The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 21 of 36

Carbonic oxide ,

Nitrous oxide ,

defiant gM

Marsh gas

Carbonic bisulphide Sulphurous acid ....

Ether

Ethyl chloride

Ethyl bromide

Boltzmann.

Klemencio.

Refractive index.

1000295

l-000?93

1000295

1-000132

1-000132

1-000139

1000473

1-000492

1-000464

1-000345

1-000547

1-000335

1-000497

1-000579

1-00 516

1-000656

1-000729

1-000720

1000472

1-000476

1-000442

—

1001450

1-001478

—

1-0C477

1-000703

—

100372

1-00154

100776

1-001174

—

1-00773

1-00122

The specific inductive capacity of a vacuum is taken as unity, and Bolts - mann's values are given for comparison.

• Su Dr. J. Hopkinson, PhU. Tran§. Royal Society, Vol. CLXXIL, 1881, pk372.

364 DYNAMICAL THEORY OF INDUCTION.

§ 5. Velocity of Propagation of an Electromagnetie Difr- tnrbance. — There is another line of experimental enquiry which leads to an important relation between electric and optic phenomena. This is the comparison of electrostatic and electromagnetic measurements. If two very small spheres are electrostatically charged and placed with their centres at a unit of distance apart, the stress between them may be mechanically measured. If the conductors are equally charged with opposite kinds of electricity, and the stress when at a unit of distance in air is one unit, the electric quantities are said to be unit electrostatic quantities. If such uuit quantities are discharged through a conductor at the rate of one discharge per second, the resulting flow or current is called an electrostatic unit of current.

In the above defiDition we suppose the dielectric to be a va.cuum or some substance such as air, of which the dielectric constant does not differ sensibly from unity. If q and q^ be two quantities measured electrostatically, and then be placed on smaU conductors separated by a distance r in a dielectric of constant E, the dynamical force between them will be nu-

numerically equal to |^.. ; and if g = j^,then the force is =^.

Hence, if r is always taken equal to unity, the real quantity of electricity producing by its action on another equal quantity a unit of force will vary as the square root of E when the experi- ment is performed in various dielectrics. In other words, the absolute magnitude of the electrostatic unit of quantity, and therefore also of the current, will vary as the square root of the specific inductive capacity of the medium in which the charges exist. There is another mode in which a unit of current may be defined, and this depends on the definition of a unit magnetic pole. If two magnetic poles of equal strength, m, are placed at a distance r apart in a magnetic medium of permeabiUty fi, the

stress or force between them will be numerically equal to — :>

in which expression it is seen that m and fi appear as quantities analogous to q and K in the electrostatic analogue. Hence, when r is unity, we see that to produce a unit stress between the poles 771 the pole strength must vary as the square root of fi, or the absolute magnitude of the unit magnetic pole varies directly-

DYNAMICAL THBOBY OF INDUCTION. 365

as the square root of the magnetic inductive capacity of the medium in which the experiment is performed, the absolute unit magnetic pole being defined as a pole which at a unit of distance acts on another like pole with a unit of force in a magnetic medium, assumed to be vacuum, or some standard substance of unit permeability.

Since an electric current produces a magnetic force, it may be defined as to magnitude by agreeing that the unit of current is to be one which, when flowing in a circular circuit of unit radius, acts for every unit of length of that circuit with a unit of force on a unit magnetic pole placed at the centre of that circle The magnitude of the force on the magnetic pole is proportional to the product of the strength of the pole and the strength of the current. Hence, if the magnitude of the unit pole is varied the magnitude of the unit of current will vary inversely as the magnitude of the strength of magnetic pole which is taken as the unit pole. When the medium is varied, the magnitude of the unit magnetic pole, or of the pole which fulfils the condi- tion of acting on another equal pole at a unit of distance with a unit of force varies directly as the square root of the permea- bihty of the medium. It follows, then, that the magnitude of the electro-maftnetic unit ofcuirent varies inversely as the square root of the magnetic permeability of the medium in which the experiment is made.

