The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 5 of 35

See also Dr. J. Hopkinson, " Magnetisation of Iron," Trans. Royal Soc., Part II., 1885, p. 455.

A very complete summary of recent research in magnetism is to be found in Prof. Chrystal's article, " Magnetism," in the Encyclopaedia Britannica, Ninth Edition.

See also '' On the Lifting Power of Electro-Magnets and Magnetisation of Iron," Shelford Bidwell, Proc. Royal Soc , June 10, 1886.

Other references to valuable Papers on the magnetisation of iron are —

Lord Rayleigh, " On the Energy of Magnetised Iron," Phil. May., August, 1886, 'p. 175.

Lord Rayleigh, " On the Behaviour of Iron and Steel under the Opera- tion of Feeble Magnetic Forces." Phil. Mag., March, 1887, p. 225.

Ewing and Low, " On the Magnetisation of Iron in Strong Fields '' Proc. Royal Soc., March 24, 1887, Vol. XLIF., p. 200.

ELECTRO-MAGNETIC INDUCTION. 59

force, and the induction is ascertained when we know the quantity of electricity discharged through the galvanometer on a reversal of the primary current. If the secondary coil makes N' turns round the iron, and if the whole resistance of the secondary circuit, including galvanometer, is equal to R', and if S is the cross-section of the rod, then 2 B S N' is the total change in induction through the secondary circuit obtained on reversing the primary current, and this is equal to R,' Q, where Q is the quantity of electricity discharged through the galvanometer. After finding in this way a series of values of B and the corresponding values of H, we have the means of determining the values of p. for varying values of B. If, as in the above case, the iron rod is not endless, the values of p, so determined will be smaller than the correspond- ing values of p. for an iron ring of the same iron, and will be less as the iron rod is made shorter, because a larger propor- tion of the magnetic circuit is then formed of air and a less portion of iron.

jected to a cycle of magnetising force in which the force beginning at zero rises up graduaUy to a maximum in one direction, and is then reversed and made a maximum in the other direction, and finally reduced again to zero, we find that the following phenomena exhibit themselves. The induction in the iron — and, therefore, its magnetisation — has a higher value at all points during the descent of the force than during its ascent. Hence, if a curve is plotted in which horizontal abscissae represent magnetic force and vertical ordinates induc- tion or magnetisation, we obtain a curve of the kind shown in Fig. 20, which is a loop or encloses an area. If at any point in the cycle we stop and reverse the magnetism a small or subsidiary loop (see Fig. 21) is formed on the principal curve. This phenomena is called magnetic hysteresis, because the mag- netism or induction "lags behind" the magnetic force. If the induction B and the magnetic force H are the variables in terms of which the curve is plotted, the curve is called a B H curve of hysteresis. We may next consider the physical meaning of this curve. Consider as before a ring of iron of cross-section S and mean perimeter I wound over with N

60 ELECTRO-MAGNETIC INDUCTION.

turns of a magnetising coil. Let a current of A amperes be sent through this coil, there are then N A ampere-turns acting

on it, and the magnetic force operating on the iron is _ — .

At any instant let the difference of potential at the ends of the magnetising coil be V volts, and let the coil have a resistance of K ohms. If the induction in the iron has a value B, the total number of lines of induction linked with the coil is BSN.

C M><jn<lilinq Fore. H

Complete Magnetisation Curve for Soft Iron Ring carried from strong positive to strong negative magnetisation. The arrows show the direction of the magnetising operation, and the shaded area the work done due to

hysteresis.

If we make a change in the potential and increase the volts to V + 8V, and at the same time increase the current from A to A + 8 A, and the induction from B to B + SB, in the small time 8 1 the following relations between these increments will exist : The time rate of change of the induction is in the

ELECTRO-MAGNETIC INDUCTION. 61

limit equal to f_, and the time rate of change of the whole number of lines of induction linked with the circuit is

7 R

S N _-. Hence this last is numerically equal to the induced electromotive force set up in the circuit by this change ; and by considering the direction of this induction it will be seen that this induced electromotive force is opposed in direction to the

FIG 2'.

