The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 3 of 36

A final, practical, and useful case is that of the predetermina- tion of the wagnetic force in the interior of a circular closed solenoid or endless helical current. Let a wooden ring of circular cross-section be wound over closely with insulated wire so that the turns of the wire are contiguous and one or more layers are put on. This is called a circular solenoid, and we can calculate the magnetic force for points in the

BlBCTRO^MAGlfETlO INDUCtlOlf. 23

interior when the circalar solenoid is traversed by a current. Let B be the mean radius of the solenoid, and a that of the mean circular section. If the wire is wound in one layer on a wooden ring, then a will be the mean between the half diameter of the section of the ring and the half diameter measured over all after the wire is wound on it.

The magnetic force is not the same at all points over the circular cross-section of the solenoid. To find out what it is at any point we may proceed as follows : — A solenoid of any size, meaning by that a spiral current with turns closely adjacent, is electrically equivalent to a bundle of elementary solenoids or spiral currents of exceedingly small cross-section. Consider such a very small-sectioned solenoid, which may be called a spiral filament. It may be obtained in practice by winding insulated wire of small size on a very fine knitting needle as a core, and then withdrawing the needle. The section of this solenoid being very small, the magnetic force in its interior is everywhere nearly the same over the cross-section, and if the spiral is long the force in the centre in the interior is equal to 4irnl, where I is the absolute current flowing in the wire, and n is the number of turns per unit of length. Let this long elementary solenoid be bent round into a circle so as to form a closed or endless solenoid, let x be the mean radius of the circle which it forms and let n^ be the number of turns of wire of the spiral in an arc of the solenoid equal to one unit angle in circular measure. Then, the number of turns per xmit of length of the spiral being n, we have n2;»np and we may write the expression for the magnetic force in the interior of the solenoid as

^^^^ (11)

X

A little consideration will then show that the magnetic effect of any circular solenoid must be the same as that of a bundle of elementary solenoids, so wound and arranged as that the number of turns of wire of each spiral per length of arc subtending one unit angle in circular measure is the same. The magnetic force in the interior of the circular solenoid at any point in the cross-section is then equal to the product of 4irn| I, and the reciprocal of the perpendicular distance

24 ELECTBO-MAQNETIC INDUCTION.

of this point from the axis of the circular solenoid. The force over the cross-section is not uniform, but has a particular value for every point, but the same value at all points at an equal radial distance from the axis of the circular solenoid or ring coil.

§2. MagnetomotlTe Force and Magnetic Induction. — We

have in the foregoing section defined magnetic force, and shown how it can be determined in a few simple cases from a fundamental principle. If any line is drawn in a magnetic field of force, and we sub-divide this line into very small elements of length, and estimate the magnitude of the mf g- netio force at the centre of each element resolved in the direction of this element, and then sum up all the products obtained by multiplying the length of each element by the strength of the magnetic force along its direction, we obtain the line integral of magnetic force along that line. This is also called the magnetomotive force along that line.

In mathematical language, if H is the magnetic force at any point on the line, and 0 the angle this force makes with the line and ds, bjx element of length of that line at that point, then

JH cos Sdaia the line integral of magnetic force along that

line. From the definition of magnetic force, it is clear that this line integral is the work done in carrying a free unit magnetic pole along that line. This magnetomotive force along a line is likewise called the difference of magnetic potential between the two ends of the line. In those cases in which the magnetic force has a uniform value and is in the direction of the path chosen, it becomes a simple matter to calculate the magnetomotive force along that line, for it is the simple product of the numerical values of the magnitude of the magnetic force and the length of the line.

Let us consider two simple cases. First, when the line integral is taken along a closed line or loop in a magnetic field drawn in air or other non-magnetic medium, but not linked with or encircling a circuit conveying a current. In this case the value of the line integral is zero, because no work is done in carrying a free pole around a closed path in an air field. Second, when the line integral is taken along a path which is

ELECTRO'MAGNETIC INDUCTION. 25

a closed loop, and whieh surrounds or is linked with a circuit conveying an electric current. Consider the simplest case. Let a straight wire convey an electric current C, the return being at a great distance. Describe a circular line round the wire at a distance r from the axis of the wire. The length of this line is 2ir r ; the magnetic force at a distance r from a

2C straight wire is — units ; and the line integral along this

T

20 line is 2?rrx -'^iirG. Hence the line integral of the

magnetic force taken once round the circuit is 4?r times the total current through the line of force. This can be shown to be generally true, and is the general relation between magnetic force and current.*

