Electromagnetic Theory, Vol. 1 (1893) — part 5 of 31

§ 46. A second way of arriving at the motional electric force is by a consideration of the work done in moving a con- ducting circuit in a magnetic field. It results from Ampere's researches, and may be independently proved in a variety of ways, that the forcive (or system of forces) acting upon a conducting circuit supporting a current, may be accounted for by supposing that every element of the conductor is subject to what Maxwell termed " the electromagnetic force." This is a force perpendicular to the vector current and to the vector induction, and its magnitude equals the product of their ten- sors multiplied by the sine of the angle between them. In short, the electromagnetic force is the vector-product of the current and the induction. Or, by the definition of a vector- product,

F = VCB, ...... (9)

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 45

if P is the force per unit volume, 0 the current, and B the induction. Here F is the force arising from the stress in the magnetic field. Its negative, say f, is therefore the impressed mechanical force, or

f=VBO, ...... (10)

to be used when we desire to consider work done upon the electromagnetic system.

The activity of f is fq, if q be the velocity ; or, by (10),

fq = qVBC (11)

This is identically the same as

fq = CVqB, ..... (12)

by a fundamental formula in vector-analysis.* Here, on the left side, the activity is expressed mechanically ; on the right side, on the other hand, it is expressed electrically, as the scalar product of the current and another vector, which is the corresponding force; it is necessarily an electric force, and necessarily impressed. So, calling it e, we have

e = VqB (13)

again, to express the motional electric force.

It should be observed that we are not concerned in this mode of reasoning with the explicit connection between e and C; and in this respect the process is remarkably simple. As it, however, rests upon a knowledge of the electromagnetic force, we depart from the method of deriving relations previously pursued. But, conversely, we may by (11) and (12) derive the electromagnetic force from the motional electric force.

Variation of the Induction through a Moving Circuit.

§ 47. A third method of arriving at (13) is by considering the rate of change of the amount of induction through a moving circuit. We need not think of a conducting circuit, but, more generally, of any circuit. Let it be moving in any way whatever, changing in shape and size arbitrarily. The induction through it is altering in two entirely distinct ways. First, there is the magnetic current before considered, due to the time-variation of the induction, so that, if the circuit were

• Proved, with other working formulae, in the chapter on the Algebra of Vectors.

46 ELECTROMAGNETIC THEORY. CH. II.

at rest in its momentary position, we should have the second law of circulation true in its primitive form

-curlE = G, (14)

when expressed for a unit circuit. But now, in addition, the motion of the elements of the circuit in the magnetic field causes, independently of the time-variation of the field, addi- tional induction to pass through the circuit. Let its rate of increase due to this cause be g per unit area. If, then, we assume that the circulation of the electric force E (of the flux) equals the rate of decrease of the induction through the circuit always, whether it be at rest or in motion, the equation (14) becomes altered to

-curlE = G + g, ..... (15)

where the additional g may be regarded as a fictitious magnetic current. That it is also expressible as the curl of a vector is •obvious, because it depends upon the velocity of each part of the circuit, and is therefore a line-integral. Examination in •detail shows that

g=-curlVciB, (16)

BO that we have, by inserting (16) in (15),

-curl(E-e) = G, .... (17)

the standard form of the second law of circuitation, when we use (13) to express the impressed force.

The method by which Maxwell deduced (13) is substantially the same in principle ; he, however, makes use of an auxiliary function, the vector-potential of the electric current, and this rather complicates the matter, especially as regards the physical meaning of the process. It is always desirable when possible to keep as near as one can to first principles. The above may, without any formal change, be applied to the case of assumed magnetic conductivity, when G involves dissipation of energy; the auxiliary g in (15), depending merely upon 'the motion of the circuit across the induction, does not itself involve dissipation.

Modification. Circuit Fixed. Induction moving

ectuivalently.

