The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 31 of 39

1. Two wires are arranged in parallel, their resistances being R and S, and their coefficients of induction being L, M, N. Shew that for an alternating current of frequency p the pair of wires act like a single conductor of resistance R and self-induction L, given by

R

RS(R + S)+p*{R(N-My + S(L-Mf}

L 1

NR* + LS2 + 2MRS+p'i(LiV-M2)(L + tf-2M) {R + Sf+p-^L + N-lMf'

2. A  conductor  of  considerable  capacity  S  is  discharged  through  a  wire  of  self-induc- 
   

tion L. At a series of points along the wire dividing it into n equal parts, (n-) equal conductors each of capacity *S" are attached. Find an equation to determine the periods of oscillations in the wire, and shew that if the resistance of the wire may be neglected the equation may be written

2 tan ^0 (S - %S') = S' cot ?«£, where the current varies as e~ , and sin2<£=>S"X2Z/4?j.

Examples 509

3. A "Wheatstone bridge arrangement is used to compare the coefficient of mutual

induction M of two coils with the coefficient of self-induction L of a third coil. One of the

coils of the pair is placed in the battery circuit AC, the other is connected to B, D as a

shunt to the galvanometer, and the third coil is placed in AD. The bridge is first balanced

for steady currents, the resistances of AB, BC, CD, DA being then Ru R2, R3, i?4: the

resistance of the shunt is altered till there is no deflection of the galvanometer needle at

make and break of the battery circuit, and the total resistance of the shunt is then R.

Prove that

LRR1=M{R1+Rif.

4. Two circuits each containing a condenser, having the same natural frequency when at a distance, are brought close together. Shew that, unless the mutual induction between the circuits is small, there will be in each circuit two fundamental periods of oscillation given by

„_ 1 1

where C1 , C2 are the capacities, Lx , L2 the coefficients of self-induction, and M the coefficient of mutual induction, of the circuits.

5. Let a network be formed of conductors A, B, ... arranged in any order. Prove that when a periodic electromotive force Fcospt is placed in A the current in B is the same in amplitude and phase as the current is in A when an electromotive force F cos pt is placed in B.

CHAPTEK XVII

DISPLACEMENT QUERENTS AND ELECTROMAGNETIC WAVES

Maxwell's Equations.

569. Our development of the theory of electromagnetism has been based upon the experimental fact that the work done in taking a unit magnetic pole round any closed path in the field is equal to 4nr times the aggregate current enclosed by this path. But it has already been seen (§ 534) that this development of the theory is not sufficiently general to take account of phenomena in which the flow of current is not steady: "the aggregate current enclosed by a path" is an expression which has a definite meaning only when the flow of current is steady. Before proceeding to a more general theory, which is to cover all possible cases of current flow, it is necessary to deter- mine in what way the experimental basis is to be generalised, in order to provide material for the construction of a more complete theory.

The answer to this question has been provided by Maxwell. According to Maxwell's displacement theory (§171), the motion of electric charges is accompanied by a "displacement" of the surrounding medium. The motion produced by this displacement will be spoken of as a "displacement-current," and we have seen that the total flow which is obtained by compounding the displacement-current with the current produced by the motion of electric charges (which will be called the conduction-current), will be such that the total flow into any closed surface is, under all circumstances, zero. Thus if S1} $2 are any two surfaces bounded by the same closed path s, the total flow of current across Si is the same as the total flow, in the same direction, across S2, so that either may be taken to be the flow through the circuit s. Maxwell's theory proceeds on the supposition that in any flow of current, the work done in taking a unit magnetic pole round s is equal to 4>tt times the total flow of current, including the displacement-current, through s. The justification for this supposition is obtained as soon as it is seen how it brings about a complete agreement between electromagnetic theory and innumerable facts of observation.