Magnetisation Curve for Very Soft Iron Ring, showing loops due to hysteresis on the descending branch, and enclosed area due to a complete cycle (Evving).

impressed electromotive force V producing the current. This induced electromotive force reckoned in volts is equal to

—^ ^--?. Hence the current A in amperes must be equal at

any instant to the resultant electromotive force divided by the resistance of the circuit ; or

V- SN (/B

10s dt

62 ELECTRO-MAGNETIC INDUCTION.

Therefore, multiplying all through by A and d t, we have

V Kdt = EA?dt + ^ A<7B. But the magnetising force H in the iron at that instant is

equal to - A. Hence

. . (18)

The first term of this equation represents the whole energy in joules given to the ring coil and core in the small time dt ; the second term represents the energy wasted in that time in heating the coppeT magnetising coil ; and the third term must therefore represent the whole energy, measured in joules, absorbed or wasted in the iron core in that same element of time. Accordingly it is easily seen that the integral

-L f H d B,

4;r J

(19)

taken between any limits, must represent the whole energy measured in ergs, dissipated by a unit of volume (viz., one cubic centimetre) of the iron core in the time limits of the integral. If the time limit is the interval of time occupied in making one complete magnetic cycle, then the above integral will represent the energy dissipated per unit of volume of the iron in this cycle estimated in ergs. But if the diagram of the magnetic cycle is drawn in terms of B and H, the integral

f H d B taken over the whole limits of the cycle is the value

of the area enclosed by the induction curve. Finally, there- fore, we reach this rule. If a mass of iron is taken once round a magnetic cycle, and an H B diagram is drawn, showing the relation of induction to magnetising force during the cycle, one centimetre or one unit of length along the hori- zontal being taken as equal to one C.G.S. unit of magnetic force, and one centimetre or unit of length along the vertical being taken for one C.G.S. unit (one line) of induction ; then l/4:r of the area of this closed loop in square units is equal to the energy wasted in such single magnetic cycle measured in ergs. The physical meaning of these loops or enclosed areas in magnetisation curves of complete magnetic

ELECTROMAGNETIC INDUCTION. 63

cycles was first pointed out by Prof. E. Warburg,* but also independently by Ewing. In the first place, we may remark that however slowly the magnetic cycle may be performed, this waste of energy always takes place, and it is therefore seen to be dependent essentially on the reversal or change of magnetism and not upon the production of local electric currents in the iron. This energy waste is called the hysteresis loss in the iron, and it cannot be got rid of by any amount of lamination or division of the iron. Careful measurements have shown what is the value of this hysteresis loss in iron of various kinds.

In a Paper entitled " Researches in Magnetism " (Phil. Trans., Part II., 1885), Prof. Ewing has given the values of the energy dissipated in ergs per cubic centimetre, experi- mentally determined for complete magnetic cycles performed on various samples of iron, as follows : —

Energy dissipated in ergs per

Sample of iron operated cubic centimetre during a coin-

upon. plete cycle of doubly-reversed

strong magnetisation.

Very soft annealed iron 9,300 ergs.

Less soft annealed iron 16,300 ,,

Hard drawn steel wire 60,000 ,,

Annealed steel wire 70,500 ,,

Same steel, glass hard 7G,000 „

Pianoforte steel wire, normal temper. . 116,000 ,,

Same, annealed 94,000 ,,

Same, glass hard 117,000 ,,

If we make one hundred complete cycles of magnetisation per second, the power absorbed per cubic centimetre of metal

estimated in watts is as follows : —

Power wasted in watts

per cubic centimetre for

Cample. 100 ( oj ) cycles per second.

Very soft annealed iron 0 '093

Less soft annealed iron 1 63

Hard drawn steel wire 6 '00

Annealed steel wire 7 '05

Same steel, glass hard 7'GO

Pianoforte steel wire, normal temper ... 11 '600

Same, annealed 9 '40

Same, glass hard 11700 _

rVViedemann's Annalen, XIII., p. 141, 188 L.

64 ELECTRO-MAGNETIC INDUCTION.

From the above we can deduce that, roughly speaking, it requires 18 foot-pounds of energy to make a double-reversal of strong magnetisation in a cubic foot of soft iron. The energy so expended can take no other form than that of heat diffused throughout the mass.