§ 48. Perhaps the matter may be put in a somewhat clearer light by converting the case of a moving circuit into that of a

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

47

oircuit at rest, and then employing the law of circuitation in its primitive form. The moving circuit has at any instant a definite position. Imagine it to be momentarily fixed in that position, by stopping the motion of its parts. In order that the relation of the circuit to the induction should be the same &s when it was moving, we must now communicate momentarily to the lines of induction the identically opposite motion to the (abolished) motion of the part of the circuit they touch.

We now get equation (15), on the understanding that g means the additional magnetic current through a fixed circuit due to a given motion of the lines of induction across its boundary, such motion being the negative of the (abolished) motion of the circuit. The matter, is, therefore, simplified in treatment. For, in the former way, the process of demonstrating (15) which I have referred to as an " examination in detail," is really considerably complex, involving the translation, rotation, and distortion of an elementary circuit (or equivalently for any •circuit). Fixing the circuit does away with this, and we have merely to examine what happens at a single element of the cir- cuit, as induction sweeps across it, in increasing the induction through the circuit, and then apply the resulting formula to every element.

In the consideration of a single element, it is immaterial what the shape of the circuit may be; it may, therefore, be chosen to be

a unit square in the plane of the paper, one of whose sides, AB, is the element of unit length. Now, suppose the induction at AB is perpendicular to the plane of the paper, directed downwards, and that it moves from right to left perpendicularly across AB. Let also from A to B be the positive sense in the circuit. It is evident, without any argumentation, that the directions chosen for q and B are the most favourable ones possible for

48 ELECTROMAGNETIC THEORY. CH. II.

increasing the induction through the circuit, and that the rate of its increase, so far as AB alone is concerned, is simply qB, the product of the tensors of the velocity q of transverse motion and of the induction B. Further, if the velocity q be not wholly transverse to B as described, but still be wholly transverse to AB, we must take, instead of q, the effective transverse com- ponent q sin 6y if 0 be the angle between q and B, making our result to be qB sin 0. Now, this is the tensor of VqB, whose direction is from A to B. The motional electric force in the element AB is therefore from B to A, and is VBq, because it is the negative circuitation which measures the magnetic cur- rent through a circuit. Lastly, if the motion of B be not wholly transverse to AB, we must further multiply by the cosine of the angle between VBq and the element AB. This merely amounts to taking the effective part of VBq along the circuit. So, finally, we see that VBq fully represents the impressed electric force per unit length in AB when it is fixed, and the induction moves across it, or that its negative

e = VqB

represents the motional electric force when it is the element AB that moves with velocity q through the induction B. Now, apply the process of circuitation, and we see that e is such that its curl represents the rate of increase of induction through the unit circuit due to the motion alone.

This may seem rather laboured, but is perhaps quite as much to the point as a complete analytical demonstration, where one may get lost in the maze of differential coefficients, and have some difficulty in interpreting the analytical steps electromag- netically.

The fictitious motion of the induction above assumed has nothing to do with the real motion of the induction through the medium. If there be any, its effect is fully included in the term G, the real magnetic current.

The Motional Magnetic Force.

§ 49. The motional magnetic force h may be similarly deduced. First we have the primitive form of the first law of circuitation,

curlH=J, (18)

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 49

when the unit circuit is at rest, where J is the complete electric current-density, and next

curlH = J+j, ..... (19)

when the circuit moves ; where the auxiliary j is a fictitious electric current equivalent to the increase of displacement through the circuit by its motion only. Next show that

j = curlh, ...... (20)

and h = VDq, ...... (21)

by similar reasoning to that concerning e ; so that by insertion in (19) the first law of circuitation is reduced to the standard form

curl (H-h) = J, ..... (22)

with the special form of the impressed force h stated.

Comparing the form of h with that of e we observe that there is a reversal of direction in the vector-products, the flux being before the velocity in one and after it in the other. This arises from the opposite senses of circuitation of the electric and the magnetic force to represent the magnetic and electric currents.

The "Magneto-electric Force."

§ 50. The activity of the motional h is found by multiplying it by the magnetic current, and is, therefore,

by the same transformation as from (11) to (12).

