UNIVERSITY OF CALIFORNIA AT LOS ANGELES GIFT OF a.m. to I h 00 m p.m. June 14, 12 12 p.m. to 5 12 p.m. 3.8 7.4 +0.4 -1.4 4.2 6.0 15,308 38,100 128 127 0.000274 .000157 These values, while by no means as concordant as could be desired, are but approximately one-fifth to one-tenth of that found in the single test made in Middletown, Connecticut. Subsequent tests made with ergometer II (see p. 29) indicate that these values are probably not far from correct, though it should be stated that a calorimeter designed to measure the heat production of a man is not best adapted to measuring so small an amount as 1 or 2 calories per hour. Exactly what use of this value is justified in an experiment with a man it is not the province of this paper to discuss. These tests are given primarily to show that the earlier value was entirely wrong and hence all calculations made with it should be regarded as worthless. CALIBRATION TESTS OF ERGOMETER II. The second ergometer was constructed during the summer of 1911 and, after preliminary tests as to the winding of the magnets, the ap- paratus was substantially installed in the chair calorimeter for a series of tests. These tests covered wide ranges of speed and magnetizing current. A further variant was introduced in that in some of the ex- periments the relative position of the disk and pole-faces was changed so that the disk rotated much nearer one pole-face than the other. Sub- sequently, the entire magnet was moved towards the hub in a straight line, so that in a few experiments the pole-faces were nearer the hub by about 20 mm. In this new position the disk was at times in the center of the space between the pole-faces and at other times it was as near as possible to one of the pole-faces without actual contact with it. These tests with varying positions of the magnet were all incidental to a study 1 Benedict and Carpenter, loc. cit., p. 15. CALIBRATION TESTS 23 of the peculiar behavior of the magnetic field when the copper disk was rotated at different speeds. The results of the several calibration tests are reported in abstract in table 5. TABLE 5. Results of calibration tests ofergometer II. Date. Duration of period. Cur- rent. Heat meas- ured. Corr. for change of calorime- ter temp. Corr. for heat of magneti- Nation. Heat pro- duced. No. of revolu- tions of pedals. No. of revolu- tions per min. Heat per revolu- tion. 1911. h. m. amp. cols. cols. cals. : cats. cal. Oct. 28 5 1.25 636.6 -0.2 60.8 ; 575.6 36,148 120 0.0159 30 6 1.25 621.1 + .2 72.7 548.6 29,072 81 .0189 31 6 1.35 705.5 - .2 85.8 619.5 29,228 81 .0212 Nov. 1 6 1.50 761.6 +2.1 107.3 656.4 29,947 83 .0219 2 6 1.50 563.3 + .4 106.9 456.8 21,100 59 .0216 3 5 1.50 716.2 +2.1 89.1 629.2 30,228 101 .0208 4 6 1.50 937.1 - .8 106.9 829.4 42,438 118 .0195 6 6 1.35 490.7 + 1.8 85.4 407.1 20,834 58 .0195 7 6 1.35 752.5 + .4 85.4 667.5 36,376 101 .0184 8 6 1.35 834.4 -f .2 85.1 749.5 42,870 119 .0175 9 6 .25 455.9 - .6 72.4 382.9 20,858 58 .0184 10 3 1.25 356.5 + .2 36.2 320.5 18,471 103 .0174 10 3 .25 345.1 - .4 36.2 308.5 17,119 95 .0180 11 6 1.10 629.5 55.2 574.3 43,832 122 .0131 13 6 1.10 581.6 +Y.8 55.2 528.2 36,573 102 .0144 14 6 1.10 489.9 -1.0 55.2 433.7 27,638 77 .0157 15 6 1.10 417.2 55.5 361.7 21,853 61 .0166 16 6 1.35 655.2 +".4 85.4 570.2 28;840 80 .0198 17 6 1.35 767.8 +1.6 85.0 684.4 37,052 103 .0185 20 4 1.50 377.4 +0.2 70.9 306.7 14,300 60 .0214 21 6 1.50 588.8 -1.4 106.4 481.0 21,888 61 .0220 22 5 .95 251.8 + .6 34.0 218.4 15,932 53 .0137 23 6 .95 293.7 + .4 40.8 253.3 18,739 52 .0135 24 4 1.50 195.3 + 1.4 71.2 ! 125.5 7,821 33 .0160 25 6 1.50 381.3 +2.9 106.8 1 277.4 15,186 42 .0183 Dec. 1 6 1.35 362.7 - .2 85.4 277.1 15,162 42 .0183 2 6 1.25 329.8 +2.3 72.0 260.1 14,945 42 .0174 4 6 .95 441.9 40.8 401.1 27,502 76 .0146 5 6 .95 519.3 +1.6 40.9 479.4 36,311 101 .0132 6 6 .95 548.8 - .2 40.8 507.8 44,232 123 .0115 7 6 1.10 587.9 + .6 55.7 532.8 1 30,389 84 .0175 8 6 1.10 638.7 + .6 55.7 583.6 37,880 105 .0154 9 6 1.10 476.4 + .8 55.6 421.6 24,020 67 .0176 11 6 1.25 492.5 + .2 72.4 420.3 22,554 63 .0186 12 5 1.25 507.1 - .2 60.3 | 446.6 i 23,577 79 .0189 12 3 1.25 180.2 - .4 36.2 143.6 8,173 45 .0176 13 6 1.25 725.9 + .8 72.4 654.3 39,910 111 .0164 1912. Jan. 1 4 1.25 284.8 -1.4 48.5 234.9 11,987 50 .0196 1 4 1.25 451.0 +6.2 48.5 408.7 21,630 90 .0189 1 4 1.25 496.2 +3.9 48.5 451.6 26,482 110 .0170 6 5 1.25 341.8 + 1.4 60.6 282.6 14,880 50 .0190 9 5 1.25 340.7 -3.3 60.6 276.8 13,970 47 .0198 11 5 1.25 317.9 +2.5 60.6 259.8 13,753 46 .0189 15 5 1.25 443.9 + .8 60.6 384.1 18,232 61 .0211 17 5 1.25 636.5 +2.0 60.6 577.9 32,937 110 .0175 18 5 1.25 473.6 - .4 60.3 412.9 21,791 73 .0189 19 5 1.25 508.4 -1.8 60.6 446.0 22,335 74 .0200 19 5 1.25 294.9 -4.1 60.6 230.2 12,682 42 .0182 20 4 1.25 414.1 +2.5 48.0 368.6 | 19,124 80 .0193 20 5 1.25 471 .4 - .2 60.3 410.9 21,039 70 ! .0195 21 3 46 1.25 456.4 +1.2 45.4 412.2 i 22,909 101 i .0180 21 5 1.25 610.0 + .4 60.3 550.1 30,283 101 .0182 22 5 .95 465.5 - .4 34.0 431.1 36,949 123 .0117 22 5 .95 462.2 .8 34.0 429.0 37.482 125 .0114 24 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE With ergometer I the magnetizing current ranged from 0.70 ampere to 1.25 amperes, but inasmuch as the winding of the magnet in ergometer II was somewhat different, the current ranged from 0.95 ampere to 1.50 amperes in the calibration tests of this ergometer. It was planned to secure calibrations of the ergometer at each current with variations in speed ranging from approximately 50 to 120 revolutions of the pedals per minute. For a given speed, the highest values of heat per revolution were obviously found with the largest magnetizing current, namely, 1.50 amperes. As a matter of fact, however, the experiments of November 4 and 6 show that with less current (1.35 amperes) through the field- coils but with a low speed, the heat per revolution was exactly the same as with a current of 1.50 amperes and with twice the number of revolu- tions, namely, 118 revolutions per minute. It is impossible, however, to analyze satisfactorily the varying conditions without recourse to a series of curves plotted for each intensity of magnetizing current. 