The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 30 of 39

We obtain an equation of the form

X -[— — — (^)+Xal + pb1 + ...j 89, = 0 (503).

492 Dynamical Theory of Currents [ch. xvi

Let us assign arbitrary values to 80m+l, 80m+2, ... 80n, and then assign to the m quantities 80 !, 802, ... 80 m the values given by the m equations (501), (502), etc. In this way we obtain a system of values for 80l} 802, ... 80n which is permitted by the constraints of the system.

The m multipliers X, fi, ... are at our disposal : let these be supposed to be chosen so that the m equations

d-^-^-(~)+\as + fMbs+...=0, (s=l, 2, ... m) (504)

dds at \9ty

are satisfied. Then equation (503) reduces to

m+i {ous at Vdtv )

and since arbitrary values have been assigned to 80m+1, ... 80n, it follows that each coefficient in this equation must vanish separately. Combining the system of equations so obtained with equations (504), we obtain the complete system of equations

^-^(5T)+Xa* + /i&*+-" = 0' (s==1>2> ■••») (506).

oUs at Vdty

Lagrange's Equations for General (including Non-conservative) Forces.

551. If the system of forces is not a conservative system, we cannot replace the expression

2(ZxSai+F1Sy1 + Z1&1)

in § 545 by — 8 W where W is the potential energy. We may, however, still denote this expression for brevity by — {STF}, no interpretation being assigned to this symbol, and equation (496) will assume the form

[T(BT- {BW})dt = 0 (507).

Jo

By the transformation used in § 548, we may replace I 8Tdt by

Jo i Xde, dAddJ)

Now — [8W] is, by definition, the work done in moving the system from the configuration 0lf 02, ... 0n to the configuration 0X + 80lf 02 + 802, ... 0n + 80n. It is therefore a linear function of 801} 802, ... 80n, and we may write

-{8W} = ®1801 + ®2802+... + ®n80n, where ®lt S2, ••• ®n are functions of 01} 02, ... 6n.

550-552] Lagrange's Equations 493

We now have equation (507) in the form

Vi 130! dt\ddj J As before each integrand must vanish. We have therefore at every instant

i (dd, dt \ddj \

If the coordinates dly 62, ... 9n are all capable of independent variation, this leads at once to the system of equations

I©-f.=0" <"^--> (608)-

while if the variations in 0lt #2, ... are connected by the constraints implied in equations (501), (502), ... we obtain, as before, the system of equations

d /dT\ dT

dt

r-±-<LL=®g + Xas + pbs+..., (s = l,2, ... n) ...(509).

The quantities ©i, ®2, ... are called the "generalised forces" correspond- ing to the coordinates 0i} 62, ....

Lagrange's Equations for Impulsive Forces.

552. Let us now suppose that the system is acted on by a series of impulsive forces, these lasting through the infinitesimal interval from t = 0 to t = t. If we multiply equations (508) by dt and integrate throughout this interval we obtain

'dT'

L90J

t=T fTdT fT

_ ^-dt= %sdt. t=o Jo do* Jo

7)T The interval t is to be considered as infinitesimal, and ^- is finite.

Thus the second term may be neglected and the equation becomes

dT fT

change in — = ®sdt (510).

dds Jo

We call \®sdt the generalised impulse corresponding to the generalised Jo force ©«, and then, from the analogy between equation (510) and the equation

change in momentum = impulse,

8JT coordinate 6

we call — the generalised momentum corresponding to the generalised dO.

494 Dynamical Theory of Currents [oh. xvi

Application to Electromagnetic Phenomena.

553. We have already obtained expressions for the energy of an electro- static system, a system of magnets, of currents, etc., and in every case this energy can be expressed in terms of coordinates associated with " accessible " parts of the mechanism. We can also find the work done in any small change in the system, so that we can obtain the values of the quantities denoted in

the last section by ©x, ©2, All that remains to be done before we can

i t ~ >_ i.: :„.: n_ i~e c? za>7\ j._ ±1 • i. . j. j_- c

apply Lagrange's equations provisionally (cf. § 547) to the interpretation of electromagnetic phenomena is to determine whether the different kinds of energy are to be regarded as kinetic energy or potential energy.

Kinetic and Potential Energy.