If a looped line is taken through a helical current which links itself round the line n times, then, if A amperes traverse the conductor, the total quantity of current flowing through the loop is n A (equal to the ampere-turns), or in absolute

O.G.S. measurement is ^; hence the line integral of the

magnetic force taken along any closed line threading n times

through the circuit of a current A is r^r n A, or 1^ times the

ampere-turns of the current which are linked with the closed

4ir line. It is useful to remember that the value of -^ is very

nearly 1-25. ^^

We have here introduced the student to the notion of a

line integral. Another similar mathematical idea which has

to be grasped is that of a mr/ace integral. If, in any field of

magnetic force, we describe a surface of any form bounded by

a closed line, the magnetic force at all points of this surface

will have a certain value, call it H. Let the surface be

supposed to be divided in a number of very small elements

of surface each equal to d S. At the centre of each element

estimate the value of the magnetic force perpendicularly or

normaUy to that element, take the product of the value of

this normal value and that of the element of area. If 6 is

the angle between the direction of the force and that of the

• See EUctrieian, YoL X., p. 7 : Mr. Oliver Heaviside oq " The Relation between Magnetic Foroe and Electric Current."

26 mlmcthO'Magnmtic induction.

normal to the surface, then the product of the normal force, or H cos ^ and dS is to be taken for all elements of the surface. The sum of all such products is called the surface integral of the force, and is expressed in mathematical language by the integral/H cos0d8.

This is also called the flux of the force through the area, for if we suppose that, instead of dealing with magnetic force, we were considering the velocity of a moving fluid, the sm&ce integral of the velocity over any area would represent the whole quantity of liquid which flows in one second through the line bounding the area or the flux of the fluid.

Returning to the measurement of magnetomotive force, the reader will notice that, as a consequence of the above general theorem, in those cases in which we are dealing with the magnetic force due to a spiral current or solenoid making a number of turns round, or linkages with, the line of the magnetomotive force, the measurements of this magneto- motive force is practically made in ampere-turns, or by the products of the number of turns of the wire and the ampere current conveyed by it.

Owing to the fact that the circumference of a circle is 2v

4ir times its radius in length we get a numeric — r introduced

which makes the magnetomotive force, measured along a line linked with a line of current making n turns round it, numerically equal to 1^ times the ampere-turns, but it is not difficult to remember or to use this simple factor.

When magnetomotive force acts on any body, whether magnetic like iron or non-magnetic like wood or air, it produces in it an e£fect called magnetic induction. The student must think of magnetic induction as something which is produced by magnetomotive force, just as electric current is produced by electromotive force. Magnetic indifction is a quantity which is called a flux, and the magnetic induction in magnetic bodies results from magnetomotive force or difference of magnetic potential, just as the flow of water results &om difference of pressure or head of water, and the flow of electricity in conductors from difference of electric potential, and the flow of heat in thermal conductors from difference of temperature.

BLEGTRO-MAGlfBTtG IHTDUCTION. 2?

The quality in virtue of which magnetomotiye foroe can produce magnetic induction in a magnetic body is called its magnetic induetivity, or magnetic permeability, A body having laxge permeability is one in which a given magnetic force produces a relatively large magnetic induction. Similarly, we might say that the quality of bodies in virtue of which a hydrostatic force or pressure produces a flow of fluid through them is called their porosity^ and a very porous body is one in which a given hydrostatic pressure produces a relatively gr€At flow of liquid. Quite similarly we define electric and thermal cofiductivity as those qualities of bodies in virtue of which electromotive force and difference of temperature produces in them electric current or flow of heat. Hence magnetic permeability, electric conductivity, thermal conductivity, porosity, and, we may add, specific inductive capacity, in electrostatics are all analogous qualities of bodies numerically capable of being measured which are of importance in that they determine the amount of the flux produced by a unit of force of the corresponding kind.