We have, then, that the electrostatic unit of current is a quantity which varies directly as the square root of the electro- static inductive capacity of the medium, or as /K, and the electromagnetic unit of current is another unit of current which varies inversely as the square root of the magnetic induc- tive capacity of the medium, or as V'/x. The electrostatic unit of current represents a much smaller quantity of electricity per second than the electro-magnetic — in other words the value of the ratio of the magnitude of the unit electromagnetic current based on the definition of a unit magnetic pole, to the magnitude of the unit electrostatic current, based on the definition of a unit of electrostatic quantity, is an integer number, and a large one. This ratio of the two units of current varies when the fundamental inductive capacities of the medium is changed, but so that the ratio of the electro- magnetic to electrostatic unit varies inversely as the square

aa2

356 DYNAMICAL THEORY OF INDUCTION.

root of the product of E and /x. If 0,^ is the magnitude of the electro-magnetic unit of current, and G, is that of tke electro- static unit for the standard dielectric, in which E s 1 and /e« -• 1,

then, when the dielectric is changed, -^ is changed in the ratio

^f

of 1: vE/iA. Let H^ae denote the value of the ratio for vacuum or for a standard dielectric, of which E ^i 1 and /i -■ 1, and B„^ denote its value for any other medium of which the dielectric constant is E and the magnetic constant fi, then

n ^««^

We have next to consider what is the physical meaning of this ratio of the electro-magnetic and electrostatic units.

The degree in which one quantity is greater or less than another, or to put it more precisely, that amount of stretching or squeezing which must be applied to the latter in order to produce the former, is called the ratio of the two quantities.* The ratio of two physical quantities is therefore the expres- sion of the operation which must be performed on the one to make it the physical equivalent to the other. What operation must be performed on an electrostatically measured unit of electricity to make it the equivalent in every way of an electro- magnetically measured unit of electricity 7 The reply is, it must be set in motion with a definite velocity. The electric current produces a magnetic field. The electro-magnetic mea- sure of current is obtained by defining the field by stating its dynamical effect on a defined magnetic pole, and the unit of electric quantity measured electro-magnetically is the quantity conveyed by the unit current so measured in a unit of time. If we imagine a circular or other conductor conveying a unit (electro-magnetic) current to have stretched alongside of it another closely adjacent conductor of like form, each unit of length of which is charged electrostatically with a unit (electro- static) of electric quantity, we might submit the foUowing question : — The current flowing in the first named conductor transmits a unit (electro-magnetic) quantity of electricity across each section of it per unit of time : with what velocity must electricity in the second conductor Be set flowing in order that

• W. K. Clifford, " The Common Sense of the Exact Scienoes," p. 9a

DYNAMICAL THEOBY OF INDUCTION. 367

there may be an equality in the quantities flowing past any sections in each of the oonduotors, as evidenced by equality in the magnefeio fields produced by the first-named current and the moving electric charge? This velocity is evidently a concrete velocity, which depends on the very nature of the qualities of the medium which determine magnetic and electrostatic attraction, and this velocity may be called the ratio of the maguitude of the electro-magnetic to the electro- static unit of quantity. This velocity is evidently one which is determined by the nature of the medium, and not by the particular units of length, time, and mass selected for use in the measurements. This comparison assumes that a moving electrostatic charge is in effect the equivalent of an electric current. This has been put to the test of experiment by Prof. Rowland.* A rigid gilt ebonite disc was fixed to an axis, and could be rotated between two gilt glass discs. One member of a very delicate astatic system of magnetic needles was placed near the disc and shielded from electrostatic disturbance. On charging the gilt ebonite disc and setting it in rapid rotation it was found to affect the magnetic needle whilst rotating just as a current of electricity would have done if flowing in a circular conductor coinciding in form with the periphery of the disc. Since 1876 Prof. Rowland has again in the United States repeated the experiment and confirmed the general result. There is, therefore, experimental founda- tion for the view that a static charge of electricity conveyed on a moving body creates a magnetic field tchiht it is in viovemenU This kind of electric current, in which a static charge is bodily moved on a conductor, is called a convection current. The experiment of comparing the magnitudes of an electrostatic and an electro-magnetic unit of electric quantity as above defined was first made by Profs. Weber and Eohlrausch, and the value of that ratio for a medium such as air, ill which approximately we have E and m both equal to unity, gave as a result a velocity very nearly identical with the velocity of light. Since that time very many experimentalists have determined the value of this ratio, which is denoted by