570. Let  us  first  put  the  hypothesis  of  the  existence  of  displacement- 
     

currents into mathematical language. Let u, v, w be the components of the

569, 570] Disjilacement Currents 511

ordinary current at any point which is produced by the motion of electric charges, and let this be measured, as before, in electromagnetic units (cf. § 484). Let the components of the displacement, which has been shewn to be identical with Faraday's polarisation (§ 172), be denoted as before by f, g, h. On Maxwell's theory of displacement, f, g, h are the quantities of electricity of the second kind which have crossed unit areas perpendicular to the coordinate axes at any point. The corresponding rates of current-flow, or quantities which cross unit area per unit time, are of course

df dg dh di' di' 'di'

These are accordingly the components of Maxwell's "displacement- current." They are, however, measured in electrostatic units. If we suppose there to be C electrostatic units of charge in one electromagnetic unit, the displacement current, measured in electromagnetic units, will have com- ponents

ldf ldg \dh

Gdt' Cdt' Gdt {0Z >'

and Maxwell's total current, measured in electromagnetic units, will have components

1 df Ida 1 dh

U + Cdi> V+Cdt>W+Cdi'

Maxwell's hypothesis is that the work done in taking a unit magnetic pole round a closed circuit is equal to 4)7r times the total current flowing through that circuit. This hypothesis is, as we have seen, self-consistent, because the total current behaves like an incompressible fluid, and conse- quently the total flow through a circuit has a definite meaning which is independent of the particular surface we select, closing up the circuit, over which to measure the current.

The hypothesis may be transformed into mathematical language by

following the procedure of § 533. It is found to be represented by the equations

. ( 1 df\ dy d/3\

dy

4nr{V+Cdt)"'dz dx 1 dh\ d/3 da

.(523).

V C dt J dx dy /

These are the equations which must replace equations (473) — (475) in the most general motion of electricity. If we differentiate the three equations with respect to x, y, z and add, we obtain

du dv dw _ 1 d /df dg dh\

dw dv d«/_ 1 d /df dg dh\ dx+dy~ fe~~Cdt[dx~+dy"+dz')'

512 Displacement Currents [ch. xvii

Since, by equation (63),

df_ dg dh_ dx dy dz "

this may be written in the form

(!:+!+!")=-! <*

Now G ( — + o~ + o~ ) dx dydz simply expresses the rate at which currents

of ordinary electricity, measured in electrostatic units, flow out of a small

d element of volume dx dy dz, and so is necessarily equal to — -y- (p dx dydz).

We accordingly see that equation (524) is true, quite independently of the truth of Maxwell's displacement-theory. It follows that equations (523) form a consistent scheme, independently of the truth of the hypothesis from which they have been derived. The displacement-theory may be regarded merely as scaffolding, and Maxwell's theory may be regarded as being simply the theory expressed by equations (523), independently of any physical in- terpretation that may be assigned to the various terms in these equations. Although we may, if we please, discard Maxwell's interpretation, it will be convenient to continue to use the name " displacement-current " to designate the vector whose components are given by formula (522).

We proceed to examine the consequences implied in Maxwell's equations (523). Since the truth of the equations must ultimately rest on something more substantial than the displacement-theory by the help of which they were derived, it is important to seize every opportunity of comparing the results of the theory with observation.

Maxwell's Equations for a non-conducting Medium.

571. In a non-conducting medium there can be no ordinary currents of electricity, so that we put u — v = w = 0, and Maxwell's equations assume the

form

4Trdf_dy_d/3' G dt dy dz

.(525).

4>tt dg _da dy G dt dz dx

4-7T dh _ 3/3 da G dt dx dy t

We notice that the whole of the left-hand members arise entirely from the " displacement-current." If the displacement-current were omitted, we should have

dy dz

570-572]

Maxwells Equations

513

so that the magnetic forces (a, /3, 7) would be derivable from a potential, and the only magnetic field in a dielectric in which no currents flowed would be one arising from permanent magnetism.

Maxwell's hypothesis, as expressed in equations (525), implies that there will be a magnetic field in a dielectric whenever the electric field changes, and enables us to calculate the forces in this field.

Magnetic Field of a Moving Charge.

