The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 29 of 39

We now see that the second force is the force arising from the ordinary electrostatic field, so that we may identify "W with the electrostatic potential when no changes are occurring. The meaning to be assigned to Sf when changes are in progress is discussed below (Chapter xx).

532. If the medium is a conducting medium, the presence of the electric forces sets up currents, and the components u, v, w of the current at any point are, as in § 374, connected with the currents by the equations

X = TU, Y=TV, Z = T1V,

these equations being the expression of Ohm's Law, where t is the specific resistance of the conductor at the point.

On substituting these values for X, Y, Z in equations (464) — (466) or (470) — (472), we obtain a system of equations connecting the currents in the conductor with the changes in the magnetic field.

533. There is, however, a further system of equations expressing rela- tions between the currents and the magnetic field. We have seen (§ 480) that a current sets up a magnetic field of known intensity, and since the whole magnetic field must arise either from currents or from permanent magnets, this fact gives rise to a second system of equations.

In a field arising solely from permanent magnetism, we can take a unit pole round any closed path in the field, and the total work done will be nil. Hence on taking a unit pole round a closed circuit in the most general magnetic field, the work done will be the same as if there were no perma- nent magnetism, and the whole field were due to the currents present. The amount of this work, as we have seen, is 4nrXi, where %i is the sum of all the currents which flow through the circuit round which the pole is taken. If u, v, w are the components of current at any point, we have

St =

1 1 (lu + mv + nw) dS,

the integration being over any area which has the closed path as boundary. Hence our experimental fact leads to the equation

476 Induction of Currents in Continuous Media [ch. xv

Transforming the line integral into a surface integral by Stokes' Theorem (§ 438), we obtain the equation in the form

'd& da

//K6-I-H-(M-

irv ) + n [ ^ — = 4>7rtu I [ dS — 0.

ox oy

As with the integral of § 529, each integrand must vanish for all values of I, m, n, so that we must have

4™=i-i -<«3>-

^Ji-% («5>-

534. If  we  differentiate  these  three  equations  with  respect  to  x,  y,  z 
     

respectively and add, we obtain

•(476),

du dv dw _

2- + ~-+j- = 0

ox oy dz

of which the meaning (cf. § 375, equation (311)) is that no electricity is destroyed or created or allowed to accumulate in the conductor.

The interpretation of this result is not that it is a physical impossibility for electricity to accumulate in a conductor, but that the assumptions upon which we are working are not sufficiently general to cover cases in which there is such an accumulation of electricity. It is easy to see directly how this has come about. The supposition underlying our equations is that the work done in taking a unit pole round a circuit is equal to An times the total current flow through the circuit. It is only when equation (476) is satisfied by the current components that the expression " total flow through a circuit " has a definite significance : the current flow across every area bounded by the circuit must be the same. We shall see later (Chapter xvn) how the equations must be modified to cover the case of an electric flow in which the condition is not satisfied. For the present we proceed upon the supposition that the condition is satisfied.

Currents in homogeneous media.

535. Let  us  now  suppose  that  we  are  considering  the  currents  in  a 
     

homogeneous non-magnetised medium. We write

a = fjLa, etc., X = tu, etc.,

in which fx and t are constant. The systems of equations of §§ 529 and 533 now become

da. (dw dv^

dt

-A4j7 = tI — ~ — I . etc-

dy dz

4,iru = — —--, etc. oy oz

.(477), .(478).

533-537] General Equations 477

Differentiating equation (478) with respect to the time, we obtain

du d ( dy\ d ( d/3\ ^dt=Zy?Tt)-dz?Tt)

f 3 fdv du\ d /du dw \dy \dx dy/ dz \dz dx

~ T {dx~* + df + dzV dx\dx dy+dz

= tV2u,

in virtue of equation (476).

Similar equations are satisfied by the other current-components, so that we have the system of differential equations

\

4777* du = V2 r dt~

4>7ru dv _„ t dt

(479).

4t7ra dw „, t dt I

If we eliminate the current-components from the system of equations (477) and (478), we obtain

E£^=Va (480),

t dt

and similar equations are satisfied by b and c.