554. At first sight it might be thought obvious that the energy of electric charges at rest and of magnets at rest ought to be treated as potential energy, while that of electric charges or magnets in motion ought to be treated as kinetic. On this view the energy of a steady electric current, being the energy of a series of charges in motion, ought to be regarded as kinetic energy. We have also seen that this energy is to be regarded as being spread throughout the medium surrounding the circuit in which the current flows, and not as concentrated in the circuit itself. Thus we must regard the medium as possessing kinetic energy at every point, the

amount of this energy being, as we have seen, ^ — per unit volume.

But we have also been led to suppose that the medium is in just the same condition whether the magnetic force is produced by steady currents or by magnetic shells at rest. Thus, on the simple view which we are now considering, we are driven to treat the energy of magnets at rest as kinetic — a result which is inconsistent with the simple conceptions from which we started. Having arrived at this contradictory result, there is no justification left for treating electrostatic energy, any more than magnetostatic energy, as potential rather than kinetic.

555. Abandoning this simple but unsatisfactory hypothesis, let us turn our attention in the first place to the definite discussion of the nature of the energy of a steady electric current.

Let us suppose that we have two currents i, i' flowing in small circuits at a distance r apart. As a matter of experiment we know that these circuits exert mechanical forces upon one another as if they were magnetic shells of strengths i, %'. Let us suppose that a force R is required to keep them apart, so that initially the circuits attracted one another with a force R, but are

553-555] Kinetic and Potential Energy 495

now in equilibrium under the action of their mutual attraction and this force R acting in the direction of r increasing.

cos e

If M is the quantity 1 1 dsds, we know that the value of R is

R = -H'd-§ (511),

this value being found directly from the experimental fact that the circuits attract like their equivalent magnetic shell (cf. § 499).

The energy of the two currents is known to be

E = ±(Li2 + 2Mii' + Ni'2) (512).

Let us suppose, for the sake of generality, that this consists of kinetic energy T and potential energy W. Then, assuming for the moment that the mechanism of these currents is dynamical, in the sense that Lagrange's equations may be applied, we shall have a dynamical system of energy T + W , and one of the coordinates may be taken to be r, the distance apart of the circuits.

•(513),

The Lagrangian equation corresponding to the coordinate r is found to be (cf. equation (508)),

d fdT\ d(T-W)_R dt\dr] dr

and since we know that, in the equilibrium configuration,

d (dT\ n p .,dM

we obtain on substitution in equation (513),

d(T-W)__ ..,m

dr dr

From equation (512) we see that the right-hand member is the value of

dE ed(T+W) dW A , ,. ,

^— , or or jr . Hence our equation shews that -r — = 0, from which we

deduce that W = 0. In other words, assuming that a system of steady currents forms a dynamical system, the energy of this system must be wholly kinetic.

This result compels us also to accept that the energy of a system of magnets at rest must also be wholly kinetic. We shall discuss this result later. For the present we confine our attention to the case of electric phenomena only. We have found that if the mechanism of these pheno- mena is dynamical (the hypothesis upon which we are going to work), then the energy of electric currents must be kinetic.

496 Dynamical Theory of Currents [ch. xvi

Induction of Currents.

556. Let us consider a number of currents flowing in closed circuits. Let the strengths of the currents be i1} i2, ... and let the number of tubes of induction which cross these circuits at any instant be N1} N2, ..., so that if the magnetic field arises entirely from the currents, we have (cf. § 502)

Ar _ . _ . \ (514).

J\2 = L2li1 + L22i2 + ..., etc. J

The energy of the currents is wholly kinetic so that we may take

= %(L11i* + 2L12i1i2 + ...) as before (§ 503).

In the general dynamical problem, it will be remembered that T was a quadratic function of the velocities. Thus i1} i2, ... must now be treated as velocities and we must take as coordinates quantities x1} x2, ..., defined by

• CciZ-j • CLu/2

h = W l2 = ~dt'etc'

Clearly xx measures the quantity of electricity which has flowed past any point in circuit 1 since a given instant, and so on. Thus in terms of the coordinates x1} x2, ... we have

T=^(Lux12 + 2Lni1d>a + ...) (515).