572. As a simple but important example of the use of Maxwell's equations (525) let us calculate the magnetic field produced by a single point charge e moving with a velocity U.

Let the direction of motion of the charge at any instant t be taken for axis of x, the position of the charge being taken for origin.

Let 0 (fig. 137) be the position of the charge at time t, and 0' its position at time t-dt; then 0'0 = udt.

Let P be the point at which we wish to evaluate the mag- netic force. Draw PQ parallel and equal to 00'. Then the electric field at P at time t will be the same as the electric field at Q at time t — dt, so that

Fig. 137.

the increase in the electr]

lc field at P in

time dt will be the

same as

the in-

crease produced by moving a distance - we have

— =— D dt

• V dt parallel

4- etc. ox

to the

axis of x.

Thus

and equations (525) may

be put in the form

4nrU df G dx'

dy ~dy

a/3

dz

4>7rudg G dx'

da

dz~

dy dx

4>irudh G dx'

dx

da. dy'

We have here three equations from which to determine the three com- ponents of magnetic force, a, /3 and 7.

A solution which obviously satisfies the last two equations is

o = 0, £ =

47T6T

h,

4<ttU

7--C-9-

33

514 Displacement Currents [oh. xvii

This solution is also seen to satisfy the first equation in virtue of the relation (cf. Equation (64)),

it is therefore the required solution of the problem.

For the electric field of a single point charge, we have *

A J! eX A eV A 7 eZ

and on substituting these values for/, g, h, the solution becomes

a a U ez UeV ,cnm

O = 0, P=~Q^, 7 = 0^ (526)-

These equations give the components of magnetic force at any point. The lines of magnetic force are circles about the path of the electron, and the intensity at distance r from the electron is

%*¥ <527>>

where 6 is the angle between the distance r and the direction of motion.

572 a. If a small element ds of a circuit in which a current i (measured in electromagnetic units) is flowing contains Nds electrons moving with an average forward velocity u0) we have (cf. equation (b) of § 345)

Ne uQ = Ci.

The magnetic force at distance r produced by the motion of the electrons in the element ds of the circuit is (cf. expression 527))

eu0 sin 6 . , sin 8

Nds -~ — or ids-

G 7

?-

This is exactly identical with the force given by Ampere's Law (§ 497). But Ampere's formula was only proved to be true when integrated round a closed circuit, whereas it is now seen that Maxwell's theory implies that the formula is true for every element of a circuit.

Experimental Confirmation.

573. The possibility that a moving electric charge might produce a magnetic field occurred to Faraday and was noted by him in his Experimental Researches (1837); the effect was observed by Rowland in 1876 and again by Rontgen in 1885. Maxwell's equations, as we have just seen, predict the actual amount of this effect. The only quantity other than the measurable electric charge which appears in Maxwell's formulae is G, the ratio of the electric units, and this can be determined in other ways (cf. § 582 below), its value being found to be almost exactly 3 x 1010.

• This is not quite accurate, for the motion of the magnetic field (a, /3, 7) induces an electric field which ought to be taken into account in evaluating (/, g, h). Equations (526) are, however, ver}' nearly accurate except for very rapidly moving charges. The exact solution will be given later (cf. §§ 624, 647, 656).

572-574] Experimental Confirmation 515

The first attempt to measure the effect quantitatively was made by Rowland and Hutchinson in 1889. They used discs charged to a potential of 5000 volts, which were made to rotate at 125 revolutions a second. The motion of the charged discs may be regarded as the motion of a succession of electric charges, and the magnetic force predicted by Maxwell's theory can be calculated from formula (527). On comparing the observed effect with that predicted by theory, values for G were found which varied from 2-26 x 1010 to 374 x 1010, the mean being 3"19 x 1010. More exact experiments of a similar type performed by H. Pender in 1901 gave for G an average value of 305 x 1010; a second set, with slightly modified apparatus, gave G= 2-96 x 1010. These values will be seen to agree very closely with the known value for C, 3*00 x 1010, so that the experiments not only prove the existence of the magnetic field produced by moving charges, but also confirm Maxwell's theory quantitatively.