We conclude that VGD is an impressed mechanical force, per unit volume, and, therefore, that VDG- is a mechanical force, that is, of the Newtonian type, arising from the electric stress. By analogy with the electromagnetic force it may be termed the magnetoelectric force, acting on dielectrics support- ing displacement when the induction varies with the time. Of this more hereafter.

Electrification and its Magnetic Analogue. Definition of

"Divergence."

§ 51. So far nothing has been laid down about electrification. But the laws of circuitation cannot be completed without including electrification and its suggested magnetic analogue.

50 ELECTROMAGNETIC THEORY. CH. II.

Describe a closed surface in a dielectric, and observe the net amount of displacement leaving it. This, of course, means the excess of the quantity leaving over that entering it. If the net amount be zero, there is no electrification within the region bounded by the surface. If the amount be finite, there is just that amount of electrification in the region. This is indepen- dent altogether of its distribution within the region, and of the size and shape of the region.

More formally, the surface-integral of the displacement leaving any closed surface measures the electrification within it.

This being general, if we wish to find the distribution of electrification we must break up the region into smaller regions, and in the same manner determine the electrifications in them. Carrying this on down to the infinitely small unit volume, we, by the same process of surface-integration, find the volume- density of the electrification. It is then called the divergence of the displacement.

That is, in general, the divergence of any flux is the amount of the flux leaving the unit volume.

And in particular, the divergence of the displacement measures the density'^pf electrification.

Similarly, the divergence of the induction measures the " magnetification," if thu;e is any to measure, which is a very doubtful matter indeed. There is no evidence that the flux induction has any divergence ; it is purely a circuital flux, so far as is certainly known, and this is most intimately connected with the other missing link in a symmetrical electromagnetic scheme, the (unknown) magnetic conductivity.

Divergence is represented by div, thus : —

divD = />, (1)

divB = o-, (2)

if p and cr are the electrification and magnetification densities respectively.

In another form, electrification is the source of displacement, and magnetification the source of induction. How these fluxes are distributed after leaving their sources is a perfectly indif- ferent matter, so far as concerns the measure of the strength of the sources. In an isotropic uniform medium at rest, the fluxes naturally spread out uniformly and radially from point-sources

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 51

of displacement or of induction. The density of the fluxes then varies as the inverse square of the distance, because the concentric spherical surfaces through which they pass vary in area directly as the square of the distance. Thus

D- ? B- * .... (3)

~

are the tensors of the displacement and induction at distance r from point-sources p and a-.

If the source be spread uniformly over a plane in a uniform isotropic medium to surface-density p or cr, then, by the mere symmetry, we see that half the flux goes one way and half the other, perpendicularly to the plane, so that

(4)

at any distance from the plane. But if we by any means make the source send all the flux one way only, then

D = />, B = o-, ..... (5)

at any distance.

A Moving Source equivalent to a Convection Current, and makes the True Current Circuital.

§ 52. The above being merely to concisely explain the essential meaning of electrification in relation to displace- ment, and how it is to be measured, consider a point- source or charge to be in motion through a dielectric at rest. Starting with the charge at rest at one place, the displacement is radial and stationary. When permanently at rest in another place, the displacement is the same with reference to it. In the transition, therefore, the displace- ment has changed its distribution. There must, therefore, be electric current. Now, the only place where the dis- placement diverges, however the source may be moving, is at the source itself, and therefore the only place where the displacement current diverges is at the source, because it is the time-variation of the displacement. The displacement current is therefore circuital, with the exception of a missing

E2

52 ELECTROMAGNETIC THEORY. CH. 1L

link at the moving charge. If we suppose that the charge p moving with velocity u constitutes a current Tip, that is, in the same sense as the motion, and such that the volume- integral of the current density is u/>, then the complete system of this " convection " current, and the displacement current together form a circuital flux.

Thus, suppose the charge to be first outside a closed surface and then move across it to its inside. When outside, if the displacement goes through the surface to the inner region, it leaves it again. On the other hand, when the charge is inside, the whole displacement passes outward. Therefore, when the charge is in the very act of crossing the surface, the displacement through it outward changes from 0 to /o, and this is the time-integral of the displacement current outward whilst the charge crosses. This is perfectly and simultaneously com- pensated by the convection current, making the whole current always circuital.