536. The equation which has been found to be satisfied by u, v, w, a, /3 and 7 is the well-known equation of conduction of heat. Thus we see that the currents induced in a mass of metal, as well as the com- ponents of the magnetic field associated with these currents, will diffuse through the metal in the same way as heat diffuses through a uniform conductor.

Rapidly alternating currents.

537. The equations assume a form of special interest when the currents are alternating currents of high frequency. We may assume each component of current to be proportional to eipt (cf. § 514), and may then replace the

operator -j- by the multiplier ip. The equations now assume the form

in^E«-v>» (48i),

^wa=V2a>etCti

478 Induction of Currents in Continuous Media [oh. xv

and if p is so large that it may be treated as infinite, these equations assume the simple form

u = v = w = 0,

a = b = c = 0.

Thus for currents of infinite frequency, there is neither current nor magnetic field in the interior. The currents are confined to the surface, and the only part of the conductor which comes into play at all is a thin skin on the surface.

Equations (481) enable us to form an estimate of the thickness of this skin when the frequency of the currents is very great without being actually infinite.

At a point 0 on the surface of the conductor, let us take rectangular axes so that the direction of the current is that of Ox while the normal to the surface is Oz. If the thickness of the skin is vezy small, we need not consider any region except that in the immediate neighbourhood of the origin, so that the problem is practically identical with that of current flowing parallel to Ox in an infinite slab of metal having the plane Oxy for a boundary.

Equation (481) reduces in this case to

4s7rfiip d2u

Tu = d?>

and if we put — = «2, the solution is

u = Ae-KZ + BeK*. The value of k is found to be

so that u = Ae v T e v T + Bey T ev T

and the condition that the current is to be confined to a thin skin may now be expressed by the condition that u = 0 when z = oo , and is accordingly B = 0. The multiplier A is independent of z, but will of course involve the time through the factor eipt; let us put A =u0eipt, and we then have the solution

-n/¥v("-v/2:f*)

u = une e

537] General Equations 479

Rejecting the imaginary part, we are left with the real solution

u = u0e V t cos (pi — a/ z)>

from which we see that as we pass inwards from the surface of the con- ductor, the phase of the current changes at a uniform rate, while its amplitude decreases exponentially.

We can best form an idea of the rate of decrease of the amplitude by considering a concrete case. For copper we may take (in c.o.s. electromagnetic units) fi = l, t = 1600. Thus for a current which alternates 1000 times per second, we have

p = 2nx 1000, a/ — — = 5 approximately.

It follows that at a depth of 1 cm. the current will be only e~5 or -0067 times its value at the surface. Thus the current is practically confined to a skin of thickness 1 cm.

rz= oo The total current per unit width of the surface at a time t is I u dz, of

J e=0

which the value is found to be

n0 cos (pt — j

4>7T/U.p

Thus, if we denote the amplitude of the aggregate current by U, the value of uQ will be U a/ — .

The heat generated per unit time in a strip of unit width and unit length is

•t=l rz = cc

I u2dtdz

t = Q J 2 = 0

= hTU<?

J z=0

e

-V5^

.-. -2x/2-^lz

dz

2~/j,p

T

Thus the resistance of the conductor is the same as would be the

resistance for steady currents of a skin of depth 2/

The results we have obtained will suffice to explain why it is that the conductors used to convey rapidly alternating currents are made hollow, as also why it is that lightning conductors are made of strips, rather than cylinders, of ruetaL

480 Induction of Currents in Continuous Media [ch. xv

Plane Current-sheets.

538. We  next  examine  the  phenomenon  of  the  induction  of  currents 
     

in a plane sheet of metal.

Let the plane of the current-sheet be taken to be z = 0. Let us introduce a current-function <I>, which is to be defined for every point in the sheet by the statement that the total strength of all the currents which flow between the point and the boundary is <I>. Then the currents in the sheet are known when the value of <E> is known at every point of the sheet. If we assume that no electricity is introduced into, or removed from, the current-sheet, or allowed to accumulate at any point of it, then clearly <E> will be a single- valued function of position on the sheet.