It may be objected that the foregoing experiments only test the magnetic field produced by a continuous chain of electric charges moving in a closed circuit, but this objection cannot be urged against experiments performed by E. P. Adams in 1901. In these experiments charged brass spheres were made to pass a suspended magnetic needle at the rate of about 800 per second and the apparatus was arranged so that the effect of one sphere had almost disappeared before the needle came under the influence of the next. From a series of such experiments Adams determined values for C ranging from 2-6 x 1010 to 3-1 x 1010, the mean being 28 x 1010.

Further confirmation of the existence of the displacement-current is pro- vided in a great number of indirect ways, particularly through the electro- magnetic theory of light and the electromagnetic mass of the electron. For the present we shall assume the truth of Maxwell's hypothesis and proceed to examine its consequences.

The General Equations of the Electromagnetic Field.

574. In  §  529,  we  obtained  the  system  of  equations 
     

da = dZ_d7

dt dy dz

in which all the quantities were expressed in electromagnetic units. If the electric forces are expressed in electrostatic units, X, Y, Z must be replaced in these equations by CX, CY, CZ, and the system of equations becomes

1 da_dZ_dY G dt dy dz

^db^_X_dZ

G dt dz dx

}Ldc_dY_dX

G dt dx dy ,

33—2

.(528).

516 Displacement Currents [ch. xvii

These three equations together with equations (523), namely

, / 1 df\ dry d/3 ^

ldg\ da_dy u ~ n j+ ) s~ s™ r \ />

V ' G dtj dz dx

( L ^ - ?£ _ ?? \ C dtj dx dy

constitute a system of six equations giving the rate of changes in the electric and magnetic fields in terms of the field at any instant. With them may be associated the two equations (63) and (362), namely

jjf + ge+j* (530),

dx dy dz r

p + ^ + d^ = 0 (531).

dx dy dz

The eight equations (528) — (531) form the most general system of equations of the electromagnetic field. In these equations u, v, w, a, b, c, a, /3, 7 are expressed in electromagnetic units, while f, g, h, X, Y, Z are expressed in electrostatic units.

Localisation and Flow of Energy.

575. We have already considered the hypothesis that electromagnetic energy may not be confined to the regions occupied by electric charges, magnets and currents, but may be spread through the whole of space. On this hypothesis the kinetic (magnetic) energy T and the potential (electric) energy W of an isotropic medium are given by

T=-^jfjfi (a2 + j32 + 72) dxdydz,

W=^-jjJK (X2 + F2 + Z2) dxdydz,

and the energy is supposed to be localised in space in the way indicated by these integrals. Knowing the kinetic and potential energies of the system, it ought to be possible to determine its equations of motion by the general dynamical methods explained in Chapter XVI.

The quantities a, /3, <y which enter in the kinetic energy must be funda- mentally of the nature of velocities. Let us denote them by £, rj, £, so that £, 7], % may be treated as positional coordinates.

Similarly u, v, w which express the rates of flow of electricity at any point are of the nature of velocities. If qx, qy, qz denote the total quantity of electricity, measured in electrostatic units, which have crossed unit areas perpendicular to Ox, Oy, Oz at any point since a specified instant, then

Cu = qx, Cv = qy, Cw = qz.

574, 575] Localisation and Flow of Energy

Maxwell's equations (529) now assume the form

4tt/. df\ a£ drj .

-G^+dt)=d-y-dzetQ'

giving on integrating, and replacing 4<irf by KX,

517

.(532).

dy dz

This relation connects the various positional coordinates qx, X (regarded as a " displacement"), £ etc.