The electric current is, therefore, made up of three parts, the conduction current, the displacement current, and the con- vection current ; thus,

(6)

p being the volume-density of electrification moving through the stationary medium with the velocity u.

If the medium be also moving at velocity q. referred to- fixed space, we must understand by u above the velocity also referred to fixed space. The velocities q. and u are only the same when the medium and the charge move together. Thus it will come to the same thing if we stop the motion of the charge altogether, and let the medium have the motion equiva- lent to the former relative motion.

Similarly, if there should be such a thing as diverging in- duction, or the " magnetification " denoted by <r above, then we- shall be obliged to consider a moving magnetic charge as con- tributing to the magnetic current, making the complete mag- netic current be expressed by

if w be the velocity of the magnetification of density o-.

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Examples to illustrate Motional Forces in a Moving Medium with a Moving Source. (1.) Source and Medium with a Common Motion. Flux travels with them undisturbed.

§ 53. In order to clearly understand the sense in which motion of a charge through a medium, or motion of the medium itself, or of both together with respect to fixed space, is to be understood, and of the part played therein by the motional electric and magnetic forces, it will be desirable to give a few illustrative examples of such a nature that their meaning can be readily followed from a description, without the mathematical representation of the results. It does not, indeed, often happen that this can be done with profit and without much circumlocution. In the present case, however, it is rather easier to see the meaning of the solutions from a description, than from the formulae.

In the first place, let us start with a single charge p at rest at any point A in an infinite isotropic non-conducting dielectric — ether, for example — which is also at rest. Under these circumstances the stationary condition is one of isotropic radial displacement from the charge at A according to the inverse- square law, and there is nothing to disturb this distribution.

Now, if the whole medium and the charge itself are supposed to have a common motion (referred to an assumed fixed space, in the background, as it were), no change whatever will take place in the distribution of displacement referred to the moving charge. That this should be so in a rational system we may conclude from the relativity of motion (the absolute motion of the universe being quite unknown, if not inconceivable) com- bined with our initial assumption that the electric flux (and the magnetic flux not here present) represent states of the medium, which may be carried with it just as states of matter are carried with matter in its motion. But as the charge, and with it the displacement, move through space as a rigid body without rotation, the changing dis- placement at any point constitutes an electric current, and therefore would necessitate the existence of magnetic force, if we treated the first law of circuitation in its primitive form, referred to a stationary medium. Here, however, the motional magnetic force, which is (§§ 44, 49) the vector-product of the

54 ELECTROMAGNETIC THEORY. CH. II.

displacement and the velocity of the medium, comes into play, and it is so constructed as to precisely annul all magnetic force under the circumstances, and leave the displacement (referred to the moving medium) unaffected ; or, in another form, it changes the law of circuitation (curl H = J) referred to fixed space, so as to refer it in the same form to the moving medium.

The result is H = 0, and D moves with the medium.

Similar remarks apply to other stationary states. They are unaffected by a common motion of the whole medium and the sources (or quasi-sources), and this result is mathematically obtained by the motional electric and magnetic forces.

(2.) Source and Medium in Relative Motion. A Charge suddenly jerked into Motion at the Speed of Propaga- tion. Generation of a Spherical Electromagnetic Sheet ; ultimately Plane. Equations of a Pure Electromagnetic Wave.

§ 54. But the case is entirely altered if the charge and the medium have a relative translational motion.

Start again with charge and medium at rest, and the dis- placement stationary and isotropically radial. Next, introduce the fact (the truth of which will be fully seen later) that the medium transmits all disturbances of the fluxes through itself at the speed (MC)~~£, which" call -y; and let us suddenly set the charge moving in any direction rectilinearly through the medium at this same speed, v. Ths question is, what will happen ?