A similar table of experimental results has been given by Dr. J. Hopkinson (Trans. Roy. Soc., Part II., 1885, p. 463), in which Paper the chemical analysis of the samples operated upon is given. The highest value of specific hysteresial dissi- pation was found for Tungsten steel, oil hardened, in which the value of the energy in ergs per cubic centimetre dissipated in a complete magnetic reversal was 216,864.

Hysteresis is therefore a quality of iron in virtue of which reversal of magnetisation is accompanied by dissipation of energy. The energy so wasted is, of course, converted into heat. This dissipation of energy into heat during magnetisa- tion is something quite apart from any production of heat by eddy (or so-called Foucault) electric currents induced in the mass, and would take place in iron so perfectly divided that no eddy currents could exist.

One result of Prof. Swing's researches has been to show that if the iron is kept in a state of mechanical vibration hysteresis is greatly diminished, and the value of the energy dissipated in a complete cycle is ?nuch reduced. The removal of strong residual magnetism from soft iron by slight tapping or twisting has also been noticed and commented on by Prof. Hughes.*

Hysteresis, therefore, is a source of dissipation of energy in the armature of dynamos. For in this case we have a mass of soft iron, viz., the armature core, which has its direction of magnetisation reversed every revolution. Suppose the core has a volume of 9,000 cubic centimetres, and that it makes 15 revolutions per second. Taking the specific hysteresis for this sample of iron at 13,356 ergs, we find that the dissipation of energy in ergs per second is equal to 9,000 x 15 x 13,356 = 180x10" =180 joules, or a loss of about a quarter of a horse-power-hour.

• Prof. D. E. Hughes, "On the Cause of Evident Magnetism in Iron," /Voc. Soc. Tel. Engineers, Maj- 24, 1883, p. 3,

ELECTRO-MAGNETIC INDUCTION. 65

Experiments were at one time made by Joule and others to determine by direct observation the heating effect of magneti- sation upon iron, but in these early experiments it is probable that the results were mostly impure, and qualified largely by the production in the iron of heat by local or eddy electric currents.*

Since about 10,000 ergs per cubic centimetre are dissipated by a double reversal of strong magnetisation of soft iron, it is not difficult to show that the consequent rise of temperature, even if all the heat is retained in the iron, is 0-000284°C., or that some 4,000 reversals would be required to raise the temperature 1°C., even provided all the heat generated is retained in the metal.

If the iron is subjected to very rapid reversals of induction, and if it is in one solid mass, then, in addition to the hysteresis waste of energy, eddy electric currents are set up in the mass of the iron and create heat. In such cases it must be noted that the source of waste of energy in the iron is twofold : first, that due to true magnetic hysteresis, and, second, that due to eddy currents. The last source of waste can be prevented by sufficiently and properly sub- dividing the iron into very thin plates or wires, which are rusted or painted so as to prevent the production to any degree of eddy currents. The first source of waste cannot be prevented by any such lamination. The question has been very much debated and considered whether the true hysteresis loss in iron depends upon the speed at which the magnetic cycle is described — whether the dissipation of energy at, say, 100 reversals per second is or is not more than one hundred times that of one slowly performed cycle.

The matter seems now decided as follows : — It appears evident from the researches of J. and B. Hopkinson,f that if the induction density is moderate in amount (for example, not more than 3,000 or 4,000 C.G.S. units) then, whether the reversal of magnetism or cycle is made very slowly

• See Joule's " Scientific Papers," Vol. I., p. 123, " On the Calorific Effects of Magneto-Electricity," and " On the Mechanical Value of Heat.' Also Phil. Mag., Series 3, Vol. XXJIL, p. 263.

t The Electrician, September 9, 1892. See also Proc. Roy. Soc., London, April P.O, 1893, Vol. LIII., p. 352.

66 ELECTRO-MAGNETIC INDUCTION.

or at the rate of a hundred complete cycles per second, the area of the hysteresis curve remains practically unaltered. The same fact has been noted by Messrs. Evershed and Vignoles,* and also by Mr. Steinmetz. We may say, therefore, as the result of the latest work, that the difference between slow cycle and quick cycle hysteresis, once supposed to exist, is found to be non-existent, and that, at any rate for such induc- tions as are used in transformers, we can apply the results of the hysteresis losses obtained by slow cycle methods to the cases of reversals at about a frequency of one hundred or more.