The magnetic induction, like magnetic force, has a definite direction as well as magnitude at every point where it exists* Both magnetic induction and force belong to that class of quantities which in mathematics are called vector quantities^ and possess both magnitude and direction. In order to define the direction and amount of magnetic induction we fall back upon the fundamental discovery of Faraday. The ground fact of all his investigations is that if a conducting circuit is placed in a field of magnetic induction any change in the magnitude or strength of the magnetic induction will create an electromotive force in that circuit urging, or tending to urge, an electric current round it, provided that the jplane of this circuit has any btit one particular direction. We may, therefore, use a small conducting circuit to explore a field of magnetic induction, just as we employed a firee magnetic pole to explore a field of magnetic force. Let a small circular conducting circuit, formed, say, of one turn of very thin wire, and enclosing one unit of area, be placed in a field of magnetic induction. Let this smaU loop oi: unit circuit be held in various positions, and let changes be made in the induction by varying the magnetomotive

28 ELECTBO-MAGNETIG INDUCTION.

force. It will be found that there is a particular position or positions of the circuit in which no change in the strength of the induction produces any electromotive force in this smaU circuit. The direction of the induction at the centre of the circuit is then parallel to the plane of the circuit. The axis round which the circuit can be revolved without affecting this inactive condition of the circuit is the direction of the induction at that place. Hence we can map out at all points of the field of induction the direction of Hie magnetic induction. If the circuit is turned into any other position such that a change in the induction does produce an electro- motive force acting in the circuit, we may find by trial another position of the circuit at any point in the field in which the total suppression of the induction, or its instantaneous reversal in direction, produces the maximum electromotive force in the circuit. The direction of the induction is then normal to the plane of that exploring circuit. At every point in the field of induction the induction has a certain magnitude as well as direction. If we suppose any plane sur&ce placed normally to the direction of the induction in a field of uniform induction, the product of the strength of the induction and the area of the surface is called the total induction through that area. If we take any surf&ce placed in any position, and suppose it divided up into small elements of surface d S, and at the centre of each element of the surface estimate the magnitude of the induction in a direction normal to the surface at the centre of the element, and sum up all the products obtained by multiplying the normal value of the induction and the area of the element, the sum so obtained is called the surface integral of induction. If we draw in a field of induction a line such that the direction of its tangent at any point is the direction of the induction at that point, such a line is called a line of induction. Suppose a surface of any kind, for simplicity a plane surface, placed in a field of induction, and let it be divided up into small areas such that over each of them the surface integral of induction is equal to unity, and if through the centre of each of these small areas we draw a line of induction, then we may make the following statements: — The bounding line of the sur&ce is said to be perforated by induction, or to have a flux of

ELECTRO-MAGNETIC INDUCTION. 29

induotion taking place through it, or to have lines of induction passing through it. Since the total surface integral of induc- tion through this surface is, by definition, equal numerically to the number of lines of induction passing through the boundary of the surface, we may also speak of the nufnher of lines of induction passing through the surface. Faraday used the phrase miimber of lines of magnetic force instead of induction. Hence the student should notice that the following expressions all denote the same thing :

1. The surface integral of induction over a surface.

2. The total induction through the surface (Maxwell).

3. The flux of induction through the surfEU^.

4. The number of lines of induotion passing through the surfeuse.

5. The number of lines of force (Faraday) passing through the surface.

If a circuit be placed normally in a field of uniform induction, then the numerical product of the area of the circuit and the strength of the induction gives us the total induction through that surface, or total number of lines of induction (force) passing through that circuit.

In the twenty-eighth series of his << Experimental Besearches on Electricity," Faraday examined afresh with elaborate care the notion of lines of magnetic force which had guided him at all stages of his electromagnetic discoveries. He there gathers together his ideas on this subject, and by a series of researches, inimitable for physical insight and exquisite experimental skill, he has shown how the definition of a line of force or induction can be raised from a merely qualitative or directive one into a quantitative conception by which not only the direction but the magnitude of the induction can be denoted. Having placed clearly before his mind the idea of the surface integral of induction or the total induction through any surface or circuit as the important one to hold in view, he proceeds in the latter part of his investigations (§ 8,152 and § 8,199 <' Exp. Bes.'') to show experimentally that when any closed circuit, such as a l6op of wire, is placed in a field of magnetic induction, and if in any way the total induction through that circuit is changed, there is a flow of electricity round the circuit, and the total quantity so set

A CFTHK '

30 ELECTRO-MAGNETIC INDUCTION,

flowing is proportional to the oonducting power of the circuity and to the change in the total magnetic induction passing throagh it. Employing for this purpose a ballistic galvano- meter, or a galvanometer with a needle having along periodic time of vibration inserted in the circuit, he placed circuits of various forms in fields of induction, and exhausted every possible method of experimental proof that in every case any change which altered the total induction through the circuit was accompanied by the production of an electro- motive force in the circuit, and that the product of the total quantity of electricity so set in motion, and the number repre- senting the resistance of the whole circuit, was in every case proportional to the change in the total induction through the circuit. This provides us with the means of defining the strength or magnitude of induction and stating what is meant by a unit of total induction.