There is no potential energy in the present system, but the system is acted on by external forces, namely the electromotive forces in the batteries and the reaction between the currents and the material of the circuits which shews itself in the resistance of the circuits. We have therefore to evaluate the generalised forces ®1} ®2> ••••

Consider a small change in the system in which xx is increased by 8xlf so that the current ^ flows for a time dt given by i1dt = Sx1. The work per- formed by the battery is E1Bx1, the work performed by the reaction with the matter of the circuit, being equal and opposite to the heat generated in the circuit, is — R^dt. Thus if Xj is the generalised force corresponding to the coordinate xlf we have

Xx hxx = E^ 8xt — R^ifdt,

so that X1 = Ex — R^.

The Lagrangian equation corresponding to the coordinate x1 is

d_ (d_T\ _dT^ dt [dij dx, ly

or ~(Lui1 + L12i2+...) = E1-R1i1 (516),

or again jfc, — ■^- = K1%1.

556, 556 a] Induction of Currents 497

The equations corresponding to the coordinates x2, x3> ••• are

h2 — = K2%2, etc.

Thus the Lagrangian equations are found to be exactly identical with the equations of current-induction already obtained, shewing not only that the phenomenon of induction is consistent with the hypothesis that the whole mechanism is a dynamical system, but also that this phenomenon follows as a direct consequence of this hypothesis. In this system the accessible parts of the mechanism are the currents flowing in the wires; the inaccessible parts consist of the ether which transmits the action from one circuit to another.

556 a. On the electron theory, the kinetic energy must be supposed made up partly of magnetic energy, as before, and partly of the kinetic energy of the motion of the electrons by which the current is produced.

Let the average forward velocity of the electrons at any point be u0 (cf. § 345 a), and let U + uQ be the actual velocity of any single electron, so that the average value of u is nil. The kinetic energy of motion of the electrons, say Te, is then

The first term represents part of the heat-energy of the matter, and this does not depend on the values of the currents xx, x2, — To evaluate the second term we use equation (b) of § 345 a,

Ne u0 = i = x,

and obtain the kinetic energy o£ the electrons in the complete system of currents in the form

s + x2'j^ds +

m

Thus the total kinetic energy may still be expressed in the form (515) if we take

'<n = L'n+j -jy-;ds, etc (517),

and in this the first term is the contribution from the magnetic energy (cf. § 503), and the second term is the contribution from the kinetic energy of the electrons.

Equation (516) assumes the form

j.

--BiV

-(/IH§"

(517 a). 32

498 Dynamical Theory of Currents [ch. xvi

If the induction terms on the left are omitted, we have as the equation of a circuit in which induction is negligible

ft-Rih-ljjfeds

dt=°'

This, with the help of the formulae of § 345 a, may be expressed in the form

jxds-i.j^ds-^f^ds-O,

which in turn is seen to be exactly identical with equation (c) of § 345 a, integrated round the circuit.

Thus we see that the analysis of § 556 applies perfectly to the electron theory of matter, provided Ln, L^, ... are supposed to have the values given by equation (517), and equation (517 a) is then the general equation of induction of currents, when the inertia of the electrons is taken into account.

Electrokinetic Momentum.

557. The generalised momentum corresponding to the coordinate xx is =-r or Nx. Thus the generalised momenta corresponding to the currents

OX\

in the different circuits are Nx, N2, ..., the numbers of tubes of induction which cross the circuits. The quantity Nx is accordingly sometimes called the electrokinetic momentum of circuit 1, and so on.

If we give to Ln the value obtained in equation (517) of § 556 a, the value of the electrokinetic momentum is (cf. equations (514))

(L'nh + L12i2 +...) + hj -y- ds,

in which clearly the last term comes from the momentum of the electrons, and the remaining terms from the momentum of the magnetic field.

Examples. I. Discharge of a Condenser.

558. As a further illustration of the dynamical theory, let us consider the discharge of a condenser. Let Q be the charge on the positive plate at any instant, and let this be taken as a Lagrangian coordinate. The

current i is given by i = - -^- = — Q. In the notation already employed

(§ 516) we have

556a-559] Electric Oscillations 499

and Lagrange's equation is

d fd_T\ dT d_W= dt \dQj dQ+ dQ~ *'

which is the equation already obtained in § 516, and leads to the solution already found.

II. Oscillations in a network of conductors

559. The equations governing the currents flowing in any network of conductors when induction is taken into account can be obtained from the general dynamical theory.