The equation of the current-lines will be <l> = constant, and the line <1> = 0 will be the boundary of the current-sheet. Between the lines <I> and <& ■+• d<& we have a current of strength d<& flowing in a closed circuit. The magnetic field produced by this current is the same as that produced by a magnetic shell of strength d<& coinciding with that part of the current- sheet which is enclosed by this circuit, so that the magnetic effect of the whole system of currents in the sheet is that of a shell coinciding with the sheet and of variable strength <£. This again may be replaced by a distribution of magnetic poles of surface density <l>/e on the positive side of the sheet, together with a distribution of surface density — <£/e on the negative side of the sheet, where e is the thickness of the sheet.

Let P denote the potential at any point of a distribution of poles of strength <E>, so that

P = Jjjdafdyf (482),

where doc' dy' is any element of the sheet. The magnetic potential at any point outside the current-sheet of the field produced by the currents is then

dP tt = -gj (483).

If a is the resistance of a unit square of the sheet at any point, and u, v the components of current, we have, by Ohm's Law,

X = au, Y=cv. The components u, v are readily found to be given by

538, 539] Plane Current-sheets 481

so that we have the equations

z = ^' ¥=-"te <484)

true at every point of the sheet. Hence, by equation (466),

dc_dY dX /32<I> 323>\

The total magnetic field consists of the part of potential II due to the currents and a part of potential (say) O', due to the magnetic system by which the currents are induced. Thus the total magnetic potential is O + Of, and at a point just outside the current-sheet (taking /x = 1)

dc d 3 ._ _.,.

the equation (485) becomes

M«°+«>— £+S) «"*

The function P (equation (482)) is the potential of a distribution of poles of surface density <t> on the sheet. Hence P satisfies Laplace's equation at all points outside the sheet, and at a point just outside the sheet and on its

positive face — — = 2tt<£>.

Hence,

at a point just

outside the positive

face of the sheet,

823>

32<J>

1 / d3P

2tt \dx2dz

9»Py

1 dy2dz)

dx2

dy2 '

1 d3P

2tt dz3

i d2n

2tt dz2 '

by

equation (483),

so that

equation

(486) becomes

d d ,~

. -r>#x <r

d2n

and similarly, at the negative face of the sheet, we have the equation

d 3 /r. _./N a- 32fi

S5-(fl + O0 — 3-ct (488).

Finite Current-sheets.

539. Suppose  that  in  an  infinitesimal  interval  any  pole  of  strength  m 
     

moves from P to Q. This movement may be represented by the creation of a pole of strength -m at P and of one of strength + m at Q. Thus j. 31

482 Induction of Currents in Continuous Media [ch. xv

the most general motion of the inducing field may be replaced by the crea- tion of a series of poles. The simplest problem arises when the inducing field is produced by the sudden creation of a single pole, and the solution of the most general problem can be obtained from the solution of this simple problem by addition.

From equations (487) and (488) it is clear that -j- ^- (fl + fl') remains finite on both surfaces of the sheet during the sudden creation of a new

pole, so that ^- (H + O') remains unaltered in value over the whole surface

of the sheet. Let the increment in — (H + CI') at any point in space be

denoted by A, then A is a potential of which the poles are known in the space outside the sheet, and of which the value is known to be zero over the surface of the sheet. The methods of Chapter viii are accordingly available for the determination of A : the required value of A is the electrostatic potential when the current-sheet is put to earth in the

on'

presence of the point charges which would give a potential -=— if the sheet

02

were absent.

Physically, the fact that ^- (O + X2') remains unaltered over the whole

surface of the sheet means that the field of force just outside the sheet remains unaltered, and hence that currents are instantaneously induced in the sheet such that the lines of force at the surfaces of the sheet remain unaltered.

The induced currents can be found for any shape of current-sheet for which the corresponding electrostatic problem can be solved *, but in general the results are too complicated to be of physical interest.

Infinite Plane Current-sheet.

540. Let the current-sheet be of infinite extent, and occupy the whole of the plane of xz, and let the moving magnetic system be in the region in which z is negative. Then throughout the region for which z is positive the potential Cl + Cl' has no poles, and hence the potential

dtdz{iL + n) 2^3^

• See a paper by the author, "Finite Current-sheets," Proc. Land. Math. Soc. Vol. xxxi. p. 151.