A unit of magnetic indtiction is a flux or amount of induction such iliaty when passing through or linked once with a circuit of unit resistance, it gives rise, if suppressed, to a flow of a unit quantity of electricity round tliat circuit.

In other words, the above is the definition of what is meant by one line of induction (or force), linked once with a single turn of a circuit of unit resistance. We can then define the meaning of the term density of magnetic induction or induction density. It is the total induction through a unit of area taken normally to the lines of induction at that place. Instead of the term density of induction, it is usual to speak simply of the induction at any point in a field of induction, or the number of lines of induction (force) passing normally through a unit of area.

We shaU employ the letter B to stand for tlie induction at any point in a field of magnetic induction. Hence in the notation of the differential calculus d B will stand for any small change in the induction. Let a circuit formed of one single turn of wire whose resistance is B be placed in a uniform field of induction of strength B, and let S be the area of that circuit ; the total induction through the circuit will be B S. Let the induction be changed by an amount d B, then the total induction is changed by an amount d (B 8). According to Faraday^s experiments the obange

ELECTROMAGNETIC INDUCTION. 31

will set a small quantity of electricity, which may be denoted by dq, flowing round the circuit, and we have d{BS)=^Bdq.

If the whole of this change of induction takes place in a very short interval of time, which may be denoted by dt, then during that time the total flow of electricity is equiva- lent to an average current of strength i, and we have

idt^dq. Hence also d{B S)-='B%dt.

But B i is the instantaneous value of the induced electro- motive force in the circuit; let this be denoted by e, and we have

d{Ba)«'edt.

<» S^ = . (12)

Therefore the magnitude of the induced electromotive force in the circuit is at any instant expresssed by the time-rate of change of the total induction passing through the circuit.

If we bear in mind clearly the meaning which Faraday attached to the phrase '' a line of force " or " the number of lines of force " as expressing the total induction through any area or conducting circuit, we can express in his language the above fact in a statement which may be called Faraday's Law of Induction ; it is as follows : —

If there be in any field of magnetic induction a circuit which is traversed by induction, then any change either of the size or position of the circuit or of the direction or strength of the induction which changes the number of lines of force (induction) passing through the circuit creates an induced electromotive force in this circuit which is numeri- cally equal at any instant to the rate of change in the number of lines of force (induction) so passing through it. The above defines the magnitude of the electromotive force ; we have next to define its direction in the circuit. To do this we must make certain conventions. If a watch-face is held in front of the observer, then the positive direction through that watch-fEuse is away from the observer, and the positive direction round that disc is in the direction in which the hands rotate. The connection between positive direction round and positive

32 ELECTBO-MAGNETIC INDUCTION.

direction through can be also fixed by thinking of the direction of the thrust and twist of a corkscrew or other right-handed screw. The negative direction of rotation is therefore the counter-clockwise direction.

In regard to the direction of the induced electromotive force, the following is the law: — The insertion of lines of induction into a circuit in the positive direction, or the in- crease of positively-directed lines of induction, gives rise to negatively-directed electromotive force. Let the reader bear in mind the following rule : Imagine a magnetic north pole and a magnetic south pole placed opposite to each other, as in the case of a horse-shoe magnet or dynamo field-magnet. The positive direction of the magnetic force in the interspace is by convention from the north pole to the south pole. This field of force creates magnetic induction in the air-space, and the positive direction of the lines of induction is in the same direction from the north pole to the south pole in the air-space. Place a watch (nonmagnetic) in this space with its watch-face facing the north pole; the watch-face is traversed by a certain flux of induction. Let the rim of this face be thought of as a conducting circuit. Then demagne- tise the magnets or withdraw them so as to diminish the induction perforating through the watch- face ; it will generate an induced electromotive force in the rim which will tend to set an induced current flowing round the rim in the direc- tion in which the hands of the watch usually rotate, or in the positive direction. Hence as time increases induction diminishes, and we get positively-directed electromotive force. If, then, N represents the number of lines of induction passing

positively at any instant through the watch-face and ----

a t

the rate of decrease of induction at any instant, since this creates positively-directed electromotive force, we must write

_ rfN _^ dt

The differential coefficient must, therefore, have the negative sign.

The above rules accordingly settle the magnitude and the direction of the induction and of the rate of change of induc- tion, and hence of the induced electromotive force.