• See Phil, Mag., 1876, Vol. II., Fifth Series, p. 233 : Dr. Helmholtz, <' On the Electro-Bfagnettc Action of Electric Convectioxu" Theae ezperi- m«Qto of Prof. Howland were carried out at Berlin,

358 DYNAMICAL THEORY OF INDUCTION.

file symbol <' v." The names and the results of the observa- tions made by some of the principal observers are set oat in the Table on opposite page.

One of the best determinations of the velocity of light is that made by Prof. Newcomb, at Washington, in 1882. The method employed was the revolving mirror method of Foacaalt, the distance between the revolving and fixed mirror being in one portion of the experiments 2,550 metres, and in the other portion 8,720 metres. The resulting velocity of light in vacuo is 2-99860 x 10"^ centimetres per second.

The following results of other observations are abstracted from Prof. Everett's book, " Units and Physical Constants," 2nd edition : —

Ohaerver Velocity in oenUmetres

Michelson, at Naval Academy, 1879 2-99910xl0*®

MichelaoD, at Cleveland, 1882 2*99853 x 10^<>

Newoomb, at Washington, 1882 (beat results) ... 2*99860 x 10^>

Newcomb (other results) 2-99810 xlC®

Foucault, at Paris, 1862 2*98000 xlOi'

Comu, at Paris, 1874 298600xlO»o

Cornu, at Paris, 1878 3004 xlO"

Last result discussed by Listing 2*9999 xlOi<^

Young and Forbes, 1880-81 3*01382xl0i»

Earlier observations gave as follows : —

Boemer*s method, by Jupiter's satellites 3 '000 x 10^^

Bradley's method, by stellar aberration 2977 xlO»

Fizeau 3142 xlO^'

The general result of the best determinations is that the velocity of light is very close to 80(X) x 10° centimetres per second, or nearly one thousand million feet par second.

We have, therefore, the following facts : — The velocity V„ of light of definite wave length in any medium is connected with the velocity V„of the same ray in vacuo by an equation—

V -^'

where fx is the refractive index of that medium for the par- ticular wave length considered, and also that the velocity V is very nearly 8 x 10" centimetres per second. Also we find that the ratio of the electro-magnetic to the electrostatic unit of electric quantity or current in any dielectric and magnetic

DYNAMICAL THEORY OF INDUCTION. 359

X

I

X

X

X

%

X

I

X

I

to

X

a

»o

2>

X

I

X

I

I

I

4

I

s

I 1

I

i'

1

o

i

I

i

^ 8

i

9

A I

a"

^1

15

4

I I

I

3

^S

s

Is

OQ

is

QQ

1

a'

CQ

tJ SB

s

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800 . DYNAMICAL THEORY OF INDUCTION.

medium B^ is oonneoted with the same ratio measured in yaeno Eg by an equation —

where E is the dielectric constant and fi the magnetic per- meability.* Experiment has also indicated that within narrow limits, taking best results, K. and V« have the same value, namely, 8 x 10^^ centimetres per second, and that ^K has the same value as [i (refractive index) for media, for which /x (per- meability) has the value unity. We are led, therefore, to infer that this close relationship is not a matter of accident, but that it indicates a very intimate connection between electricity and light, and that the hypothesis that light is a disturbance propagated through an elastic medium may be supplemented with some considerable show of reason by the hypothesis that electro-magnetic phenomena are the result of actions taking place in identically the same medium or ether. There are no transparent media for which the magnetic permeability differs by more than a very small quantity from unity, and hence the approximate identity of the values of the ratio of the units compared in air with the value of the velocity of light waves of very long wave-length ; and the approximate identity for true dielectrics of the value of the refractive index and of the square root of the dielectric constant furnishes a test of the proba- bility of the truth of the electro-magnetic theory of light. Maxwell's mathematical method of arriving at this theory consisted in forming certain equations expressing the velocity of propagation olvector potential, and noticing that these equations were mathematically of the same form as those which determine the velocity of propagation of a disturbance through an elastic medium. The physical meaning of this term, vector potential, may be arrived at as follows : —