From the researches of the Messrs. Hopkinson it is also clear that, for higher induction densities, there is a difference between the hysteresis loops for very slow cycles and for rapid ones, and this difference is chiefly in that part of the the curve preceding the maximum induction. As Prof. J. A. Ewing has observed, after sudden changes of magnetising force the induction does not at once attain its full value, but there is a slight increase going on for some seconds. Dr. Hopkinson has remarked that this small difference between the curve as determined by very slow reversals and that as determined by very rapid reversals is a true time effect, the difference being greater between a frequency of 5 per second and 72 per second than between 5 per second and exceedingly slow cycles. There may be, there- fore, a true time lag of magnetism at the higher speeds, but for all such frequencies as are employed in ordinary transformer work, we may take it that the hysteresis loss is constant per cycle and equal to that obtained by slow reversals.

The reader should carefully note that, if the hysteresis diagrams are taken for a solid iron ring and an equal sized ring made of iron wire in a way afterwards to be described with the wattmeter, when rapidly alternating currents are employed, the area of the diagram will be greater for the solid than for the divided ring. The area of the hysteresis diagram, then, gives us the energy loss due both to eddy currents set up in the iron and also that due to true hysteresis. Hence, in such experiments as above described, the greatest

• The Electrician, September 16, 1892,

ELECTRO-MAGNETIC INDUCTION. 67

care has to be taken to eliminate all eddy current loss before we can draw any conclusions as to hysteresis proper. When a solid mass of iron is magnetised the eddy currents set up in the mass have to die away first before the iron attains its maximum magnetisation, because at every point in the interior of the iron the magnetic force due to the eddy current is in opposition to the external impressed magnetising force. Hence the effect of eddy currents is to delay the rise of magnetism. Over and above this, however, for strong in- ductions there seems to be a slow increase of magnetisation after the magnetising current has become constant. Prof. Ewing says : "I repeatedly observed that when the mag- netising current was applied to long wires of soft iron there was a distinct creeping up of the magnetometer deflection after the current had attained a steady value."

This time lag appears to be most manifest in the softest iron, and to be especially noticeable near the beginning of the steep part of the magnetisation curve.

In an investigation on the magnetisation of iron under feeble magnetic forces, Lord Rayleigh has also drawn attention to the fact that the settling down of iron when very soft or annealed into a new magnetic state is far from instantaneous.* He has shown that if the strength of the earth's horizontal magnetic field is called h, for unannealed iron and steel magnetising forces ranging from i h to y^oTy ^l caU forth pro- portional magnetisation — in other words, the susceptibility is constant over this range, and the value of the corresponding permeability is from 90 to 100, this small proportional magnetisation taking place independently of what may be the actual magnetisation of the iron, provided it is not very near the condition usually called saturation. The moment, how- ever, that the magnetising force is pushed beyond these limits the phenomena of hysteresis and retentiveness make their appearance.

The subject of hysteresis in iron is by no means yet entirely explored. The chemical and physical states of the iron exercise the greatest influence on its magnetic hysteresis, and high specific electrical resistance seems in general to be an

• Lord Rayleigh, " On the Behaviour of Iron and Steel Under the Operation of Feeble Magnetic Forces," Phil. Mag., March, 1887, p. 225.

F2

C8 ELECTRO-MAGNETIC INDUCTION.

index of large hysteretic power in iron ; and those elements, like manganese, which, when added to iron, increase its specific electrical resistance, have also an effect in increasing its hysteresis waste.

One important point in connection with magnetic hysteresis is the effect of rise of temperature of the iron upon the hysteresis loss. Experiments made recently by W. Kunz,* are instructive on this point. This experimentalist investi- gated in 1892 and in 1894 the effect of rise of temperature in producing a diminution of magnetic hysteresis in iron, and the following are the results obtained from a long series of observations of this phenomenon : —

Four kinds of iron, two of steel, and one of nickel have been the subject of investigation. Special difficulties occurred in maintaining the wire samples at the high temperature for the required length of time, and in the measurement of these temperatures, and therefore the methods are more particularly described.