ELECTHO'MAONETIC INDUCTION. 33

§3. The Magnetic Oircoit. — Magnetio Besistance. — We

have in the foregoing section defined magnetic force, magneto- motive force, and magnetic induction, and explained how they are measured. We have in the next place to examine the conditions under which magnetic induction is produced hy magnetomotive force. The same magnetomotive force will not always produce the same magnetic induction. What it will produce depends upon the nature ot the magnetic circuit. The path of a line of induction is always a closed loop — that is, it is an endless line. In its path it passes either through a magnet or is linked vnih an electric circuit, and the path of a line of induction is called a magnetic circuit. This circuit may pass wholly through air, or it may pass partly or wholly through iron masses, or of masses pajtly of iron, air, brass, wood, &c. There are three principal kinds of magnetic circuits. First, those in which the path of a line of induction lies wholly in iron or msbgnetic metals. Second, those in which it travels wholly in air or in non-magnetic metals. Third, those in which it passes partly through iron masses and partly through air or non-magnetic materials. This distinction is founded upon the fact that there is a very great difference between circuits in which the lines of in* duction lie wholly in iron, wholly in air or non-magnetic materials, or partly in magnetic and partly in non-magnetio materials.

A given magnetomotive force produces an enormously greater induction when the path of the lines of induction is wholly in iron masses than it does when they are wholly in air. This is analogous to the fact that a given electromotive force produces a much greater current when the electric circuit of given dimensions is composed wholly, say, of copper, than it does when it is composed wholly, say, of carbon. The number expressing the ratio between the numerical value of the electromotive force and that of the toted current produced by it in any electric circuit is a measure of the value of the electrical resistance of that circuit, and briefly we may say that for unvarying or continuous currents the electrical resistance of a circuit

The electromotive force acting in the circuit The total current in circuit *

34 ELECTRO-MAGNETIC INDUCTION.

Similarly the term inagnetic resistance is applied to that quality of the magnetic circuit in virtue of which a given magnetomotive force produces a certain definite total mag- netic induction, and the numerical measure of this magnetic resistance of a circuit is obtained by taking the quotient of the numbers representing the magnetomotive force and the total induction, or the magnetic resistance of a circuit

— The magnetomotive force acting in the circuit The totaJr induction in circuit

There is, however, a great difference between electric and magnetic circuits. In the case of electric circuits, though the electric resistance is affected by temperature changes and other physical alterations, yet, when all necessary corrections are made, it is found practically that the electric resistance of a circuit does not depend upon the current flowing through it, whereas in the case of magnetic circuits formed wholly or partly of magnetic metals the magnetic resistance of the circuit does depend essentially upon the value of the induction.

It has not yet been found necessary to coin words analogous to volt, ampere, and ohm, as names for the practical units of magnetomotive force, magnetic induction, and magnetic resistance.

All magnetic circuits which are wholly composed of non- magnetic substances, such as air, wood, brass, &c., have the same magnetic resistance for the same dimensions, and the specific iruignetic resistance of these substances, or the magnetic resistance per unit length and unit cross-section, is taken as unity. As an instance of the great difference in magnetic resistance between an air circuit and an iron circuit let us consider the following simple case.

Let two rings be made, one of wood and the other of the best soft iron. Let these rings have a circular cross-section of radius a, and let the mean radius of the circular axis of the ring be denoted by B, the value of B being large compared with that of a. Let each of these rings be wound uniformly and closely over with insulated wire, and let there be N turns on each ring. We have then two circular solenoids, and if a current is passed through each coil of which the strength is A amperes, we have equal magnetomotive forces acting round the axis of

ELECTRO-MAGNETIC INDUCTION. 86

■!«aoh droular solenoid. This magnetomotive force along the .tircnlflr axis of the ring is equal to the product of the length ,2*- J( of the path of the line of induction and the strength of the field along this line, and it is therefore approximately equal

*® U NA

10

^^ x2irB = liNA.

The sectional area of the magnetic circuit is ir a^, since all the lines of induction are confined to the inside of the circular solenoid and from endless circular lines. The mean length of the magnetic current is 2jrB. The total magnetic resistance is numerically obtained by multiplying the specific magnetic resistance (which in the case of the wooden ring is equal to unity) by the length and dividing by the section of the path.