Let us suppose that the currents in the different conductors are h> h, • •• iny and let the corresponding coordinates be xlt x2, ... xn, these

being given by \ = -77- , etc. If any conductor, say 1, terminates on a

condenser plate, let xx denote the actual charge on the plate, and let the

dx current be measured towards the plate, so that the relations ix = -~, etc.

will still hold. Let conductor 1 contain an electromotive force Ex and be of resistance Rr.

The quantities xlt x2, ... may be taken as Lagrangian coordinates, but they are not, in general, independent coordinates. If any number of the conductors, say 2, 3, ... s meet in a point, the condition for no accumulation of electricity at the point is, by Kirchhoff 's first law,

i2±h± ... ± is = 0,

from which we find that variations in x2, x3, ... are connected by the relations

Bx2 ± Sx3 + ... + Sxs = 0.

Let us suppose that there are m junctions. The corresponding con- straints on the values of Sxl} 8x2> ... can be expressed by m equations of the form

a18x1 + a2Sx2+ ... +anBxn = 0) .....

\ (518)>

&! Sxx + b28x2 + ... + bn Sxn = 0>

etc., in which each of the coefficients ax, a2) ... an, b1} ... has for its value either 0, + 1 or — 1.

The kinetic energy Twill be a quadratic function of xx,x2, etc., while the potential energy W (arising from the charges, if any, on the condensers) will

32—2

500

Dynamical Theory of Currents

fCH. XVI

be a quadratic function of x1; x2, .... The dynamical equations are now n in number, these being of the form (cf. equations (509))

d fdT\ BT dW „ „ . , . , _ _ . /tin,

a(a)-s+^-*-^+x^+^+";('-1^-"»)-"(619)-

These equations, together with the m equations obtained by applying Kirchhoff's first law to the different j unctions, form a system of m+n equa- tions, from which we can eliminate the m multipliers \ /x, ..., and then determine the n variables %1, x2, ... xn.

560. As an example of the use of these equations, let us imagine that a current / arrives at A and divides into two parts ilt i2, which flow along arms

C t

ACB, ADB and reunite at B. Neglecting induction between these arms

and the leads to A and B, we may suppose that the part of the kinetic energy

which involves ij and i2 is

±Li1* + Mi1i% + lN'if.

There are no batteries and no condenser in the arms in which the

currents ix and %2 flow. The currents are, however, connected by the

relation

ij + i2 = I

so that the corresponding coordinates xY and x2 are connected by

8xi + Sx2 = 0.

The dynamical equations are now found to be (cf. equations (519))

d

dt dt

(Liy + Mi2) = - Rix + X, (Mii + Ni2) = - Si2 + .

If we subtract and replace i2 by I — i'i , we eliminate \ and obtain

di dT

(L + N-2M)a£ + (M-N)<^ = SI-(R + S)i1.

If I is given as a function of the time, this equation enables us to deter- mine i1} and thence i2.

559-561] Electric Oscillations 501

For instance, suppose that the current I is an alternating current of frequency p\2ir. If we put I = i0eipt, the solution of the equation is

. S-(M-N)ip

h~(L + N-2M)ip + (R + S) '

while similarly i2 =

(L + N-2M)ip + (R + S)

When p — 0, the solution of course reduces to that for steady currents. As p increases, we notice that the three currents il3 i2 and / become, in general, in different phases, and that their amplitudes assume values which depend upon the coefficients of induction as well as on the resistances. Finally, for very great values of p, the values of ix and i% are given by

N^M= T^M = L + N - 2M ' shewing that the currents are now in the same phase and are divided in a ratio which depends only on their coefficients of induction. For instance, if the arms AGB, ABB are arranged so as to have very little mutual induction (M very small), the current will distribute itself between the two arms in the inverse ratio of the coefficients of self-induction.

It is possible to arrange for values for L, M and N such that the two currents it and i2 shall be of opposite sign. In such a case the current in one at least of the branches is greater than that in the main circuit. Let us, for instance, suppose that the branches consist of two coils having r and s turns respectively, arranged so as to have very little magnetic leakage. Then LN — M2 is negligible (cf. § 525) and we have approximately

r- rs s2 ' The equations become

h. _ *2 I

s —r s — r'

so that the currents will flow in opposite directions, and either may be greater than the current in the main circuit. By making s nearly equal to r and keeping the magnetic leakage as small as possible, we can make both currents large compared with the original current.