A part of the result can be foreseen without mathematical investigation ; the remainder is an example of the theory of the simplest spherical wave given by me in "Electromagnetic Waves." Let A (in Fig. 1) be the initial position of the charge when it first begins to move, and let AC be the direction of its sub- sequent motion. Describe a sphere of radius AB = vt ; then, at the time t the charge has reached B. Now, from the mere fact that the speed of propagation is v, it follows that the dis- placement outside the sphere is undisturbed. It is clear that there cannot be any change to the right of B, because the charge has only just reached that place, and disturbances only

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. i>5

travel at the same speed as it is moving itself. Similar con- siderations applied to the expanding sphere through this charge at every moment of its passage from A to B will show that no disturbance can have got outside the sphere. The radial lines, therefore, represent the actual displacement, as well as the original displacement, though of course, in the latter case, they extended to the point A.

We have now to complete the description of the solution. There is no displacement whatever inside the sphere BTCDF.

The displacement emanating from the charge at B, therefore, joins on to the external displacement over the spherical surface. We can say beforehand that it should do so in the simplest conceivable manner, by the shortest paths. On leaving the pole B it spreads uniformly in all directions on the surface of the sphere, and each portion goes the shortest way to the opposite pole D. But it leaks out externally on the way, in such a manner that the leakages are equal from equal areas. The displacement thus follows the lines of longitude.

56 ELECTROMAGNETIC THEORY. CH. II.

This completes the case so far as the displacement is con- cerned. But the spherical surface constitutes an electro- magnetic sheet, and corresponding to the displacement there is a distribution of coincident, induction. This induction is perpendicular to the tangential displacement, and therefore follows the lines of latitude. Its direction is up through the paper above A (at E, for example), and down through the paper below A (at F, for example). The tangential displace- ment and induction surface-densities (or fluxes per unit area of the sheet), say, D0 and B0, are connected by the equation

or, 0 = cv0.

Or, if E0 and H0 be the equivalent forces got by dividing by c and by ft respectively, then, since fj.cv2 = 1,

E0 = /xvH0. Or, expressing the mutual directions as well,

E0 = VB0v;

where v is the vector velocity of the electromagnetic sheet at the place considered. These last are, in fact, as we shall see later, the general equations of a wave-front or of a free wave, which though it may attenuate as it travels, does not suffer distortion by mixing up with other disturbances.

Now, as time goes on, the charge at B moves off to the right, the electromagnetic sheet simultaneously expanding. The ex- ternal displacement, therefore, becomes infinitesimal ; likewise that on the D side of the shere. Practically, therefore, we are finally left with a plane electromagnetic sheet moving perpendicularly to itself at speed v, at one point of which is the moving charge, from which the displacement diverges uniformly in the sheet, following, therefore, the law of the inverse first power (instead of the original inverse square), ac- companied by a distribution of induction in circles round the axis of motion, varying in density with the distance according to the same law, and connected with the displacement by the

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above equations. In the diagram, AB has to be very great, and the plane sheet is the portion of the spherical sheet round B, which is then of insensible curvature.

(3.) Sudden Stoppage of Charge. Plane Sheet moves on. Spherical Sheet generated. Final Result, the Stationary Field.

§ 55. Having thus turned the radial isotropic displacement of the stationary charge into a travelling plane distribution, let us suddenly reduce the charge to rest. We know that

FIG. 2.

after some time has elapsed, the former isotropic distribution will be reassumed ; and now the question is, how will this take place ?

Let B in Fig. 2 be the position of the charge at the moment of stoppage and after. Describe a sphere of radius vt, with B for centre ; then the point C is where the charge would have got at time t after the stoppage had it not been stopped, and the plane DOE would have been the position of the plane

58 ELECTROMAGNETIC THEORY. CH. IL

electromagnetic sneer. Now, the actual state of things is- described by saying that : —

(1.) The plane sheet DOE moves on quite unaltered, except at its core C, where the charge has been taken out.

(2.) The stationary radial displacement of the charge in its new position at B is fully established within the sphere, with- out any induction.

(3.) The internal displacement joins itself on to the external in the plane sheet, over the spherical surface, by leaking into it and then following the shortest route to the pole C. That is, the tangential displacement follows the lines of longitude.

