a few extra turns of wire. As the actual distance between the windings of the two coils (including the end-plates and the space occupied by the small guiding ring) in the above experiments was about 5 millim., it was clear that the observed weights were on this account smaller than the theoretical values; in other words, that the induction, as calculated from the magnetization-curve, was greater than the actual induction across the surface of contact.
This result can be easily explained; for when the induction
is low the permeability of iron is great, and the tubes of induction pass more readily through the iron; but when the induction is high and the permeability small the absence of coil-windings near the surface of contact causes a greater proportion of tubes to pass out into the air; in other words, causes a greater spreading of the tubes. The effect of the absence of a given number of turnings near the plane of section might be calculated [see Neumann: "Ueber die Magnetisirung eines Drehungsellipsoids," Crelle, Bd. xxxvii. (1848)], but it was deemed better first to diminish the gap between the coils as much as possible. With this object the coils were somewhat altered, the end-plates being altogether removed and the small guiding ring being let into one of the coils. The gap-width could then be reduced to zero, but a width of about 1.5 millim. was necessary in order to examine the contact of the pole-faces. The effect of this was to increase the Tractive Force by several hundred grammes at all inductions, the increase being greatest at high inductions. The magnetization-curve was carefully re-determined and the results calculated out as before. The attraction of the coils alone was also remeasured, but found not to have been appreciably increased by the shortening of the gap.
The results now agreed with Maxwell's theory to one-half per cent. for inductions up to 19,000 units, but between 19,000 and 20,000 units the square roots of the observed weights are rather more than 1 per cent. below the calculated values.
This difference at high inductions might well be due to the fact that there was still a gap of 1.5 millim. between the coils, which would have a spreading effect at high inductions; to the effect of the stress on the magnetization; or to temperature effects, the coils being considerably heated by a current of 10 amperes. A better agreement with the theory was therefore hardly to be expected.
No readings could be taken at inductions below 6000, since the weight of the lower coil and half-core was sufficient to overcome the stress at this induction. The following table gives the values of the intensity of magnetization, induction, and the square roots of the theoretical and observed Tractive Forces measured in grammes weight; and the diagram (fig. 2) shows the values of the Induction and the square root of the observed Tractive Force, the theoretical values being represented by the straight line whose inclination is determined by the factor