III. Rapidly alternating currents.

561. This last problem illustrates an important point in the general theory of rapidly alternating currents. In the general equations (519),

d (dT\ dT 3W „ D . . ,

dmy^8+wrE'~Bela+Xaa+tih'+---'

let us suppose that the whole system is oscillating with frequency pj2-rr, which is so great that it may be treated as infinite. We may assume that every

502 Dynamical Theory of Currents [ch. xvi

d_ dt

d variable is proportional to eipt, and may accordingly replace -7- by the multi

plier ip. The equations now become

. (dT\ dT dW „ _ . ^ .

%P[^~) ? + d — s = s + ^ s + '"'

and all the terms on the left hand may be neglected in comparison with the first, which contains the factor ip. The terms on the right cannot legitimately be neglected because , fi,... are entirely undetermined, and may be of the same large order of magnitude as the terms retained. If we replace A,, fi, . . by ipX', ipfi, ..., the equations become

— + 'as + ix'bs 4- ... =0, etc.

OXg

in which V, //,... are now undetermined multipliers. These, however, are exactly the equations which express that T is a maximum or a minimum for values of cb1} x2, ... which are consistent with the relations (cf. § 559) necessary to satisfy Kirchhoff" s first law. Since T can be made as large as we please, the solution must clearly make T a minimum. Thus we see that

As the frequency of a system of alternating currents becomes very great, the currents tend to distribute themselves in such a way as to make the kinetic energy of the currents a minimum subject only to the relations imposed by Kirchhoff 's first law.

This result may be compared with that previously obtained (§ 357) for steady currents. We see that while the distribution of steady currents is. determined entirely by the resistance of the conductors, that of rapidly alternating currents is, in the limit in which the frequency is infinite, determined entirely by the coefficients of induction.

It follows that, in a continuous medium of any kind, the distribution of rapidly alternating currents will depend only on the geometrical relations of the medium, and not on its conducting properties. In point of fact, we have already seen that the current tends to flow entirely in the surface of the conductor (§ 537). We now obtain the further result that it will, in the limit, distribute itself in the same way over the surface of this conductor, no matter in what way the specific resistance varies from point to point of the surface.

IV. Transmission of Signals along a wire.

562. Imagine a signal being sent along a wire, initially free from all electrical disturbance. At any instant let i denote the current at a point distant x from the end of the wire, and let q denote the total quantity of electricity which has flowed past this point. Then i and q are functions of x and t.

561, 562] Electric Oscillations 503

Let q be measured in electrostatic units, but let i be measured in electro- magnetic units. Then the rate of flow past any point will be iC electrostatic units per second, where C denotes the number of electrostatic units in one electromagnetic unit (cf. § 484). Thus

L% df

If L is the self-induction of the wire per unit length, the total kinetic energy of the currents is

where the integral is taken along the wire. In any element dx of the wire the charge is — ^ dx, so that if K is the electrostatic capacity of the wire per unit length, the potential energy W is given by

Let R be the resistance of the wire per unit length in electromagnetic units, then the rate of generation of heat is

R \i2dx.

The values of q at different points of the wire may be taken as Lagrangian coordinates, for they suffice to specify the position of each element of current. The Lagrangian equation corresponding to the coor- dinate q at any distance x will be (cf. § 556)

dt[dq) dq+'dq~ Mh

m which we have ~-r = -^r2 q and — = 0 .

dW To evaluate -=— , let us imagine q changed to q + 8q at every point of the

wire, subject to Bq vanishing at the two ends. The increment in W, say 8W, is given by

'(d(q + Bq)* fdqV

\ dx j \dx)

sw=m

dx=Ti!a£iL(Sq)dx-

and, on integrating by parts, this becomes

Thus at any point x,

dW ld*q

dq ~ Kdx2

504 Dynamical Theory of Currents [ch. xvi

The Lagrangian equation accordingly becomes

O2 dt* K dx* ~ dt'

K.?2

Since Gi=-~ , it is at once seen that the current i at any point satisfies dt

the same differential equation, and this is also true of the potential V, since

dx namely

^- = — KV. Thus q, i and V all satisfy the same differential equation,

Ek*±. + KR*±-»± (520)

This equation is the general equation for the transmission of electric signals along a wire. It is called the " Telegraphic equation " by Poincare and others.