Suppose a regiment of soldiers to set off marching down a street, the ranks being well spaced out. At any place in the street let two lines be drawn across the street parallel to each other and a few yards apart. Let two observers take

• It ifl unfortunate that usage has consecrated the same Qreek letter ft for refn^tivity in optics and magnetic inductivity in electro-magnetics. In some respects it would be an advantage in electro-optics if these quautitiei were differently symboliied.

DYNAMICAL THEORY OF INDUCTION. 361

note of how many soldiers cross each line. At any instant the total number of soldiers which are contained between the two lines is equal to the difference between the numbers which have crossed each line respectively. However irregular the movement may be, the total number of soldiers at any instant in the area or the product of the area, and the number of soldiers per unit of area within the boundary, will be equal to the number obtained by reckoning the algebraic sum of the soldiers which have from the beginning of the time crossed the whole ^ boundary line, calling those numbers positive when soldiers have stepped into the area and negative vfhen they have stepped out of it. We have here a simple example of the way in which a line intefjral may be the equivalent of a surface integral. If the area be irregular in shape and contain A square yards, and if the perimeter be I linear yards, then if n^ 7ia, &c., are the number of men which have stepped across each yard length of the boundary, and if Ni N,, &c., are the number of men in respective square yards within the area at any instant, then Ni + Na + , &c., to A terms or ^N is called a surface integral and will be equal to ni + n^ + , &c., to I terms, which is a line integral, provided that each n is reckoned positive when men step in, and negative when men step out of the area over each yard of the boundary. The algebraic sum of all the stepping over the boundary all the way round the area is equal to the sum of the men per square yard all over the area. We have here given an illustration of an important proposition in mathematical physics, viz., that a surface integral, or the sum- mation of a certain quantity over an area, can be replaced by a line integral, or the summation of another relative quantity all along the boundary line of that area. We proceed to illustrate it from an electrical point of view.

Let G (Fig. 182) be the circular cross-section of an infinite straight wire conveying a current G. Bound G describe a

circle of radius r. The magnetic force at p is known to be

o n equal to — units, and is directed along the circumference

of the circle ; the line integral of the magnetic force along the

dotted line is equal to — x27rr=49rO, and the surface

r integral of the current through the area enclosed by the dotted

362 DYNAMICAL THEORY OF INDUCTION.

circle is 0. Hence we have generally that the line integral of the magnetic force is equal to 4ir times the snrfiace int^al of the current. This proposition is generally true, and it is easy to show that if A be any area {see Fig. 188) traversed normally by a current, such that the current density is u over any element

Fio. 132.

of area ds, then the integral otudseXL over the area, or jtids,

is equal to the line integral of the magnetic force taken along the boundary line. The mathematical operation of taking a line integral has been called by Maxwell curling^ and we express

Fio. 133.

the above proposition by saying that itr times the total current through the area is equal to the curl of the magnetic force round it. On the theory that lines of magnetic force do not spring suddenly into existence in a field, but are propagated onwards from point to point in the field, it is possible to show

DYNAMICAL THEORY OF INDUCTION. 863

that just as the current is the curl of the magnetic force so the magnetic force is the curl of another quantity called the vector potential.