To measure the temperature of the wire, thermo-electric junctions were used, consisting of a platinum wire twisted for about 1-5 cm. of length round a wire of platinum con- taining 10 per cent, of rhodium. Two such couples were used, whose free ends were brought well insulated to a mercury switch, by means of which each couple could be connected to a Deprez-d'Arsonval galvanometer having about 200 ohms resistance. The temperature of the junctions at switch and galvanometer was always the same — about 20° C. The calibration of these couples up to 300°C. was done by means of an accurate mercury thermometer, plunged with one junction of the couple into oil, which was warmed up in large beakers surrounded with asbestos and kept well stirred. The galvanometer deflections corresponding to the tempera tures from 40°C to 300°C were noted. From 300°C upwards, the calibration was done by utilising the known fusing points of certain substances melted by a gas furnace : the junction was placed in the molten material, the flame lowered, and the deflection noted when solidification began. Each couple was thus separately calibrated — the outside junctions being kept at a constant temperature by immersion in petroleum at 20°C.

• See Elektrotechnische Zgitschrift, 1894, No. 14., p. 194,

INDUCTION. 69

The substances used and temperatures measured, or assumed from Bornstein-Landolt's tables, were as follows :

Substance Warmed or Melted.

Temperature or Melting Point.

Galvanometer Deflection.

Oil

Deg. 40

Ifi-fi

73

27-2

104

38-0

140

50 .g

200

77 -fl

300

125 '4

KC1O,...

359

155 6

PbCl

498

244 '4

IK

634

330 2

KC1

734

393 '6

NaCl

772

417 '8

Na2SO4

861

474*2

To heat the wire under test a method already employed by Ledeboer was adopted— namely, winding an insulated platinum wire round the test wire, and heating it by passing a current through the platinum wire. The iron wire was contained in a porcelain cylinder having a suitable opening to take the thermo-couple, and round this cylinder was wound (non-inductively) the platinum wire. The insulation between the couple and the platinum was tested before each observa- tion. This platinum coil was surrounded by layers of asbestos, among which the wires of the thermo couple were led out. The tube thus formed was placed inside a glass tube and accurately centred by suitable packing. This tube was placed again in another glass tube with asbestos distance pieces. A current of hydrogen passing through the inner tube protected the wires from oxidation. To protect the magnetising coil against the high temperatures, a hollow tube of pure copper was placed between, and a stream of water kept passing through it, and this had to be insulated by asbestos from the glass tubes to prevent their breaking. Observation showed that no heat passed through this jacket. The two couples always agreed in their indications, thus showing that the wire was evenly heated.

As indicated above, the test wire was a long straight piece ; its magnetic condition was observed by magnetometer, by the "single pole" method. The magnetising coil is placed

70

ELECTRO-MAGNETIC INDUCTION.

vertically in the direction east and west from the magneto- meter, and the top end of the magnetised wire is on a level with the instrument. The alteration in the position of the pole due to differences in induction density affect the distance between the magnetometer and wire so little as to be negligible. The vertical component of the earth's field must be taken into account. In magnetising, the cycle was always performed a few times, until the curve became regular. Then a series of observations at varying temperature of a certain cycle was taken, generally with maximum B = 3,590 (about), until at the high temperature (830° C. or so) the magnetism dis- appeared. Then another cycle after cooling. Suitable arrangements were made for compensating for the effect of the magnetising coil itself on the magnetometer, and for demagnetising by reversals of a gradually diminishing current.

The strength of field in the middle of the solenoid is calculated from the well-known formula, w 47TNC H=ld T'

where N = number of turns, C = the current in C.G.S. units, and I = the length of the solenoid ; and the induction was calculated from the magnetometer deflection by means of the known value of the horizontal component of the earth's field.

The following Tables are for a maximum induction density of 3,590, and are the mean of four series : —

Material.

Temp.

Hysteresis Loss in Ergs.

Material.

Temp.

Hysteresis Loss in Ergs.

Deg.

Deg.

20

2,350

/

20

2,690

German annealed charcoal

290 470 656

728

1,600 1,204 710 550

Swedish J iron \

270 460 650 742

2,080 1,550 905 825

836

316 |

812

712

20

2,107

20

Indefinite

20

3,420

20

3,100

Soft

284

2,480

275

2,270

wrought < iron

468 656 744

1,750 821 800

Puddled iron '

460 560 656

1,730 1,310 979

20

900

744

777

|i

20

2,090

ELECTRO-MAGNETIC INDUCTION.