We have seen in § 505 a, how to calculate the self-induction per unit length of any wire. If the wire is sufficiently thin in comparison with its distance from other conductors, the self-induction L per unit length becomes identical with the quantity denoted by L' in § 505, and we accordingly have the relation (cf. equation (430 e)),

KL = k/jl,

where k is the dielectric constant, and /x the magnetic permeability of the insulator surrounding the wire. Let us put

a? = —

KfA

so that a depends only on the properties of the insulating material, and the telegraphic equation becomes

a?dt* + dt " dx*'

For slow signals, the first term in this equation, which arises from the inertia of the electric current, may be neglected. The equation then reduces to equation (303) of Chapter IX which was obtained as the equation of transmission of signals along a submarine cable. Under practical conditions signals along a submarine cable are so retarded by the high electrostatic capacity of the cable that this inertia term may legitimately be neglected, but the term has to be retained when the equation is applied to telegraph and telephone problems.

When the wire is far removed from other conductors, the electrostatic capacity K will be small. If K is neglected entirely, the equation becomes

d-6 992(f>

dt" dx*'

562, 563] Mechanical Action 505

The solution of this equation is

$ =f(oc - at) + <E> {x + at),

where /, O are arbitrary functions, and the solution is seen to represent the transmission of a signal without change of type or loss of intensity, the velocity of transmission being a.

In practical telephony and telegraphy it is not usual possible to neglect entirely the value of K in the second term of the equation. Solutions of the general three-term equation have been obtained by Heaviside*, Poin- caref, PicardJ, Boussinesq§, and Riemann||.

It is found that the signal is still transmitted with the same velocity a, but that there is a change of type and loss of intensity ; there is also an electric field and current left trailing behind each signal; these would of course tend to confuse the succeeding signal if the signals are sent without sufficient interval.

Thus for rapid transmission or clear speaking it is necessary to reduce the value of KR (cf. § 369); the smaller this term is made, the smaller the amount of blurring or indistinctness will be. We see at once why telephone wires are kept as far as possible from other conductors, and can understand the difficulty of clear speaking or rapid signalling through a submarine cable.

Mechanical Force acting on a Circuit.

563. Let 6 be any geometrical coordinate, and let © be the generalised force tending to increase the coordinate 6, so that to keep the system of circuits at rest we must suppose it acted on by an external force — ©. Then Lagrange's equation for the coordinate 6 is

±^T\JT

dt Vb0/ de

and therefore, when the system is in equilibrium, we must have

*-% <521>-

If the energy of the system were wholly potential and of amount W, the

force © would be given by

dW © =

de

Thus the mechanical forces acting are just the same as they would be if the system had potential energy of amount — T.

• Phil. Mag. 1888 and Coll. Papers. t c- R- 117 (1893), p. 1027.

t C. R. 118 (1894), p. 16. § C. P. 118 (1894), p. 162.

|| Riemann-Weber, Die partielle Differentialgleichungen der Math. Physik, 4th edn. (1901), ii. p. 322.

506 Dynamical Theory of Currents [ch. xvi

564. Let us suppose that any geometrical displacement takes place, this resulting in increases WY, 802, ... in the geometrical coordinates 01} d2, ..., and let the currents in the circuits remain unaltered, additional energy being supplied by the batteries when needed.

The increase in the kinetic energy of the system of currents is

while the work done by the electrical forces during displacement is 2©c£0 which, by equation (521), is also equal to

These two quantities would be equal and opposite if the system were a conservative dynamical system acted on by no external forces. In point of fact they are seen to be equal and of the same sign. The inference is that the batteries supply during the motion an amount of energy equal to twice the increase in the energy of the system. Of this supply of energy half appears as an increase in the energy of the system, while the other half is used in the performance of mechanical work.

This result should be compared with that obtained in § 120.

565. As an example of the use of formula (521), let us examine the force acting on an element of a circuit. Let the

components of the mechanical force acting on any element ds of a circuit carrying a current i be de- noted by X, 7, Z.