Let A B (Fig. 184) be a portion of a straight conductor in which a current can be started. Let x x\ y \f be two lines drawn a unit of distance apart, parallel to each other and at right angles to the conductor. These lines bound a strip of plane space taken in the plane of the current. Draw any two transverse lines ab,cd, parallel to the conductor and separated by a small distance. We know that when a current is started in the con- ductor the lines of magnetic force F will be circles formed round A B as axis, and having their planes perpendicular to the plane x x\ y y\ Let us now assume that if a current is suddenly started in the conductor A B the magnetic force is

^/////

Fia. 154.

propagated outwards from the conductor with a finite velocity V. In other words, each circular line of force must be con- sidered to expand outwards like a circular ripple on the surface of water. When once the field has arrived everywhere at its normal value the magnetic force at a distance r from the wire

20 is — I where C is the value of the current, and we shall sup- pose, as usual, that the magnetic field is indicated as to value by the density of the lines of force, or that the number per square centimetre traversing normally the plane x x\ y y' is at any point proportional or numerically equal to the magnetic force at that point. If, then, we neglect for the moment all effect of self-induction, and suppose the current in the wire to rise up instantaneously to its full value, we may yet regard the

364 DYNAMICAL THEORY OF INDUCTION.

circular lines of force as expanding outwards with a certain velocity of enlargement, and attaining or taking up their final positions after a short interval of time. If we represent the intersections of these rings of force on the plane ofxxy y' by dots, these dots will march forward like the soldiers in the pre- vious illustration. The total number of lines of force which at any instant are found traversing the area abdcis equal numeri- cally to the difference in the number between those which from the beginning of the epoch have intersected or cut through the Jine a b and those which have cut through cd. In other words, the surface integral of the magnetic force over abed may be represented by, or is equal to, the line integral round abed of a certain quantity called the vector potential^ which, physi- cally interpreted, is the total number of lines of force which have cut through a unit element of the boundary in the process of expansion or propagation outwards. This term vector poten- tial is justified as follows : — If F be the total number of lines of force per unit of length of ab which have cut through a b from the instant of beginning the current, and if the small distance bdis called Sx, the length x b being called x, then by Taylor's theorem (Diff. Gale), the number which have cut

dF through unit of length of c d is F — -i— B a?, and hence the

a X

dF difference between F and this last quantity is -j— Sx, and this

last when multiplied by By, which we may take for the length

dF of a & or c d — that is -j— BxB y — is the total number of lines of

force included in the area abed. If we call the induction through this area B — that is to say, the number of lines of force per square centimetre is B — it follows that the number through ab cd is BBxBy. Hence, equating the two values,

d F we have --77- BxBy^^BBxBy,

dx

Hence, the mean magnetic force over the small area is numeri- cally equal to the space variation of a certain quantity F. In electrostatics the electric force X at any point in the electric

DYNAMICAL THEORY OF INDUCTION-. 365

field is the space variation of a certain quantity V, called the electrostatic or scalar potential — that is to say,

-j -^1

ax

and accordingly by anology that quantity F whose space varia- tion gives the magnetic force under the circumstances considered above is called the vector potential of the current. From Ampere's investigations it is known that the magnetic force due to an element of a current G of length 6 5 at a distance r from this

element, has the value ^^, and is along a line at right angles

to the plane containing 8 s and r. The space variation of - — -?

C 8 g ^

is -— - ; hence the vector potential of an element of current at

any point is proportional to the length of that element divided by its distance from that point.

In electrostatic phenomena we obtain the static potential at any point due to any charge Q by taking each element g of the charge, and dividing the magnitude of this element of charge by its distance from the point at which the potential is required,

and taking the sum 2 ^ of all such quotients. In electrostatics

the potential at a point is a scalar or directionless quantity, and the summation is merely an algebraic sum ; but in dealing

with currents the quotients — - are vectors, or directed quan-

r

tities, and have to be added together according to the laws for the addition of vector quantities just as forces and velocities are added. Hence the potential of a current at any point is a vector or directed quantity. The lines of vector potential of a straight current are lines described in space parallel to the current, and the lines of vector potential of a circular current are circles described on planes parallel to the plane of the cur- rent. Beturning to the simple case of a straight current, let us suppose that a unit of length is described somewhere parallel to the current, and that on starting the current suddenly cir- cular lines of magnetic force are propagated outwards with a velocity V ; these lines will, as they expand, cut perpendicularly through the element of length just as the expanding ripples on water due to a stone dropped into it would <' cut through " a