71

These values show, when plotted as curves, that the equation L = a - b t (where L is the hysteresis loss and t the temperature, and a and b constants) expresses fairly the law.

Material.

Temp.

Hysteresis Loss in Ergs.

Material.

Temp.

Hysteresis Loss in Ergs.

Deg.

Deg.

(

20

11,540

20

9,660

309 i 11,580

309

9,860

Hard patent j steel «|

526 660 790

6,040 2,200 1,180

Patent cast steel

468 560 640

4,950 1,985 1,614

!;

20

5,230

744

1,048

20

4,670

The above Tables are to be interpreted as follows : Taking a wire of the material named, it was subjected to a magnetic cycle of induction in which the maximum induction reached was 3,590 C.G.S. The wire being taken at a particular temperature, as given in the second column of the tables, a hysteresis loss, diminishing with rise of temperature, was found, the value of which per cycle is given in the third column.

The two kinds of steel referred to in the last two tables had, for ordinary temperatures, magnetic cycles in shape like a rhombus, altering in shape at about 300°, even increasing in area, and between this and 470° changing in form to that of an ordinary iron curve, and decreasing greatly in area. The character of the steel is lost after heating, as shown by the final observations, and it becomes also quite soft. There is, in these cases, no simple relation between hysteresis loss and temperature.

The following tables relate to charcoal iron : —

B.

Temp.

Loss.

B.

Temp.

Loss.

Deg.

Deg.

(

20

8,900

20

21,020

270

6,690

270

14,840

f,200 (

468 570

4,660 3,340

470 570

9,900 7,550

068

2,270

— .

—

744

2,168

—

—

72 ELECTRO-MAGNETIC INDUCTION.

For higher temperatures than 570, B = 14,400 could not be reached.

The results show that the above law also holds good, generally speaking, for higher values of induction density.

The corresponding results for steel show that the character- istics described for the lower values of induction density hold also for the higher. In all the above cases a new wire was taken for each series of observations. It is shown, therefore, that when an iron wire is subjected to repeated cycles of temperature and magnetisation the hysteresis loss decreases up to the fourth cycle of temperature, and then becomes uniform ; the results of each temperature cycle being ex- pressible as a straight line, but of different inclination. The steel wire has the first temperature cycle as already described, and the remainder behave like the soft iron.

In the case of nickel subjected to a cycle of maximum B = 3,590, it was found that the hysteresis fell with the increase of temperature, at first rapidly, and afterwards increasingly slowly : it fell from 11,420 ergs at 20° to 4,700 ergs at 288°.

One of the most important results is the fact that repeated cycles of magnetism at a high temperature reduce the hysteresis loss in iron very considerably, and it would appear that this is also the result of one cycle at a very high temperature.

The above results show that in soft iron a very marked decrease in the hysteresis loss takes place as the temperature of the iron is raised. Further reference to this matter will be made in reference to the losses of energy in transformer working.

§ 7. The Electromotive Force of Induction. — We have in § 2 enunciated Faraday's law of induction in terms of the variation of a quantity called the flux of induction through the circuit. It is possible to express the fundamental rule in a more elementary manner, and in a way which adapts it to explain every fact yet observed. It is as follows : — If any element of a conducting circuit is so placed in a field of magnetic induction that a movement of that element of the conductor or change in the field of induction causes lines of

ELECTRO-MAGNETIC INDUCTION. f3

induction to intersect it, it creates an electromotive force in the element of which the direction is perpendicular to the plane containing the lines of magnetic force and the direction of motion of the centre of the element.

This operation is called " cutting lines of magnetic force." We shall allude later to hypotheses which have been con- structed to suggest in some degree an explanation of the nature of this effect.

The simplest possible case which can be considered is when a short linear element, such as a straight wire, is made to move in a uniform magnetic field, in a direction perpendicular to the plane containing the field lines and the linear con- ductor, the direction of the length of this last being also perpenlicular to the direction of the field lines.

(

C ••''''

7]

V s'

.j

PB

t i

v \ L^>

/

A

H D

FIG. 22.