539, 540] Plane Current-sheets 483

has no poles. Moreover this potential is a solution of Laplace's equation, and vanishes over the boundary of the region, namely at infinity and over the plane z = 0 (cf. equation (487)). Hence it vanishes throughout the whole region (cf. § 186), so that equation (487) must be true at every point in the region for which z is positive. We may accordingly integrate with respect to z and obtain the equation in the form

+^f ^

no arbitrary function of x, y being added because the equation must be satisfied at infinity.

The motion of the system of magnets on the negative side of the sheet may be replaced, as in § 539, by the instantaneous creation of a number of poles. At the creation of a single pole currents are set up in the sheet such that 12 + 12' remains unaltered (cf. equation (489)) on the positive side of the sheet. Thus these currents form a magnetic screen and shield the space on the positive side of the sheet from the effects of the magnetic changes on the negative side.

To examine the way in which these currents decay under the influence of resistance and self-induction, we put D,' = 0 in equation (489), and find that II must be a solution of the equation

dt 2-jt dz ' The general solution of this equation is

and this corresponds to the initial value

H=/(aj, y, z).

Thus the decay of the currents can be traced by taking the field of potential 12 at time t = Q and moving it parallel to the axis of z with a

velocity jr- .

31— '2

484 Induction of Currents in Continuous Media [ch. xv

EXAMPLES.

1. Prove that the currents induced in a solid with an infinite plane face, owing to magnetic changes near the face, circulate parallel to it, and may be regarded as due to the diffusion into the solid of current-sheets induced at each instant on the surface so as to screen off the magnetic changes from the interior.

Shew that for periodic changes, the current penetrates to a depth proportional to the square root of the period. Give a solution for the case in which the strength of a fixed inducing magnet varies as cos pt.

2. A magnetic system is moving towards an infinite plane conducting sheet with velocity w. Shew that the magnetic potential on the other side of the sheet is the same as it would be if the sheet were away, and the strengths of all the elements of the magnetic system were changed in the ratio R/(R + w), where 2nR is the specific resistance of the sheet per unit area. Shew that the result is unaltered if the system is moving away from the sheet, and examine the case of w = — R.

If the system is a magnetic particle of mass M and moment m, with its axis perpen- dicular to the sheet, prove that if the particle has been projected at right angles to the sheet, then when it is at a distance z from the sheet, its velocity z is given by

pf(£-i2)2=C-m2/823.

3. A small magnet horizontally magnetised is moving with a velocity u parallel to a thin horizontal plate of metal. Shew that the retarding force on the magnet due to the currents induced in the plate is

to2 uR WfQiQ + R)'

where m is the moment of the magnet, c its distance above the plate, 2nR the resistance of a sq. cm. of the plate, and Q2 = u2+R2.

4. A slowly alternating current I cos pt is traversing a small circular coil whose magnetic moment for a unit current is M. A thin spherical shell, of radius a and specific resistance <r, has its centre on the axis of the coil at a distance / from the centre of the coil. Shew that the currents in the shell form circles round the axis of the coil, and that the strength of the current in any circle whose radius subtends an angle cos-1/x at the centre is

^-L-iLL2(2»+l)^ -^cose„coS(^-a

, , (2n + Do- where tanen = - : — .

4:irpa

5. An infinite iron plate is bounded by the parallel planes x = k, x——h; wire is wound uniformly round the plate, the layers of wire being parallel to the axis of y. If an alternating current is sent through the wire producing outside the plate a magnetic force Hq cos pt parallel to z, prove that H, the magnetic force in the plate at a distance x from the centre, will be given by

„ „ (cos\i2mx--Qos2mx\S ^=Hcosh2^ + cos2mA; «» (** + #'

_ sinh m (h + x) sin m (h-x) -sinh m (h-x) sin m (h + x) cosh m(h + x) cos m {h-x) + cosh m (h - x) cos in (h+x)'

where m2 = 2irnpj<T.

Discuss the special cases of (i) mh small, (ii) mh large.

CHAPTER XVI

DYNAMICAL THEORY OF CURRENTS

General Theory of Dynamical Systems.

541. We have so far developed the theory of electromagnetism by- starting from a number of simple data which are furnished or confirmed by- experiment, and examining the mathematical and physical consequences which can be deduced from these data.