The principle of least action can be expressed, as in equation (507), in the form

(\ST-{BW})dt = 0, J o

where the value of {8 W] in the present problem is

{8W} = 8W +! i (X8qx+ Y8qy + Z8qz) dxdydz,

which again, on substituting for W, can be put in the form

{SW } = ^jjj[X(K8X + 4tt%) + Y(K8Y+ 4>7r8qy)

• Z (K8Z + 4nr8qz)] dxdydz

i . - , , .- , — ^~

.dy dz J \dz dx } \ dx dy J

on using relations (532). On further transforming by Green's Theorem, this becomes

[8W] =^ff[X(m$S-n8v) + ...]dS

SL[[[[ YfiJX-dk) V(d3-dlt, 7(d8rl_d8^ 4ttJJJ l \dy dzJ + \dz dx)+ [dx dyj

dxdydz

-£/// (S*-£*K

dxdydz.

Similarly on varying T, we find

8T= 1-jfjlpaSa + /*{38/3 + fiyBj] dxdydz

= j- 1 1 1 [aS| + 6S?7 + c§£] dxdydz,

giving

TOT* = ^- IffhaSZ + b8n + c8£) dxdydz1*

— j— j dt J J j(d8^+b8r] + 68%) dxdydz.

As in § 545, we suppose the values of 8%, 8r], S£all to vanish at the instants t = 0 and t = r, so that the top line on the right hand vanishes.

518 Displacement Currents |_CH- xvn

Collecting terms, we now obtain

r.(»-i.in)-i-£r-//i}«+s-©«?

-^-[Tdtt\{{nY-mZ)^+...}d8.

If our suppositions as to the localisation of the kinetic and potential energies are correct, then £, 77, £ may be regarded as independent coordinates at every point of the field. Thus the variations &%, Stj, 8£ may have all possible values at all points of the field. It follows that their coefficients must vanish separately; hence at every point of the field, we must have

a dZ BY _

These are the equations which the principle of least action gives as the equations of motion when we assume Maxwell's equations (529). We see at once that they are identical with equations (528), so that the two sets of equations (528) and (529) are related through the principle of least action.

Poynting's Theorem.

576. If we still assume the energy to be localised in the medium in the way imagined by Maxwell, the total energy in any closed region will be given by

T + W = jjjiS (X2 + F2 + Z*) + £ (a2 + /32 + 72)| dxdydz,

whence, on differentiating, and replacing fia. by a, KX by 4nrf, etc., d(T+W) [[[[( ^df „dg „dh\ 1 f da adb dc) , . .

On substituting from equations (528) and (529), this becomes d(T+W) C [ffi^fdy d/B\ fdZ dY\

vfdy d/3\ fdZ dY\ , , ,

dt 4t7rJJJ\ \dy dzj \dy dz J *j

• G jjj(uX + vY+wZ) dxdydz.

In this equation, the last line represents exactly the rate at which work is performed or energy dissipated by the flow of currents, so that the first line must represent the rate at which energy flows into the region from outside.

575-577] Poynting' s Theorem 519

By Green's Theorem (§179), the first line

= §ir [NZ/3 ~ 7^ + m(YV Za) + n(Ya - X/3)} d#

£, ra, n being the direction-cosines of the normal inwards into the region. Thus if we put

TI^^iYy-ZP), etc (533),

j it appears that the value of -j (T+W) is the same as if there were a flow

of energy in the direction I, m, n of amount ITIX + mUy + nU2. The vector II of which 17a;, Uy, II2 are components is of amount

n = v(iv + iy + n/) = -£- rh sm e,

where R, H are the electric and magnetic intensities and 6 is the angle between them. The direction of the vector II is at right angles to both R and H, and the flow of energy into or out of the surfaces is the same as if there were a flow equal to IT in magnitude and direction at every point of space. This vector II is called the "Poynting flux of energy."

The integral of the Poynting Flux over a closed surface gives the total flow of energy into or out of a surface, but it has not been proved, and we are not entitled to assume, that there is an actual flow of energy at every point equal to the Poynting Flux. For instance if an electrified sphere is placed near to a bar magnet, this latter assumption would require a perpetual flow of energy at every point in the field except the special points at which the electric and magnetic lines of force are tangential to one another. It is difficult to believe that this predicted circulation of energy can have any physical reality. On the other hand it is to be noticed that such a circulation of energy is almost meaningless. The circulation of a fluid is a definite oonception because it is possible to identify the different particles of a fluid ; we can say for instance whether or not the particles entering a small element of volume are identical or not with an equal number of particles coming out, but the same is not true of energy.