If A B (Fig. 22) is the element of length of the conductor of length L, and if A D represent in magnitude and direction one of the lines of induction of the uniform magnetic field, H, in which it is placed, AB being at right angles to AD, and if AC represent in magnitude and direction a displacement of A B taking place uniformly in one second, so that A B moves uniformly parallel to itself from position A B to position C G in one second, we have then three lines, A B, AC, A D, mutually at right angles, and representing respectively the length L, the velocity V, and the magnetic induction H.

The result of the motion is to generate in A B an electro- motive force E, numerically equal to the product HLV in consistent units. But since the sides of the parallelopipedon, or solid rectangle, are taken to represent respectively H, L and V, their product E represents the volume of the solid and the magnitude of the electromotive force of induction.

74 ELECTRO-MAGNETIC INDUCTION.

If the directions of A B, AC and A D are not orthogonal, but inclined, the same still holds good.

For let the field lines AD (Fig. 23) be supposed to be inclined at an angle 6 with the direction of the length of the conductor A B, and let the direction of motion of A B parallel to itself, represented by A C, be inclined at an angle <£ with A B. The strength of field estimated perpendicular to A B is H sin 6, and if A C represents the actual velocity of A B, or displacement in one second, then A C sin <£, or V sin <£, is its velocity in a direction perpendicular to its own length. The magnitude of the induced electromotive force E in A B is then numerically equal to L H sin 0 V sin <£, or to H L V sin 0 sin <j> ; but this expression also represents the volume of a doubly skew parallelopipedon or solid rhomboid ; hence, as before, if vectors be drawn representing respectively the length and

c

V,

A Ho

FIG. 23.

velocity of a conducting element, and also the field strength in which it is placed, the volume of the solid rhomboid described on these vectors as adjacent sides represents the magnitude of the electromotive force induced in the element.

The magnitude of this induced electromotive force is not in any way dependent upon the nature of the material of which this conductor is made. Faraday experimentally proved this ("Exp. Ees.," §193-201) by taking a double conductor com- posed of an iron and a copper wire twisted together and united at one end. On passing this double conductor through a magnetic field no induced current was detected in it by a galvanometer. This proved that the electromotive forces set up in each separate conductor were equal and opposite, and hence, since the lengths, field, and velocities were the same, no factor entered into the production of the effect, which depended on the nature of the conductor. From further

ELECTRO-MAGNETIC INDUCTION. 75

experiments with circuits partly metallic and partly electro- lytic fluids he inferred that in all bodies, whether what are commonly called conductors or non-conductors, or elec- trolytic conductors, identically the same electromotive force is brought into existence by moving the same lengths in the same way in the same magnetic fields.

When a metallic disc is rotated in a uniform magnetic field so that its axis of rotation is parallel to the direction of the field, there is set up a difference of potential between the centre and the edge. In this case we can tap off a current by an external wire connected to the centre and the edge of the disc.

We can now show that, starting with the elementary law above stated, as to the magnitude of the induced E.M.F. in an element of a conductor, we can deduce the other principle of the relation of the induced E.M.F. to the rate of change of the induction through the circuit.

Let A B C D (Fig. 24) be a conducting rectangle, of which the plane is perpendicular to the induction lines of a uniform magnetic field of strength H, the same being shown in plan on the figure ; let the circuit be capable of revolving about an axis 0 0 in its own plane, and let it be displaced through any angle, 0, as shown in elevation and plan in Fig. 20. If the frame is so displaced it is clear that the sides A C,BD "cut" across lines of magnetic induction, but that the upper and lower sides do not. During this displacement the vertical sides alone will be the seat of electromotive forces. Imagine this frame to revolve round the vertical axis with a uniform angular velocity <a, and at any instant t to have a position such that its plane makes an angle 6 with the plane normal to the lines of force. Let the length of the side A C be L and that of A B be E : the actual velocity of the side

A C is — -, and the strength of the field, in a direction per- pendicular to its length and its direction of motion at that instant, is H sin 6. Hence the electromotive force of induction

in the side A C is — - L H sin 6, and an equal and oppo- sitely directed electromotive force acts in the side B D at the same instant. Hence the total electromotive force acting

76 ELECTKO-MAGNETIC INDUCTION.