There are always two directions in which it is possible for a theoretical science to proceed. It is possible to start from the simple experimental data and from these to deduce the theory of more complex phenomena. And it may also be possible to start from the experimental data and to analyse these into something still more simple and fundamental. We may, in fact, either advance from simple phenomena to complex, or we may pass backwards from simple phenomena to phenomena which are still simpler, in the sense of being more fundamental.

As an example of a theoretical science of which the development is almost entirely of the second kind may be mentioned the Dynamical Theory of Gases. The theory starts with certain simple experimental data, such as the existence of pressure in a gas, and the relation of this pressure to the temperature and density of a gas. And the theory is developed by shewing that these phenomena may be regarded as consequences of still more funda- mental phenomena, namely the motion of the molecules of the gas.

In our development of electromagnetic theory there has so far been but little progress in this second direction. It is true that we have seen that the phenomena from which we started — such as the attractions and repulsions of electric charges, or the induction of electric currents — may be interpreted as the consequences of other and more fundamental phenomena taking place in the ether by which the material systems are surrounded. We have even obtained formulae for the stresses and the energy in the ether. But it has not been possible to proceed any further and to explain the existence of these stresses and energy in terms of the ultimate mechanism of the ether.

486 Dynamical Theory of Currents [ch. xvi

The reason why we have been brought to a halt in the development of electromagnetic theory will become clear as soon as we contrast this theory with the theory of gases. The ultimate mechanism with which the theory of gases is concerned is that of molecules in motion, and we know (or at least can provisionally assume that we know) the ultimate laws by which this motion is governed. On the other hand the ultimate mechanism with which electromagnetic theory is concerned is that of action in the ether, and we are in utter ignorance of the ultimate laws which govern action in the ether. We do not know how the ether behaves, and so can make no progress towards explaining electromagnetic phenomena in terms of the behaviour of the ether.

542. There is a branch of dynamics which attempts to explain the relation between the motions of certain known parts of a mechanism, even when the nature of the remaining parts is completely unknown. We turn to this branch of dynamics for assistance in the present problem. The whole mechanism before us consists of a system of charged conductors, magnets, currents, etc., and of the ether by which all these are connected. Of this mechanism one part (the motion of the material bodies) is known to us, while the remainder (the flow of electric currents, the transmission of action by the ether, etc.) is unknown to us, except indirectly by its effect on the first part of the mechanism.

543. An analogy, first suggested by Professor Clerk Maxwell, will ex- plain the way in which we are now attacking the problem.

Imagine that we have a complicated machine in a closed room, the only connection between this machine and the exterior of the room being by means of a number of ropes which hang through holes in the floor into the room beneath. A man who cannot get into the room which contains the machine will have no opportunity of actually inspecting the mechanism, but he can manipulate it to a certain extent by pulling the different ropes. If, on pulling one rope, he finds that others are set into motion, he will under- stand that the ropes must be connected by some kind of mechanism above, although he may be unable to discover the exact nature of this mechanism.

In this analogy, the concealed mechanism is supposed to represent those parts of the universe which do not directly affect our senses — e.g. the ether — while the ropes represent those parts of which we can observe the motion — e.g. material bodies. In nature, there are certain acts which we can perform (analogous to the pulling of certain ropes), and these are invariably followed by certain consequences (analogous to the motion of other ropes), but the ultimate mechanism by which the cause produces the effect is unknown. For instance we can close an electric circuit by pressing a key, and the needle of a distant galvanometer may be set into motion. "We infer that there must be some mechanism connecting the two, but the nature of this mechanism is almost completely unknown.

Suppose now that an observer may handle the ropes, but may not pene- trate into the room above to examine the mechanism to which they are

541-545] Hamilton's Principle 487

attached. He will know that whatever this mechanism may be, certain laws must govern the manipulation of the ropes, provided that the mechanism is itself subject to the ordinary laws of mechanics.

To take the simplest illustration, suppose that there are two ropes only, A and B, and that when rope A is pulled down a distance of one inch, it is found that rope B rises through two inches. The mechanism connecting A and B may be a lever or an arrange- ment of pulleys or of clockwork, or something different from any of these. But whatever it is, provided that it is subject to the laws of dynamics, the experimenter will know, from the mechanical principle of " virtual work," that the downward motion of rope A can be restrained on applying to B a force equal to half of that applied to A.