866 DYNAMICAL TREORY OF INDUCTION.

stick held perpendionlarly in the water a little way from the place where the '' splash " was made. Suppose that after N lines of force have cut through the element of length this little line is made to move forward parallel to itself, so that there is no further increase in the number of lines of force which after- wards cut through it, it is evident that it must move with the velocity of propagation of the expanding rings of force. But the number expressing the number of lines of force which have cut through the element of length already is the value of the vector potential at that point where the element is at that instant ; hence the velocity of propagation of the vector potential is the velocity of propagation of an electro-magnetic disturbance. Maxwell*s general mathematical method of investigating the propagation of an electro-magnetic disturbance consisted in forming equations expressing the change of the value of the vector potential of a current or system of currents at any point in the field, and deducing equations which mathematically are of the same type as those which express the propagation of a disturbance through an elastic solid or fluid, and his result was that the velocity of propagation of the vector potential through a medium of electrostatic and magnetic inductivities

K and p, was equal to — :^, or to (K/a)-*. n/K/a

The complete proof of the above proposition as given by Maxwell in all its generality requires soma elaborate analysis, but is is not difficult to give a simple illustration by treating a reduced case, and which will exhibit the principles of the more complete problem.

Let an infinite straight conductor be supposed situated in a dielectric medium of specific inductive capacity (electrostatic inductivity) E and of permeability (maguetio inductivity) fi. We proceed to investigate the velocity of lateral propagation of electro-magnetic induction on the supposition that if a current is instantaneously started at its full value in the conductor, supposing this possible, the magnetic force travels outwards laterally from the conductor in all directions with a velocity v. This amounts to the supposition that the circular lines of magnetic force surrounding the conductor swell out or expand outwards from the surface of the conductor, so that the radius of any determinate circular line of force increases or grow3

DTNAMIGAL TREOBY OF INDUCTIOHT.

367

vriih a velocity v. It must be borne in mind that the magnetic force at any point in the field at any instant is defined by the density or concentration of the lines of force — that is, by the number passing normally through a unit of area. If we complicate the problem by supposing the strength of the current in the conductor to gradually increase, then the concentration of the lines at any point must be supposed to increase gradually, but the rate of increase of concentration — that is, of the force — is a different thing from the rate of outward movement of the lines of force.

We might in imagination suppose each line of force to be labelled so as to recognise it. All the lines travel outward from the conductor at the same rate, but some go out farther than others. The first ones shed off expand out to reach

Fia. 135.

positions in the most distant portions of the field, and the succeeding ones reach intermediate positions, and as the current strength grows up fresh arrivals or deliveries of lines of force happen which pack the space fuller, and increase the concentration at all points of the field, at a rate depending on the rate of growth of the current.

Let 0 0 (Fig. 185) be a portion of the straight conductor. In the plane of 0 G take any little rectangular area abcd^ with side ac equal to unit of length, and side ab equal to Sx^ Bx being a very small quantity compared with the distance between OC and ac, that is, let the distance Oc-d? and Od^'X + Sx, and let the distance Bxhe the distance by which the radius of any circular line of force of the conductor 0 0 increases

368 DYNAMICAL THEORY OF INDUCTION.

in a small time 8 1, At any instant the number of lines, of force which pass normally through the small area abed is equal to the difference between the number which have ''cut" across ac and those which have cut across bd in consequence of our supposition as to the outward growth or expansion of the circular lines of force. Let F be the total number of hues of force due to the current in 0 G which have from the beginning of the current flow ** cut across " a c, then, by the principles of the Differential Calculus, the number which have cut across bd is represented by the quantity

dF F - — 8xy and the number existing in, or perforating through,

dx 1 p,

the area abed is the difference between F and F--— 8x, or

dF equal to — 8x, Let B stand for the induction through dx

unit of area of the rectangle abed, ov to the number of lines of

force per unit of area, then the total number of lines of force

through abed is represented also by B 8 or, since the area of

abcdia 8x square units, ac = bd being unity.