01 b .01 4 .01 3 .012 nil ^ X-. ^ ~? \ v \ N K FIG. 9. Calibration curve of ergometer II for magnetizing current of 0.95 ampere. .018 .017 .01 6 .015 .014 6O 7O 8O 9Q 1OO 11O 12O Fio. 10. Calibration curve of ergometer II for magnetizing current of 1.10 amperes. Beginning with the lowest current, namely, 0.95 ampere, we find that the values all lie fairly close to the curve (see fig. 9). Two of the obser- vations shown on this curve were made when the disk was rotating very close to the rear pole, leaving a wide air-gap on the other side. These two values, which are indicated by small circles, lie approximately on the CALIBRATION TESTS 25 curve, and from these observations it would appear as if the rotation of the disk somewhat out of the center of the air-gap caused a very slightly larger amount of heat per revolution. The general form of the curve shows again a tendency toward a maximum heat per revolution with a speed of approximately 60 to 80 revolutions, and a tendency to fall off when the ergometer is running at a high speed. With a magnetizing current of 1.10 amperes, we have two series of observations that are by no means concordant (fig. 10), and yet both indicate a noticeable falling off in the heat per revolution at high speed. We are unable at this time to account for the marked discrepancy be- tween these two sets of observations, but since this current is not used at present in actual experimentation with man and since the curve agrees with the others in its general form, it is deemed inadvisable at this time to repeat the calibration test. The calibrations made with a current of 1.35 amperes lie for the most part on a very definite curve (fig. 11). In one single observation at 80 revolutions per minute, it is approximately 5 per cent too high. The gen- eral form of curve noted for the other calibrations is here markedly shown, namely, a low heat per revolution with a low speed, a fairly constant heat per revolution between 60 and 80 revolutions per minute, and then a falling off in the heat per revolution as the speed is increased. .021 .020 .019 .01 8 .01 7 O1 6 X A ^ ; *\ / \ ^x ^s X 5O 6O 7O 8O 9O 1OO 11O 12O Fia. 11. Calibration curve of ergometer II for magnetizing current of 1.35 amperes. The observations with the strongest current through the field, namely, 1.50 amperes, are shown in fig. 12. One observation, characterized by a circle, was made when the disk was rotating near the rear pole, i.e., out of center of the air-gap, but the variation from the normal is so slight as to make it almost imperceptible. In this curve we again find a low heat per revolution with low speed, a fairly constant heat per revolution be- tween 60 and 80 revolutions per minute, and a decrease in the heat per revolution as the speed is further increased. 26 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE By far the largest number of tests were made with a current of 1.25 amperes, which was selected for the study of the magnetic field described in Part III of the report. In order to make this study it was necessary to move the disk as far as possible toward the rear pole-face and thus provide space in the air-gap between the front pole-face and the disk for a flat bismuth spiral. It was deemed advisable, therefore, to test the machine under these conditions in order to find if there was any marked difference in the calibration test when the circular disk was somewhat off center. The points obtained in this way are surrounded by circles in the curve shown in fig. 13. It will be seen that they lie somewhat above the curve, as was also the case with figs. 9 and 12. The reason doubtless is that the magnetic field, at least near the edges of the poles, is so non-uniform that the lines of induction intercepted by the disk are somewhat denser when the latter is brought close to one pole-face. Aft0 .022 .021 .O2O .019 .018 .017 .01 6 .015 ^^-' / r X \ / X V / N / / / 3O 4O 5O 6O 7O 8O 9O 1OO 11O FIG. 12. Calibration curve of ergometer II for magnetizing current of 1.50 amperes. Tests were also made with the magnet covering more of the copper disk, i.e., pushed in about 2 cm. toward the hub. Accordingly, in fig. 13 we find a large number of points which may be classified under several groupings. In the series of observations in which the magnet was pushed farther over the copper disk, one might expect a somewhat smaller brake- effect upon the copper disk; as a matter of fact, it was found that the curve was shifted somewhat to the left, showing abnormally high values of heat per revolution at low speeds. These points are indicated by squares (cf. Part III). Since the chief use of the instrument, however, is for a regular magnetizing current of 1.25 amperes, with the disk ex- actly in the center of the air-gap and the periphery of the disk tangen- CALIBRATION TESTS 27 tial to the upper edge of the pole-face of the magnet, it seemed undesirable to make further calibrations of this instrument under these peculiar conditions, which were necessitated only by the study of the magnetic field. We have, therefore, chiefly to consider the observations made with the disk in the regular position. The heavy line 1 plotted curve represents all observations with the disk and magnet in their original positions. Here again we find with low speeds the low heat per revolution a fairly constant heat per revolution with a speed between 60 to 90, and a fall in heat per revolution as the speed increases beyond this. .022 .021 .020 .01 9 .018 .01 7 .01 6 .015 .014 .013 2O 3O 4O 5O 6O 7O SO 9O 1OO 11O 12O Fid. 13. Calibration curves of ergometer II for magnetizing current of 1.25 amperes. Black crosses: Disk in normal position between poles of magnet; curve in heavy line is based upon these. Circles : Disk close to rear pole, air-gap on other side widened. Squares: Poles moved 2 cm. in toward center of disk. Curve in light line represents values of &>(f> 2 (see p. 37). To show their similarity the curves corresponding to the five magnet- izing currents, 0.95, 1.10, 1.25, 1.35, and 1.50 amperes have been replot- ted on one diagram (see fig. 14). This series of curves is strikingly simi- lar to those found with ergometer I when calibrated in June and July of 1911, and indicates that the instruments are essentially alike in their mechanical and electrical features. The special feature to be noted here is that the curves show uniformly a low heat per revolution with a low speed, nearly constant heat per revolution between approximately 60 to 90 revolutions per minute, and a rapidly falling heat per revolution at high speeds. Since practically all experiments are made with bicycle riders at speeds between 60 to 80, it may be stated again that, in general, 1 The lighter lined curve is discussed in Part III, p. 37. 28 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE the heat per revolution is sufficiently constant between these limits, irre- spective of speed, although reference should be made to the calibration curves if the speeds are below 60 or above 80. The abnormal appearance of these curves led to much speculation as to the cause. In Part III of this report it will be shown that a complete explanation of the observed effects is found in the magnetic reaction of the eddy currents induced in the copper disk. .02 3 .022 .021 .O2O .019 .018 .017 .016 .015 .014 .013 .012 .011 <** ^, / <.SO AMP. \ X / ^s / / <3S AMP. \ x // / ^ AMP. \ X [X \ / / ^ X \ X / *uo AMP' \ X \ S, xs ^ AMP. ^x \ \ / \ Sy \ V X X ^^ \ 30 40 50 60 70 8O 9O 1OO 11O 12O FIG. 14. Calibration curves of ergometer II for magnetizing currents of 0.95 to 1.50 amperes. For physiological experimenting, the apparatus is most satisfactory, since the constant brake-effect gives a constant heat production. Although unfortunately it is not everywhere possible to determine the absolute values by means of calibration tests inside a large calorimetric chamber, yet it may be that by driving the ergometer with an electric motor of known efficiency and measuring the input of electrical power, an approxi- mate idea can be obtained of the actual power required to rotate the pedals. We have made some rough tests of this sort. The chief diffi- culty lies in running the disk at a sufficiently low speed without the various losses becoming disproportionately large. In connection with these observations it is of especial interest to note that ergometer I remained essentially constant in its electrical and mechan- ical properties over a period of some 8 years, thus showing a remarkable CALIBRATION TESTS 29 constancy in the apparatus. We feel justified, therefore, in heartily recommending its use when a constant amount of work is to be done and uniformity in muscular work is essential. Furthermore, the amounts of energy computed from the speed of the magnetizing current are accu- rate to within about 2 per cent. FRICTION TESTS WITH ERGOMETER II. Although it was doubtful if a knowledge of the heat per revolution due to friction would be of any particular value, it seemed desirable to make measurements of the friction of this apparatus if only for com- parison with those made with ergometer I, and for checking the recent experiments with the latter. Three friction tests were accordingly made with ergometer II on December 18, 20, and 22, 1911, the results being reported in table 6. As in the friction tests with ergometer I, the amounts of heat measured were so very small that but little reliance can be placed upon the results for individual periods; and it is not surprising that we find variations of 50 per cent between the heat per revolution found on December 18 and December 22 when compared with that in the test on December 20. When we consider, for example, that through a whole experiment lasting from 10 h 14 m a. m. to 2 h 15 m p. m. only a sum total of 7 calories was measured, the numerical values found are certainly not of great significance. The important thing is that these results show an average of heat per revolution not far from 0.0025 calorie, which is in reasonably close agreement with those found with ergometer I in the calibrations inside of this identical calorimeter. In general, the frictional heat per revolution is not far, therefore, from 1 to 2 per cent of the total heat produced when the apparatus is used with the field magnetized at 1.5 amperes. TABLE 6. Friction test, ergometer II. Corr. for Heat No. of Rate j u eat Time. meas- ured. change of cal. pro- duced. revolu- tions of P? r per revo- mm- lution. temp. ute. | CO/8. cals- cals. cal. 1911 Dec. 18, 2K)2>p.m. to 5 b 02">p.m. Dec. 20, 10 14 a.m. to 2 15 p.m. Dec. 22, 10 46 a.m. to 4 46 p.m. 12.4 7.08 14.89 -4.5 -1.6 + 1.0 7.9 5.48 15.89 22,276 30,121 45,235 124 0.000355 125 ; .000182 126 I .000351 I PART III. THE MAGNETIC REACTIONS PRODUCED BY A COPPER DISK ROTATING BETWEEN THE POLES OF A MAGNET. That a rotating disk exerts not merely a tangential drag, but also a repulsive force, on a magnet pole placed near it, has been known since the days of Arago. 1 Nobili 2 first discovered that the loops of induced current are displaced in the direction of rotation of the disk, though he did not understand the part played by self-induction in causing this. Indeed, as far as we are aware, no attempt has been made up to the present time to make a quantitative determination of the electric and magnetic effects. Mathematically, the problem of the currents induced in bodies rotating in a magnetic field has been attacked by Felici, Jochmann, Maxwell, Himstedt, Niven, Larmor, Gans, and especially by Hertz. 