544. The branch of dynamics of which we are now going to make use enables us to predict what relation there ought to be between the motions of the accessible parts of the mechanism. If these predictions are borne out by experiment, then there will be a presumption that the concealed mechanism is subject to the laws of dynamics. If the predictions are not confirmed by experiment, we shall know that the concealed mechanism is not governed by the laws of dynamics.

Hamilton's Principle.

545. Suppose, first, that we have a dynamical system composed of dis- crete particles, each of which moves in accordance with Newton's Laws of Motion. Let any typical particle of mass mx have at any instant t coordi- nates xx,yXy zx and components of velocity ux, vx, wx, and let it be acted on by forces of which the resultant has components Xx> Yx, Zx. Then, since the motion of the particle is assumed to be governed by Newton's Laws, we have

«i^--*i (490),

«i^*i (491),

™>^r = ^ (492).

Let us compare this motion with a slightly different motion, in which Newton's Laws are not obeyed. At the instant t let the coordinates of this same particle be xx + 8xX) yx + 8yx, zx + 8zx and let its components of velocity be Wi + Swj, vx + 8vx, wx + 8ivx. Let us multiply equations (490), (491) and (492) by 8xx, Bylt 8zx respectively, and add. We obtain

mx (^ 8xx + ^ fyt + ^ W) = Xx 8xx + Yx 8yx + Zx 8zx . . .(493). Now ^8xx = ^(ux8xx)-uXJt(8xx)

= -y (ux 8xx) — itj 8ux.

488 Dynamical Theory of Currents [ch. xvi

If we sum equation (493) for all the particles of the system, replacing the terms on the left by their values as just obtained, we arrive at the equation

-r- Smx (tij Bx1 + vx 8yx + wx Bzx) - Smj (ux Bux + vx Bv, + w1 BwJ

= S(XlScc1+Y1By1 + ZM) (494).

Let T denote the kinetic energy of the actual motion, and T+BT that of the slightly varied motion, then

so that BT = %mx (i^Bu^ + vx Bvx + wx Bw,),

and this is the value of the second term in equation (494).

If W and W + B W are the potential energies of the two configurations (assuming the forces to form a conservative system), we have

W = - S J (X, dxx + Y, dVl + Zx dzx),

and 8F = -2(Z1S«1+F18y1 + ^18^1)J

and so the value of the right-hand member of equation (494) is — B W. We may now rewrite equation (494) in the form

8 (T — W ) = -ji Srax (wj Bxx + vx Byx + wx Bzx).

This equation is true at every instant of the motion. Let us integrate it throughout the whole of the motion, say from t — 0 to t = r. We obtain

~~\t=T

B\T-W)dt = Jo

ini! (Wi&Ci + v1By1 + WxBz^

t=o

.(495).

The displaced motion has been supposed to be any motion which differs only slightly from the actual motion. Let us now limit it by the restriction that the configurations at the beginning and end of the motion are to coincide with those of the actual motion, so that the displaced motion is now to be one in which the system starts from the same configuration as in the actual motion at time t = 0, and, after passing through a series of con- figurations slightly different from those of the actual motion, finally ends in the same configuration at time t = t as that of the actual motion. Mathe- matically this new restriction is expressed by saying that at times t = 0 and t = r we must have 8x=8y=Bz = 0 for each particle. Equation (495) now becomes

: t\T-W)dt = 0 (496).

Jo

546. Speaking of the two parts of the mechanism under discussion as the " accessible " and " concealed " parts, let us suppose that the kinetic and potential energies T and W depend only on the configuration of the

545-548] Lagrange's Equations 489

accessible parts of the mechanism. Then throughout any imaginary motion of the accessible parts of the system we shall have a knowledge of Tand W at every instant, and hence shall be able to calculate the value of

(\T-W)dt (497).

Jo

We can imagine an infinite number of motions which bring the system from one configuration A at time t = 0 to a second configuration B at time t = t, and we can calculate the value of the integral for each. Equation (496) shews that those motions for which the value of the integral is stationary would be the motions actually possible for the system. Having found which these motions were, we should have a knowledge of the changes in the accessible parts of the system, although the concealed parts remained unknown to us, both as regards their nature and their motion.