Equations for a Uniform Isotropic Dielectric.

577. We return now to the general equations of § 574, and proceed to examine the form they assume in a uniform isotropic dielectric. Since there can be no electric current we put u = v = w = 0. We also put

47r/= KX etc., a = fxa etc.,

520

Displacement Cm-rents

[CH. XVII

and the equations assume the form

G dt dy dz

KdY = da_dy

G dt dz dx

fji da. Gdt

dZ dy

dY\

dz

•(A),

lid/3 _dX_d_Z>

dz dx '

C dt

.(B).

KdZ = dJ3___da ^*y = ^lr?^

G dt dx dy) G dt dx dy I

From the first equation of system (A), we have

Kjx <PX 9_ /> *y\ d_ (n d/3\ G2 dt2 ~dy\Cdi) dz\Cdt)'

and on substituting the values of ~ -j- and ^ -y- from the last two equations

of system (B), this equation becomes

K^d2X_ d_(d_Y_dX\ d_fd_X_dZ\ G2 dt2 dy\dx dy)+dz\dz dx)

d2X . d*X _ d_ (dY dZ\

dx \dy dz,

dy2 ' dz2

Since the medium is supposed to be uncharged, we have

ax dY . dZ dx dy

cz

d2X

so that the last term may be replaced by + -^-j , and the equation becomes

Kfid2X

= V2X

C2 dt2

By exactly similar analysis we can obtain the differential equation satis- fied by Y, Z, a, /3 and 7, and in each case this differential equation is found to be identical with that satisfied by X. Thus the three components of electric force and the three components of magnetic force all satisfy exactly the same differential equation, namely

~J± = a2V2V

dt2

■ (534),

where a stands for C/^K/u,. This equation, for reasons which will be seen from its solution, is known as the "equation of wave-propagation."

d2v Solutions of -^ = a2^2x

Solution for spherical waves.

578. The  general  solution  of  the  equation  of  wave-propagation  is  best 
     

approached by considering the special form assumed when the solution %

is spherically symmetrical. If ty is a function of r only, where r is the

distance from any point, we have

d-y „_, a2 d ( „dy^

— ^ = a2V2 v = \ r2 —

dt* X r2 dr V dr

577-579] Equation of Wave-propagation 521

which may be transformed into

d?(rx)_d?(rX) «o-v

S^'T

and the solution is

r%=/(r-a0 + ^(r + a0 (536)>

where /and <I> are arbitrary functions.

The form of solution shews that the value of X a^ any instant over a sphere of any radius r depends upon its values at a time t previous over two spheres of radii r — at and r + at. In other words, the influence of any value of X is propagated backwards and forwards with velocity a. For instance, if at time t = 0 the value of x is zero except over the surface of a sphere of radius r, then at time t the value of X is zero everywhere except over the surfaces of the two spheres of radii r ±at; we have therefore two spherical waves, converging and diverging with the same velocity a.

General solution (Liouville).

579. The general solution of the equation can be obtained in the following manner, originally due to Liouville.

Expressed in spherical polars, r, 6 and <£, the equation to be solved is

1<2 19 AAA , 1 d (ziuedA | ia2*-Q a2 dP ~ r2 dr \ or) r2 sin 6 dd \ ddj r2d<t>2

Let us multiply by sin 6d6dcf) and integrate this equation over the sur- face of a sphere of radius r surrounding the origin. If we put

X=f[xsineddd(p (537),

the equation becomes

a2 dt2 ~ r2dr\ dr)' the remaining terms vanishing on integration. The solution of this equation (cf. equation (536)) is

X = -{f(at-r) + ^(at+r)} (538).