3 The chief results of Hertz's work that have a bearing on the present paper may be summarized as follows: When a conducting mass is rotated in a mag- netic field, the induced currents, owing to self-induction, are distorted in the direction of rotation to an extent independent of the intensity of the magnetic field but increasing with the angular velocity. At the surface of the conductor the currents are less distorted than in the interior. At infinite angular velocity the surface of the conductor would act toward magnetic forces like a conducting surface in an electric field, screening the interior entirely from all magnetic action. These mathematical investigations were all made on the assumption of certain ideal conditions, which in general it would be hard to realize experimentally. In order to apply theoretical principles at all to the present case it is necessary to make some simple assumptions and to be content with qualitative relations. The problem would be comparatively simple if the disk were so thin that it could be regarded as a current sheet, if the magnetic induction B were uniform in the space between the poles, and if the self-induction of the disk could be neglected. Calling a> the angular velocity of the disk, 4 we would then have for the induced electro- motive force e = constant X m B 1 Arago, Pogg. Ann., 1826, 7, p. 590; Pohl, Pogg. Ann., 1826, 8, p. 369. 2 Nobili, Pogg. Ann., 1833, 27, p. 401. A very full account of the classical experi- ments on rotating disks is given in Wiedemann's "Galvanismus und Elektromagnetis- mus," Braunschweig, 1874. s Felici, Annali di sci. mat. e fis., 1853, p. 173; Jochmann, Pogg. Ann., 1864, 122, p. 214; Maxwell, "Electricity and Magnetism," 2, p. 300; Himstedt, Wied. Ann., 1880, 11, p. 812; Niven, Proc. Roy. Soc. 30, 1880, p. 113; Larmor, Phil. Mag. (5), 1884, 17, p. 1; Gans, Zschr. f. Math. u. Phys., 1902, 48, p. 1; Hertz, Inaugural Dissertation, also "Gesammelte Werke," 1, 1895, p. 37. 4 In Parts I and II speeds were expressed in revolutions per minute of the pedals, because in using the bicycle ergometer this is the important quantity. Since in Part III attention is centered chiefly on the disk, we shall, in what follows, in general refer to the angular velocity or number of revolutions per minute of the disk, obtained by multiplying all pedal speeds by 3.25, the ratio of the two sprocket-wheels. 31 32 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE Hence the currents in the disk would be proportional to a>B (T where o- is the specific resistance of the disk. The rate of production of heat would then be proportional to across the disk will thus be diminished by a certain amount which we will call ', the "counter flux" due to the eddy currents. This diminution in flux may be assumed for moderate speeds to be proportional to the intensity i of the eddy currents and to the angle 0, or *' = M* (2) Thirdly, in accordance with the fundamental principle of electro- magnetic induction, we have where the actual resultant magnetic induction through the disk is From these assumptions (1), (2), and (3), the following equations may be derived, in which the product kjc 2 k 3 is replaced by a single con- stant k : As will be seen, these assumptions do not take into account all of the variables; nevertheless, it will be shown on p. 37 that equation (4) is roughly verified. The significance of equation (5), which represents the heat per revolution of the disk, will be discussed in a later paragraph. MEASUREMENT OF MAGNETIC FIELD BY MEANS OF A BISMUTH SPIRAL. It seemed desirable to measure not simply the total magnetic flux at different speeds, but the induction at a number of points in and near the air-gap as well. Among the various practicable methods, that of the bismuth spiral seemed best adapted for our purpose. Most of the observations described below were made with a Hartmann and Braun spiral, kindly loaned us by the Worcester Polytechnic Institute. The fine bismuth wire of this spiral, coiled into a flat disk about 17 mm. in diameter, had a resistance under normal conditions of about 20 ohms. A small portion of the work was done with a second spiral, similar to the first, and the results obtained with the two instruments agreed very well. Unfortunately we did not have at our disposal a spiral of smaller diameter. In order to make it possible to introduce the bismuth spiral into the narrow gap between pole-face and disk, it was necessary to shift the electro- magnet slightly, bringing one of its faces almost into contact with the disk, while the gap on the other side was correspondingly widened. The effect of this change on the heat per revolution was considered in Part II. Even with this increased air-space on one side of the disk, it was not easy to bring the spiral into the center of the field without its being chafed by the 34 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE disk. Hence but few observations were made in the center of the field, and no reliable ones were obtained there when the disk was rotating. During the magnetic observations the ergometer was mounted inside the calorimeter, but the front of the calorimeter was open and no attempt was made to allow the thermal conditions to reach a steady state; each speed was generally maintained for only a minute or less. Hence in general the temperature of the disk was somewhat lower than during the calibra- tion tests. The bismuth spiral was clamped securely in a holder that was capable of being moved parallel to itself in various directions. In nearly all cases the exciting current in the electro-magnet was 1.25 amperes and in the few remaining cases the results have been corrected to this value. Resistances were measured with a Wolff Wheatstone bridge and sensitive galvanometer. The spiral received heat by radiation from the copper, and by con- duction from the strong current of air when the disk was in motion. As this made a direct determination of its temperature impossible, it was decided to estimate the temperature from the resistance of the bismuth when the magnetic field was off. This