547. Equation (496) has been proved to be true only for a system con- sisting of discrete material particles. At the same time the equation itself contains, in its form, no reference to the existence of discrete particles. It is at least possible that the equation may be the expression of a general dynamical principle which is true for all systems whether they consist of discrete particles or not. We cannot of course know whether or not this is so. What we have to do in the present chapter is to examine whether the phenomena of electric currents are in accordance with this equation. We shall find that they are, but we shall of course have no right to deduce from this fact that the ultimate mechanism of electric currents is to be found in the motion of discrete particles. Before setting to work on this problem, however, we shall express equation (496) in a different form.

Lagrange's Equations for Conservative Systems of Forces.

548. Let 0lt #2, ... 0n be a set of quantities associated with a mechanical system such that when their value is known, the configuration of the system is fully determined. Then 0U 02, ... 0n are known as the generalised coordi- nates of the system.

The velocity of any moving particle of the system will depend on the values

dB dd of -Tf, -tj , etc. Let us denote these quantities by 0lt 62, etc. Let x be a

Cartesian coordinate of any moving particle. Then by hypothesis x is a function of 0l} 02, ..., say

x=f(01,02,...),

so that by differentiation,

dx_df a 3/ x

Tt-Wi^de2""

490

Dynamical Theory of Currents

[CH. XVI

Thus each component of velocity of each moving particle will be a linear function of 01; 02, ..., from which it follows that the kinetic energy of motion of the system must be a quadratic function of 0,, 02, ..., the coefficients in this function being of course functions of 0,, 02, ....

Let us denote T— W by L, so that L is a function of 0,, 02, ... 0n, and of 0„ 02, ... 0n, say

L = cf)(0u 02) ... 0n, 0„ 02, ... 0n).

If L + BL is the value ol L in the displaced configuration 0l + B01, 02+B02, ••• 8n+B0n, we have

Ot>i OUn OUi

so that equation (496), which may be put in the form

BL = 0,

f

Jo

now assumes the form

— 80, + 2 ^80,

30, i 80!

We have

B0, = (0, + 80,) - 0,

.(498).

L30,

so that /;g^=/;gfw<a

Hdt\deJ

The last term vanishes since, hy hypothesis, Sft vanishes at the beginning and end of the motion, and equation (498) now assumes the form

Jo i 130, cftW) Let us denote the integrand, namely

i (30, eft Vd0,/J by /, so that the equation becomes

Idt = 0.

548-550] Lagrange's Equations 491

The varied motion is entirely at our disposal, except that it must be continuous and must be such that the configurations in the varied motion coincide with those in the actual motion at the instants t = 0 and t = t. Thus the values of 801, 802, ... at every instant may be any we please which are permitted by the mechanism of the system, except that they must be continuous functions of t and must vanish when t = 0 and when t = r. Whatever series of values we assign to 801} 802, . .., we have seen that the equation

Idt = 0

o

is true. Hence the value of 7 must vanish at every instant, and we must have

^idL_didL)

i la^ dt\ddj)

549. At this stage there are two alternatives to be considered. It may be that whatever values are assigned to 801} 802, ... 80n, the new configura- tion #i 4- 801} 02 + $02, ••• &n+ 80n will be a possible configuration — that is to say, will be one in which the system can be placed without violating the constraints imposed by the mechanism of the system. In this case equation (499) must be true for all values of 80lt 892) ... 80n, so that each term must vanish separately, and we have the system of equations

dL d /dL

Q) = 0, (s = l, % ...n) (500)

d0s dt \dd^

There are n equations between the n variables 0lt 02, ... 0n and the time. Hence these equations enable us to trace the changes in 01} 02, ... 0n and to express their values as functions of the time and of the initial values of

0, 02, •■• @n> V, 02, ••■ 0n-

550. Next, suppose that certain constraints are imposed on the values of 0lf 02, ... 0n by the mechanism of the system. Let these be m in number, and let them be such that the small increments 80lf 802, ... 80n are connected by equations of the form

al801 + a2802+...+an80n = O (501),

b1801 + b2802+... + bn80n = O (502),

etc.

Then equation (499) must be true for all values of 80x, 802, ... which are such as also to satisfy equations (501), (502), etc. Let us multiply equations (501), (502), ... by \ fi, ... and add to equation (499).