For small values of r this assumes the form

X = 1 {/(at) + 4> (at)} - r [f (at) - 4>' (at)} + ^ {/" (at) + 3>" (at)} + .. .]

(539).

In order that X may be finite at the origin through all time, we must

have

f(at) + 3> (at) = 0

at every instant, so that the function 4> must be identical with -/. On

putting r = 0, equation (539) becomes

()r=0 = -2/'(at)3

522 Displacement Currents [ch. xvii

and from equation (537), putting r = 0, we have

(V)r=o=47r(x),.=o,

so that 47r(x)r=o = -2/,(aO (540).

Equation (538) may now be written as

rX =/(at - r) —/(at + r). On differentiating this equation with respect to r and t respectively,

^(r) = -f(at-r)-/(at + r), ~(r)- f(at-r)-f(at + r),

a dt and on addition we have

-y(- + r)-|(A)+l|(rt.).

This equation is true for all values of r and i : putting t = 0, we have

-2/'(r) = l(rX,=0)+^=0

as an equation which is true for all values of r. Giving to r the special value r = at, the equation becomes

-2/(at) = jt(t\t=0) + tit=0.

The left hand is. by equation (520), equal to 47r (%)r=o- If we use %, % to denote the mean values of % an(^ X averaged over a sphere of radius at at any instant, the equation becomes

(X)r=o = fo(txt=o) + i%t=o (541).

Thus the value of x a^ any point (which we select to be the origin) at any instant t depends only on the values of % and % at time t = 0 over a sphere of radius at surrounding this point. The solution is of the same nature as that obtained in § 578, but is no longer limited to spherical waves.

General solution (Kirchhoff).

580. A still more general form of solution has been given by Kirchhoff. Let <£> and "SP be any two independent solutions of the original equation, so that

d2(& d2^

-a¥ = a2V2®> V = a2V^ (542)-

By Green's Theorem (equation (101))

-XJJ(&^-V^dS=ff[(®VV-VV®)dxdydz

r ]]\ dn dnj

dV .T. cZ3»

a2

Jflre-f)-*

579, 580] Equation of Wave-propagation 523

by equations (542). The volume integrations extend through the interior of any space bounded by the closed surfaces S1} S2, ..., and the normals to Slt S2, ... are drawn, as usual, into the space. If we integrate the equation just obtained throughout the interval of time from t = — t' to t = + t", we obtain

(543).

-f

So far M* has denoted any solution of the differential equation. Let us now take it to be - F (r + at), this being a solution (cf. equation (536)) what- ever function is denoted by F, and let F (x) be a function of x such that it and all its differential coefficients vanish for all values of x except x = 0, while

( '°F(x)dx = l.

J —oo

Q

Such a function, for instance, is F (x) = Lt — 7-= zr .

c=o jt(^2 + c2)

We can choose t' so that, for all values of r considered, the value of

1 — at' is negative. The value of r + at'' is positive if t" is positive. Thus

F(r + at) and all its differential coefficients vanish at the instants t = t" and

t = — t', so that the right-hand member of equation (543) vanishes, and the

equation becomes

rt" re, 9^ 9<^

-sJ-,*JJrar-*air;'w-0 (544)-

Let us now suppose the surfaces over which this integral is taken to be two in number. First, a sphere of infinitesimal radius r0, surrounding the origin, which will be denoted by S1} and second, a surface, as yet unspecified, which will be denoted by S. Let us first calculate the value of the contribu- tion to equation (544) from the first surface. We have, on this first surface,

V=LF(r0 + at),

^ = -^ = --,F(r0 + atH-F(r0 + at),

so that when r0 is made to vanish in the limit, we have

ft

•H-T$«r.<«ft

and therefore

/ w/(* £ - * hd ds> = - wr,*'- F(at) di

47T ,

a ^_0

since the integrand vanishes except when t = 0.