resistance was measured at fre- quent intervals and the temperature computed with the aid of the resist- ance temperature coefficient of bismuth. Thus, in a typical group of observations at each position of the spiral, the following resistances were observed: (1) magnetic field off, disk stationary; (2) magnetic field on, disk stationary; (3) field on, disk running at two or more speeds in suc- cession, in many cases repeating in reverse order; (4) field still on, disk stationary; (5) field off , disk stationary. Allowance was made whenever necessary for the drift in temperature between observations (1) and (5). In general, the mean of (1) and (5) gave iv , the resistance of the spiral in a magnetic field of intensity (practically) zero. The remaining observations gave values of w f , the resistance with field on, at various speeds. In most cases the speeds were 0, 11, 60, and 112 revolutions per minute of the pedals, or 0, 36, 195, and 364 revolutions per minute of the disk. For each speed the value of iv f w was corrected for temperature, and from this the induction in gausses was obtained from the calibration curve furnished with the spiral. At the conclusion of each set of observations the spiral was advanced a millimeter or so and the observations repeated. Most of the magnetic distortion was to be looked for along lines paral- lel to the direction in which the portion of the disk between the poles was moving, i.e., along the line AB in fig. 16. Nearly all of the observations were accordingly made along this line and they will be considered first. The results are shown in fig. 15, in which the abscissae represent dis- tances in millimeters measured from the center of the field along the line AB of fig. 16. Positive values lie in the direction in which the disk is supposed to be rotating. The heavy vertical lines G G' in fig. 15 indi- MAGNETIC KEACTIONS 35 cate the position of the edges of the magnet pole; thus G' is the "trailing tip." The curves in heavy lines show the observed induction in gausses at different angular velocities. The number of revolutions per minute of the disk is indicated on each curve. To avoid confusion, the individual observations are omitted, except in the case of one curve. The points for -40mm -20 40mm FIG. 15. Magnetic induction across air-gap. Direction of motion of disk is to right. G, G' indicate position of edges of magnet poles. 36 A BICYCLE EEGOMETER WITH AN ELECTRIC BRAKE the other curves agree among themselves to about the same degree of closeness as these. Owing to the unsatisfactory character of the obser- vations in the middle of the field when the disk was in motion, but little weight was placed on these data, and the curves are accordingly shown as broken lines in this region. Since the bismuth wire was coiled in a spiral about 17 mm. in diameter, it is clear that these curves can not show accurately the precise form of the magnetic field. A simple consideration shows that if the curves could be drawn with precision they would slope more steeply than the curves here drawn; they would then cross the lines G G' at points higher up, and the maxima would all be higher. Still, crude as they are, they show clearly the reaction of the eddy currents in the disk. The curve obtained with the disk stationary (speed 0), is quite sym- metrical, showing slight maxima close to the edges of the poles. As the speed increases, the distortion of the magnetic field and the marked decrease in flux at high speeds are very evident. From the curve for speed 364, it might be inferred that here the induced current is confined entirely to a narrow path close to the trailing edge of the pole-face. That this is the case will be shown later. Since the ordinates of the curves for speeds 36, 195, and 364 represent the resultant induction through the disk, it is evident that the algebraic difference between these ordinates and those for speed must be a measure of the magnetic field that would be produced by the induced currents alone. These differences are plotted in fine lines. Negative ordinates signify a component opposing the flux from the electro-magnet. The most striking feature of these curves is the very pronounced demag- netizing field produced in the disk at high speeds. The points where the curves cross the axis of abscissae show that the displacement of the cur- rents in the direction of rotation increases with the speed (eq. (1)), though at a lower rate. It is presumably near these points that the induced currents attain their maximum values. A few observations were made with the bismuth spiral in other posi- tions. The induction was found to be practically uniform when the spiral was moved in a radial direction, except close to the outermost edge of the magnetic field near the circumference of the disk, for example at the point P in fig. 16. Here the flux density was found to increase with increasing speed, as would be expected, for the demagnetizing effect of the currents must lead partly to a diminution in the total flux around the magnetic circuit, and partly to increased leakage around the outer edge of the disk. Indeed, the currents along the edge of the disk on the side approaching the magnet flow in such a direction as to bend the lines of magnetic induction outward around the edge of the disk. When the spiral was laid flat against the side of the magnet pole, with its plane perpendicular to the disk, it showed a decrease of about 30 per cent in magnetic induction on the "leading" side, while on the MAGNETIC REACTIONS 37 "trailing" side the induction was about doubled when the disk was run- ning at 364 revolutions per minute. COMPARISON OF RESULTS WITH THEORY. Although, for the reasons given, the curves in fig. 15 do not represent the facts quite accurately, still it is worth while to inquire how well they satisfy the conditions expressed in equations (4) and (5). In these equa- tions it is necessary to know the value of , the