To find the value of X, we have to consider a displacement in which the element ds is displaced a distance dx parallel to itself, the remainder of the

circuit being left unmoved. Let the component of magnetic induction perpendicular to the plane containing ds and dx be denoted by N, then if T denotes the kinetic energy of the whole system, the increase in T caused by displacement will be equal to i times the increase in the number of tubes of induction enclosed by the circuit, and therefore

dT=iNdsdcc.

Thus, using equation (521),

X = 7T- = iNds,

and there are similar equations giving the values of the components Fand Z.

If B is the total induction and if B cos e is the component at right angles to ds, then the resultant force acting on ds is seen to be a force of amount iB cos e ds, acting at right angles to the plane containing B and ds, and in such a direction as to increase the kinetic energy of the system. This is a generalisation of the result already obtained in § 498.

564-567] Magnetic Energy 507

Magnetic Energy.

566. We have seen that the energy of the field of force set up by a system of electric currents must be supposed to be kinetic energy. We know also that this field is identical with that set up by a certain system of magnets at rest. These two facts can be reconciled only by supposing that the energy of a system of magnets at rest is kinetic energy — a suggestion originally due to Ampere.

Weber's theory of magnetism (§476) has already led us to regard any magnetic body as a collection of permanently magnetised particles. Ampere imagined the magnetism of each particle to arise from an electric current which flowed permanently round a non-resisting circuit in the interior of the particle. The phenomena of magnetism, on this hypothesis, become in all respects identical with those of electric currents, and in particular the energy of a magnetic body must be interpreted as the kinetic energy of systems of electric currents circulating in the individual molecules. For instance two magnetic poles of opposite sign attract because two systems of currents flowing in opposite directions attract.

We have seen that the mechanical forces in a system of energy E are

BE . . BE

— -jtq , etc., if the energy is potential, but are + ^z, etc., if the energy is

kinetic. It might therefore be thought that the acceptance of the hypothesis

that all magnetic energy is kinetic would compel us to suppose all mechanical

forces in the magnetic system to be the exact opposites of what we have

previously supposed them to be. This, however, is not so, because accepting

this hypothesis compels us also to suppose the energy to be exactly opposite

in amount to what we previously supposed it to be. Instead of supposing

BE that we have potential energy E and forces — ■«— , etc., we now suppose that

we have kinetic energy — E and forces H = — - , etc., so that the amounts of

OOl/

the forces are unaltered.

To understand how it is that the amount of the magnetic energy must be supposed to change sign as soon as we suppose it to originate from a series of molecular currents, we need only refer back to § 502.

567. The molecular currents by which we are now supposing magnetism to be originated must be supposed to be acted on by no resistance and by no batteries, but if the assemblage of currents is to constitute a true dynamical system we must suppose them capable of being acted upon by induction whenever the number of tubes of force or induction which crosses them is changed. In the general dynamical equation

d(BT\ dT

508 Dynamical Theory of Currents [ch. xvi

we may put E and R each equal to zero, and ^- is already known to vanish.

7\T Thus the equation expresses that -^-. remains unaltered.

We now see that the strengths of the molecular currents will be changed by induction in such a way that the electrokinetic momentum of each remains unaltered. If the molecule is placed in a magnetic field whose lines of force run in the same direction as those from the molecule, then the effect of induc- tion is to decrease the strength of the molecule until the aggregate number of tubes of force which cross it is equal to the number originally crossing it. This effect of induction is of the opposite kind from that required to explain the phenomenon of induced magnetism in iron and other paramagnetic sub- stances. It has, however, been suggested by Weber that it may account for the phenomenon of diamagnetism.

568. Modern views as to the structure of matter compel us to abandon Ampere's conception of molecular currents, but this conception can be re- placed by another which is equally capable of accounting for magnetic phenomena. On the modern view all electric currents are explained as the motion of streams of electrons. The flow of Ampere's molecular current may accordingly be replaced by the motion of rings of electrons. The rotation of one or more rings of electrons would give rise to a magnetic field exactly similar to that which would be produced by the flow of a current of electricity in a circuit of no resistance.

It is on these lines that it appears probable that an explanation of magnetic phenomena will be found in the future. No complete explanation has so far been obtained, for the simple and sufficient reason that the arrange- ment and behaviour of the electrons in the molecule or atom is still unknown.

EXAMPLES.