524 Displacement Currents

Thus equation (544) becomes

[CH. XVII

*'-?—;£:(

• = 0

47rJ _# a

eft

dn c?i

/•<" fcj> dr

• *k&)F{r+at)-lF(-r+at)d^;}dt (545)-

Integrating by parts, we have, as the value of the first term under the

time integral,

/

-%-F'(r+at)dt

-t> r on '

a r dn

F(r+ at)

t=t"

•<" 1 drd®

t ft

-v J -f ciron at

The first term vanishes at both limits, and equation (545) now becomes We can now integrate with respect to the time, for F(r + at) exists only

at the instant t= — r/a. Thus the equation becomes

ia<£>

dn

«=o 47rJJ ar3w cfa dn\r) r

tm-Z

a

dS,

giving the value of <E> at the time t = 0 in terms of the values of <& and 4> taken at previous instants over any surface surrounding the point. The solution reduces to that of Liouville on taking the surface S to be a sphere,

so that r- = — — . on or

As with the former solutions, the result obtained clearly indicates propa- gation in all directions with uniform velocity a.

Propagation of Electromagnetic Waves. 581. It is now clear that the system of equations

C2 dt"

etc., obtained in § 577 indicate that, in a homogeneous isotropic dielectric, all electromagnetic effects ought to be propagated with the uniform velocity

C

This may be compared with the result obtained in § 562. It was

■s/Kf*

there shewn that electric signals propagated along a wire would advance with

C a velocity -7= where K, fx were the inductive capacity and magnetic

Vif

/*

580-582] Electromagnetic Waves 525

permeability of the medium surrounding the wire. It now appears that the velocity of signals along a wire is identical with the velocity of waves in the medium outside the wire.

Maxwell's displacement theory gives a simple explanation of this. A current flowing in a wire is accompanied by a displacement current in the ether. This sets up a magnetic field which is propagated with velocity G/wK/jl in the dielectric and this in turn induces a further current in the wire. On this view the actual process of propagation takes place in the medium, the wire directs the path of the electromagnetic disturbance and absorbs some of the energy.

It is to be noticed that the velocity of propagation along wires was obtained in § 562 before we had introduced the conception of " displacement- currents" at all. That the result is not inconsistent with the velocity obtained on the hypothesis of displacement-currents will be understood from the result of § 575.

Numerical Values.

582. We notice that in free air, in which K = fi — 1, the velocity of pro- pagation of electric waves, whether along wires or in the air, ought to be the same as G, the ratio of the electric units. This enables us to apply a severe test to the truth of the theory which has so far been developed, for both the value of G and the velocity of propagation of electric waves admit of direct experimental determination.

The best determinations of G, the ratio of the two units, are the following:

Rosa and Dorsey (1907) 2*9971 x 1010

Perot and Fabry (1898) 2-9973 x 1010

Hurmuzeseu (1896) 3-0010xl010

Abraham (1890) 2-9913xl010

The true value is probably very close to the value obtained by Rosa and Dorsey, namely G = 29971 x 1010.

Recent determinations of the velocity of propagation of electromagnetic waves in air are as follows :

Maclean (1899) 2-991 x 1010

Saunders (1897) 2-997 x 101C

Trowbridge and Duane (1895) ... 3-003 x 1010

The mean of these values is 2997 x 1010.

In the determinations of Saunders and of Trowbridge and Duane the waves were guided by copper wires, while the experiments of Maclean dealt with waves propagated through air without wires. The equality of velocities is of course a consequence, and also a confirmation, of the results obtained in § 562.

526 Displacement Currents [ch. xvn

The ratio of the units, G, is also equal, or at least very nearly equal, to the velocity of light in air, and this confirmed Maxwell in his suggestion that light propagation is a special case of the propagation of electromagnetic waves. Out of this suggestion, amply borne out by the results of further experiments, has grown the electromagnetic theory of light of which a short account is given in the next chapter. The best determination of the velocity of light in air at present available is that of Michelson who finds (October 1924) for the value of this velocity

299735 x 1010 cms. a second,

with a probable error of one in 22,000.

Except for small differences, which are well within the errors of the various experiments, the quantities previously mentioned are seen to agree with this in value.