resultant flux at angular velocity w, and 9', the "counter flux" at the same angular velocity. From the areas of the distorted curves in heavy lines 9 is obtained, and 9' from the areas of the curves in fine lines (algebraic sum of negative and positive lobes). The areas were taken arbitrarily between 40 and +45 mm., since outside of these limits the ordinates are small. From the areas and the measured dimensions of the pole-faces, the values shown in columns 2 and 3 of table 7 were obtained. TABLE 7. Magnetic fluxes at different speeds. ft) 4> If ft> 2 T>X !0" 3 9 ft>p 2 X 10~ 9 36 65,000 800 105 152 195 364 42,200 27,700 23,600 38,100 68 96 348 279 Since only relative values are required, 2 ^> in column 4 should, by equation (4), be equal to j-, a constant. This condition is satisfied as well as could be ex- pected, considering that the increase in a from the first to the third ob- servation is ten-fold and the increase in 9' is nearly fifty-fold. It must also be remembered that as the speed increases, the distribution of the current paths and hence also the resistance of the paths may change to a marked degree. In other words, our assumptions considered only the total flux, although theoretically the distribution of magnetic induction and of the current loops in space ought to be considered. Moreover, since the magnetic lines pass between pole-pieces of limited extent, it is possible that our first assumption, equation (1), does not hold at the higher speeds. Changes of a- with temperature can hardly have affected the result materially; but, on the other hand, the lack of reliable obser- vations in the central portions of the field lends an element of uncertainty. In equation (5) we expressed the relation 209177 38 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE The left-hand member of this equation is proportional to the heat generated per revolution of the disk, since the numerator represents the rate of production of heat, while the denominator indicates the number of revolutions per minute. We are thus in a position to obtain relative values for the heat per revolution, based on magnetic data alone, which can be compared, for the same current in the electro-magnet (1.25 am- peres), with the calibration curve of the ergometer (fig. 13). Since k 3 is a constant and the temperature of the disk changed but little during the magnetic tests, it is sufficient to compute the values of cocf> 2 at various speeds and to plot these values as functions of the speed. The values of o>$ 2 corresponding to the three observed values of are given in table 7. In order to draw the entire curve, it was necessary first to find the relation between and co. This relation, which can be derived from our fundamental assumptions, is where a and 6 are constants. The equation is roughly satisfied by our observed values of and o>, but we considered it better to obtain values of < corresponding to various values of co from a curve connecting these quantities. Since the curve was nearly a straight line over the observed range, the interpolation was simple. To facilitate the comparison with the ergometer calibration curve for 1.25 amperes, all of the values of 4> 2 curve is shown as a fine line. It has the same general form as the calibration curve, but its maximum comes at a lower speed. This is no doubt due in large measure to the sources of error already mentioned. But it may also be due partly to the fact that since no attempt was made in the magnetic tests to reach thermal equilibrium, the copper disk was, for the same speed, cooler during the magnetic tests than during the calibrations. At low speeds, where is nearly constant, the relatively small value of a- during the magnetic tests would make the heat per revolution relatively high. But at high speeds a smaller value of . Since equation (5) shows that the heat per revolution varies as the square of <, the result will be a relatively small value of o>c/> 2 . A rough calcu- lation shows that the correction from this cause would amount perhaps to 5 per cent, raising the ordinates to the right of the maximum of the &> 2 curve slightly, and reducing those to the left. Nevertheless, aside from minor discrepancies, the similarity of the two curves is very striking, proving beyond a reasonable doubt that the peculiarity in the ergometer calibrations is due almost entirely to the demagnetizing effect of the eddy currents in the disk. The increased temperature of the disk at high speeds, by reducing the intensity of the currents, enhances this peculiarity, but only to a minor degree. MAGNETIC REACTIONS 39 FURTHER EXPERIMENTS WITH THE EDDY CURRENTS. The great intensity of the currents in the disk was also made evident by the following quite elementary experiments: (I) Compass tests. A small pocket compass held near the disk showed the presence of a strong magnetic field due to the eddy currents, even at a considerable distance from the electro-magnet. One way of testing this was to trace out the magnetic lines parallel to the surface of the disk by the usual step-by-step method, holding the compass with its plane vertical like a dip needle, close to the disk near one pole of the magnet, and then advancing it by stages parallel to the disk and along the direction of the lines. In fig. 16 the heavy lines marked were thus obtained with the disk stationary, showing the direction of the stray lines from the electro-magnet. 