Hence, the magnetic resistance is equal to 1 x -^-r- = 2— , and

this is the total magnetic resistance of the circuit. The total induction is numerically equal to the quotient of the magneto-

motive force, viz., - N A, by the magnetic resistance, viz.,

2 B 2ir N A rt^

— —, and is therefore equal to — Tri— is— > ^^^ ^^^^ induction or

1 N A ifuluctian density is equal to - •-^-- , since the section is ira^,

5 A

The above is very nearly true if a is very small compared with B, but if this is not the case, we have to take account of the faci that in this last instance the magnetic force has different values at different points over the cross-section of the solenoid (see Appendix, Note B).

' ' As an actual example we may take the case of a wooden ring in which a « 1-27015 centimetres, B» 12*8291 centi- metres,' and NB62d. If a current of 1 ampere is sent

round this wire, the magnetomotive force on the axis is

Atr

jg X N A = IJ X 623 - 779, and the induction density along the

central axis is ??'^ = 1-^^^ = 9-7 C.G.S. units nearly, or

5 B 6 12-829 about 10 lines of induction per square centimetre.

If, however, a soft iron ring of the same dimensions is employed and wound over with the same number of turns and the same current employed, it would have been found that the

d2

.36

ELECTRO-MAGNETIC INDUCTION.

induoiion density on the axis would have been about 10,000* G.G.S. units or lines of induction (force) per square centi- metre. The same magnetomotive force in both cases, viz.^ 779, would in one case give rise only to an induction density of 10 G.G.S. units, and in the other case to an induction density of many thousands of C.O.S. units. This is because soft iron has a far less specific magnetic resistance than wood, and a given magnetic force produces far more induction through it than it does in wood.

;,-.*'. ,v /.. / V - vj : s > 7\ ,;^::-^, : ' *;>->• "- <> :

Fig. 6. Diagram showing the pftth of the Lines of Induction in a closed Iroit Ring when magnetised by a closed endless solenoid or coil wound on it. The dotted lines inside the ring represent the Lines of Induction* There- Is no sensible field outside the ring.

There is, very roughly speaking, about the same numerical difference between the minimum specific magnetic resistance of soft wrought iron and that of air as there is between the specific electrical resistance of copper and gas retort carbon.

If, instead of forming the core of the solenoid above mentioned wholly of iron or wholly of wood, we make it partly of iron and partly of wood, or employ an iron ring with a cut or air-gap in it, and apply the same magnetomotive

ELECTRO-MAGNETIC INDUCTION.

37

foredy the result will be that the induction in different parts of the core will be found to be different. There will be a certain number of lines of induction which will be continuous right round the core, and which are determined by the resultant ma»gnetic resistance of the path which is in part of iron and in part of wood or air, and which resultant magnetic resistance will be something intermediate between that of a ^complete iron core and a complete wooden or air core. There

-At^^v

^>"i

Fio. 7.

Diagram showing the paths of the Lines of Induction in and outside an Iron Ring with an air-gap in it when magnetised by a solenoid wound on -it. The lines exterior to the ring are supposed to be delineated by iron filings. The lines inside the ring are represented by the dotted lines.

will be, however, an additional induction in the iron, which will be represented by an extra number of lines of induction which pass through the iron but turn back and complete themselves through the space outside. In Figs. 6, 7, and 8 are shown

. I^ree diagrams illustrating the form of the lines of induction for three cases of complete and incomplete iron circuits.

' Where the lines of induction enter and leave the iron parts of the core they give rise to magnetic poles in the iron, and

38 ELECTRO-MAGNETIC INDUCTION.

this additionally complicates the case. These magnetic polefr produce a reverse magnetising force in the interior of the iron which opposes the magnetic force due to the magnetising solenoid. Such an imperfect iron circuit is often called an open magnetic circuit, whilst the complete iron ring core would be called a closed magnetic circuit. The wood, or air, or other body of unit specific magnetic resistance is said to form a gap* in the iron magnetic circuit.

Fig. 8. Diagram showing the paths of the Lines of Induction in and near an* Iron Ring with wider air-gap in it, when magnetised by a solenoid wound on it. The dotted lines show the paths of the lines of induction in the- iron core, and the exterior field is supposed to be delineated by iron filings.

Speaking generally, the subject of closed magnetic circuits is more easy to treat and deal with than that of open magnetic circuits. The particular reason why magnetic problems are more difficult to manage than the corresponding electrical problems is because the magnetic resistance even of a closed magnetic circuit is not a constant quantity. It is not only affected by temperature, but is also determined, within very wide limits, by the value of the magnetic induction itself, by

ELECTRO-MAGNETIC INDUCTION. 39