Hence, ' ^ = B (109)

a X

or the induction is represented by the space rate of change of the vector potential of the current at that point in the direc- tion of X. In this case let it be borne in mind that the vector potential signifies the number of lines of force which have from the beginning of the epoch cut through unit length taken parallel to the current. Again, since by supposition each line of force moves outwards parallel to itself through a distance

So; in a time 8ty ~- is the velocity of propagation v of the

0 t

electro-magnetic disturbance or of the vector potential. The

rate of " cutting across " a c at any instant is represented by

d F

-f- ; hence the number of lines of force added to the area ia

dt

d F a time Bt must be --j- 8t, and this must be equal to the aoca- dt

mulation of the lines in abed in the same time in the area

abed.

DYNAMICAL THEOBY OF INDUCTION. 300

If in a small time interval the rate of cutting across a c is -^ -, then the rate at which *' cutting" is taking place across a length bd, removed by a distance 5ar, is

dt dxKdtJ

and the rate at which accumulation of lines of induction is going on in the area is

dx \dt/

Hence, since B is the induction per unit of area and the area oi abed is 8x square units, the rate of increase of induction through abcdiQ

Accordingly we have

dt

|-(BM =

-dxi-dt)'""'

instant,

dt "

d /dF\ dx\dt)'

=

d /dF\di dt\dt) dx

dx dt

rfSP dt . rft« dx'

or,

but --? = v « velocity of propagation of the impulse. Hence, dt

-^?

^''^' (110)

dx d T<^

or, generally, sinco B = — - ,

we have -_. + i;2 =0 . . . . (Ill)

dt^ dx^ ^

as the equation of motion of the vector potential. This equation, which is a reduced case of the general one, is of the same type as that obtained in the theory of sound for the

B B

370 DYNAMICAL THEORY OF INDUCTION.

propagation of an impulse along a tube or canal. In the case of sound the symbol F would be the velocity potential,* In the electro-magnel-ic problem the F is the vector potential. It might perhaps be more expressively called the induction potential.

The rate of cutting, or the value of — , also expresses the

dt

electromotive force acting along the unit of length a c in tbe

dielectric. On Maxwell's hypothesis this electromotive force in

the dielectric acting parallel to the current in the conductor

produces a displacement in the dielectric, such that if E is the

electromotive force we have as above

dt K '

where D is the displacement through unit of area ; hence,

d^F 4,r dT),

rfF K dl ^"^^

and — is the rate of displacement or the displacement current dt

flowing through unit of area taken perpendicularly to the cur- rent in 0 G at the point considered. Let this displacement

current be denoted by u. We have then that -— - = -^, K

rft* K being the dielectric constant of the medium.

Consider now a small parallelopipedon (Fig. 136) or solid rectangle described in the dielectric, of which the sides are respectively ac = l, cd^Sx, c e = 8y.

The effect of the cutting across of this solid rectangle by expanding lines of induction will be to generate in it a displace- ment current such that the total displacement current parallel to a c and through cdfe will heudxdy. By a previous theorem the line integral of magnetic force round any line is equal to 4ir times the surface integral of the current through the area bounded by that line, and this is true whether the magnetic force be produced by that current, or whether it is a current produced by a certain changing magnetic force. Apply the theorem to the small rectangle bounded by the lines c e/d. The surface in- tegral of the current through c efd is udxdy. The magnetic

• See Besant'B " Hydromechauic3," p. 251 (Third Edition).

DYNAMtCAt TBEOliY OF INDUCTION.

sn

B

force along ca is -, where B is the induction at c and fi is the

magnetic permeability of the medium, since by a fundamental theorem the ma^gnetic induction B at any place is equal to fA times the magnetic force at that point. The magnetic force

along df removed by a distance 8 x from ceis -(B- — Sx)

/a\ dx /

and there is no magnetic force along c d and ef, for these sides

are perpendicular to the direction of the magnetic force of the

Fio. 136.

current in 0 G. Hence, the line integral of magnetic force round cefdia

l(B8y-(Brfy-^-?8a?5y)y ti\ dx J

or

hence,

iTru^x^y^^—hx^y,

fA d X

or

47r/AM =

dB

dx

(118)

Accordingly, in the equations (112) and (118) above, we have

dJ^F dB obtained values for the quantities -^ and -jz in terms of

the permanent constants of the medium ; and by substitution