1 The dotted lines marked 390 were obtained when the disk rotated at 390 revolutions per minute. In this figure the north pole of the electro-magnet is on the side toward the observer and the disk rotates counter-clockwise. Observations at points on the other side of the magnet pole showed a corresponding change in the direction of the resultant magnetic field when the disk was in rotation. The point Q, just outside the disk, is a neutral point, where the field due to the eddy currents is equal and opposite to that due to the magnet. (II) Galvanometer tests. The copper leads from a sensitive gal- vanometer were touched to the surface of the disk at points from 1 to 5 mm. apart, the points being so oriented that the galvanometer showed no deflection. Care was taken to reduce the effect of thermo-electric forces to a minimum. This is the old method used by Faraday and Nobili for plotting the lines of current flow. Though it can not always be as- sumed that the current flows in a direction perpendicular to the line joining these "equipotential" points, still they furnish an approximate idea of the direction taken by the current paths. A few such pairs of points are indicated in fig. 16, and with their aid some of the current lines have been constructed, the arrow-heads indicating the direction of flow. These lines must not be confused with the magnetic lines described above. Tests made close to the magnet pole proved that at 390 revolutions per minute the inwardly directed current lines were confined to a narrow band about a centimeter wide, near the trailing edge of the pole, as shown. The demagnetizing effect of the currents is here very evident. (III) Intensity of the eddy currents. The galvanometer leads were touched to the disk, as described above, at a point near the magnet pole, but oriented in such a way as to produce a maximum deflection. From the distance between the points of contact and the resistance and sen- sitiveness of the galvanometer, the potential difference between the points was found, and from this and the specific resistance of copper 1 At the time of these tests the magnetic poles were pushed in about 2 cm. from the outer edge of the disk. This can hardly have produced an appreciable change in any of the quantities observed (cf. fig. 13). 40 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE the current density was found to be of the order of 650 amp. /cm 2 . This was at about 300 revolutions per minute of the disk. As a rough check on this, the electromotive force induced in the copper was computed from the observed flux and the speed of the disk. The potential gradient was found to be of the same order of magnitude as FIG. 16. Magnetic lines and current loops on surface of rotating disk. Long arrow shows direction of rotation. that derived from the galvanometer observations above, namely, about one-thousandth of a volt per centimeter. From these data we estimate the total current in the disk to have been not less than 2000 amperes. (IV) Effect of eddy currents on the flux through the magnet coils. In order to measure the diminution in total flux when the disk was MAGNETIC REACTIONS 41 running, a single turn of wire was wrapped around one of the magnet coils and connected to a ballistic galvanometer. The throw was measured when the field current was turned on, and again when the disk was sud- denly set in rotation. The latter throw was always in the opposite direc- tion to the former; its measured value was certainly somewhat too small, since it took an appreciable time for the disk to attain full speed. The results indicated a diminution of the total flux amounting to only about 4 per cent, when the disk rotated at 320 revolutions per minute. Even allowing for the gradual acceleration of the disk, it is apparent that the reaction of the eddy currents causes chiefly an increased magnetic leak- age, without greatly diminishing the flux through the coils. The diminution of the flux on starting the disk causes a slight momen- tary increase in the current through the electro-magnet, while suddenly stopping the disk diminishes the magnetizing current for an instant. This is analogous to the momentary changes produced in the current through a coil of wire when an iron core is moved in and out. Soret 1 seems to have observed this effect first. On the other hand, Jacobi 2 asserted that the magnetizing current was diminished when the angular velocity of his disk was increased. If we understand his paper aright, this must have been an error. (V) Effect of eddy currents on permanent magnets. It is of interest to consider briefly the effect of moving masses of metal on permanent magnets. If the pole of a bar magnet is held close to a rapidly revolv- ing copper disk, its moment is permanently weakened. This method is sometimes made use of in the artificial seasoning of horseshoe magnets. In the design of at least one type of speedometer, this demagnetizing action is especially guarded against in an ingenious manner. If one of the magnet systems of a Kelvin galvanometer employing astatic needles is inclosed in a copper damper, this system undergoes a slight demagnetizing action at every swing. Thus in time the astaticism of the systems must be perceptibly impaired, unless the needles are very well hardened. The currents induced in masses of metal moving relatively to per- manent magnets must, at the beginning and end of the motion, induce eddy currents in the magnet itself. If the acceleration is the same on starting and stopping, these currents can have little to do with the demag- netization of the magnet, for they flow in a direction tending to increase the magnetization when the motion begins, and tending to decrease it when the motion ceases. The case is analogous to moving the keeper of a horseshoe magnet rapidly up against the poles, which causes de- magnetizing eddy currents to flow, while suddenly pulling off the keeper gives rise to currents in the opposite direction. 1 Soret, Comptes rendus, 1857, 45, p. 301 . 2 Jacobi, Comptes rendus, 1873, 74, p. 237. 42 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE INFLUENCE OF TEMPERATURE ON THE CONSTANCY OF THE BICYCLE ERGOMETER. From what has preceded, it is clear that the rate of heat production varies inversely as the resistance of the rotating disk, and hence that the heat per revolution varies in the same manner. Over the usual range of room temperatures, it may be assumed that the same expenditure of energy in the disk raises its temperature to the same extent above its surroundings. If the ergometer is used outside the calorimeter in a room at the same temperature as that inside the calorimeter during calibration, the results of the calibrations can be applied without correction, provided the circulation of air is approximately the same in the two cases. But if, for example, an accuracy of 2 per cent in the energy measured is desired, then, since the temperature coefficient of copper is approximately 0.004, a temperature correction will have to be applied if the temperature of the room differs by more than 5 C. from the mean temperature inside the calorimeter during calibration. In general, during the work that has been done thus far with the ergometer, no such correction has been neces- sary. The highest observed temperature of the disk (see Part II) was 43 C. at a pedal speed of 1 20 revolutions per minute, the room tempera- ture being 20 C. It was to be expected that as the speed increased the maximum temperature would occur at a higher speed than the maximum value of the heat per revolution, since the maximum temperature depends on the heat per second, i.e., it is proportional to the heat per revolution multiplied by the speed. In using the ergometer for accurate quantita- tive measurements, care should always be taken to maintain each speed long enough for the temperature of the copper disk to reach a sufficiently steady state. For practical purposes, this precaution is seldom necessary. THE DESIGN OF ELECTRIC BRAKES. In conclusion, we will summarize briefly the general principles that ought to be considered in the design of apparatus employing electro- magnetic damping, particularly with reference to the demagnetizing effects of the eddy currents. We shall base our deductions in part on the equation (6) derived from our fundamental assumptions on p. 32. is the impressed flux when the disk is stationary, as a limit. Hence in order to minimize the demagnetizing action for a given amount of power to be absorbed, it is best to use a large magnetic flux and a low speed. (e) Size and shape of pole-piece. The most important quantity is the width, measured in a direction tangential to the disk. The current paths may be regarded as consisting of two parts, one lying in a radial direction under the pole, in which the currents are induced, and the other consist- ing of the remainder of the disk, in which the circuits are completed. If the polar area is small in comparison with the area of the disk, it follows that the first portion mentioned will contain most of the ohmic resistance of the circuits, since the lines of flow are here very constricted. Hence the resistance may be assumed to be inversely proportional to the width of pole. If now the same total flux be spread out over a pole-face n times as wide, the total current will remain unchanged, while the production of heat and therefore the consumption of energy will be - as great. On the other hand, if the magnetic induction remains constant, so that the total flux varies directly as the width of pole, the consumption of energy will also vary in the same manner. The demagnetizing effect will probably be somewhat less with a broad pole, since the same angular lag will then not bring the demagnetizing 44 A BICYCLE ERGOMETER WITH AN ELECTRIC BRAKE system of current loops so directly under the pole. This is the case in the damping disk of watt-hour meters, which in addition to broad pole- faces employ thin disks and low speeds, thereby reducing the demagnet- izing factor to a minimum. Lengthening the pole-face in a radial direction will, by reasoning analogous to the preceding, cause a proportionate increase in the ex- penditure of energy if the flux density is kept constant, and a decrease in the same ratio if the total flux is constant. (/) Intensity of magnetic field. The consumption of energy varies as the square of the flux density. The percentage of demagnetization from the eddy currents is a constant for the same speed, independent of the field intensity. This explains why the maxima of the calibration curves in figs. 8 and 14 all occur at practically the same speed, whatever the current in the electro-magnet. (gr) Reluctance of the magnetic circuit. To insure a "stiff" field, re- sisting the demagnetizing action of the eddy currents, it would be advan- tageous to use a magnetic circuit of relatively large reluctance and large magnetomotive force, with strongly saturated poles. Crowding of the flux in the neighborhood of the trailing edge of the pole could be reduced by widening the air-gap on that side of the magnet, or by using split pole- pieces, like those in the Lundell generators. By inserting a variable air-gap in the magnetic circuit, the maximum of the calibration curve could probably be shifted to the right or left. (h) Location of magnet poles. These should be far enough from the outer edge of the disk to minimize magnetic leakage around the edge. The entire magnet should be shaped in such a way as to reduce the leak- age, especially in the neighborhood of the poles. This requirement is met, for example, in the permanent magnets of watt-hour meters. It is true that our calibration curves (fig. 13) do not show any less evidence of demagnetization when the poles are pushed 2 cm. nearer to the center of the disk, but this is because there was still considerable opportunity for magnetic leakage, owing to the construction of the magnet. Thus on the whole it will be seen that, for maximum expenditure of energy, it is advantageous to use small magnet poles, while to minimize the magnetic reaction the poles should be broad. The best compro- mise between these opposing factors can only be reached by experiment. In any case, the magnetic field should be as intense as possible. University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. JfxN JUL librar AT LOS ANGELE LIBRARY UC SOUTHERN REGIONAL LIBRARY FACILITY