(4.) The induction in the spherical sheet is oppositely directed to before, still, however, following the lines of latitude, and being connected with the tangential displacement by the former relations.

In time, therefore, the plane sheet and the spherical sheet go out to infinity, and there is left behind simply the radial dis- placement of the stationary charge.

(4.) Medium moved instead of Charge. Or both moved with same Relative Velocity.

§ 56. Now, return to the case of § 54, and referring to Fig. 1, suppose it to be the charge that is kept at rest, whilst the medium is made to move bodily past it from right to left at speed v, so that the relative motion is the same as before. We must now suppose B to be at rest, the charge being there origi- nally, and remaining there, whilst it is A that is travelling from right to left, and the spherical surface has a motion com- pounded of expansion from the centre A and translation with it. Attending to this, the former description applies exactly.

The external displacement is continuously altering, and there is electric current to correspond, but there is no magnetic force (except in the spherical sheet), and this, is, as before said, accounted for by the motional magnetic force.

The final result is now a stationary plane electromagnetic sheet, as, in fact, described before in § 45, where we considered the displacement and induction in the sheet to be kept up steadily by electric and magnetic forces impressed by the motion

OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 59

Now stop the motion of the medium, without altering the position of the charge, and Fig. 2 will show the growth of the radial stationary displacement, as in § 55, as it is in fact the same case precisely after the first moment.

We can in a similar manner treat the cases of motion and stoppage of both charge and medium, provided the relative speed be always the speed vt however different from this may be the actual speeds.

(5.) Meeting of a Pair of Plane Sheets with Point-Sources. Cancelment of Charges ; or else passage through one another; different results. Spherical Sheet with two Plane Sheet Appendages.

§ 57. From the two solutions of §§ 54, 55 (either of which may be derived from the other) we may deduce a number of other interesting cases.

Thus, let initially a pair of equal opposite charges +p and

• p be moving towards one another, each at speed v through the medium (which for simplicity we may consider stationary), each with its accompanying plane electromagnetic sheet. When the charges meet the two sheets coincide, the two dis- placements cancel, leaving none, and the two inductions add, doubling the induction. We have thus, momentarily, a mere sheet of induction.

Now, if we can carry the charges through one another, with- out change in their motion, the two sheets will immediately reappear and separate. That is, the plane waves will pass through one another, as well as the charges.

But if the charges cancel one another continuously after their first union, a fresh case arises. It is, given a certain plane sheet of induction initially, what becomes of it, on the understand- ing that there is to be no electrification ?

The answer is, that the induction sheet immediately splits into two plane electromagnetic sheets, joined by a spherical sheet, as in Fig. 3. For it is the same as the problem of stop- page in § 55 with another equal charge of opposite kind moving the other way and stopped simultaneously, so that there is no electrification ever after. Touching the sphere at the point F in Fig. 2 is to be placed the additional plane wave, and the

60

ELECTROMAGNETIC THEORY.

CH. II.

internal displacement is to be abolished. That is to say, in Fig. 3 the displacement converges uniformly to F in the plane sheet there, then flows without leakage to the opposite pole C along the lines of longitude, and there diverges uniformly in the other plane sheet. Each displacement sheet has its corre- sponding coincident induction, according to the former formulae. They all move out to infinity, leaving nothing behind, as there is no source left.

Fio. 6.

(6.) Spherical Sheet without Plane Appendages produced by sudden jerking apart of opposite Charges.

§ 58. Similarly, let there be a pair of coincident or infinitely close opposite charges, with no displacement, and let them be suddenly jerked apart, each moving at the speed of propaga- tion of disturbances. The result is simply a single spherical wave, without plane appendages, and without leakage of the displacement. The charges are at opposite poles, at the ends of the axis of motion, and the displacement just flows over the

OUTLINE OF ELECTROMAGNETIC; CONNECTIONS. 61

surface from one to the other symmetrically. There is the usual induction B0 = /wD0 to match.

Fig. 3 also shows this case, if we leave out the plane sheets and suppose the positive charge to be at F and the negative at C.

After a sufficient time, we have practically two widely sepa- rated plane electromagnetic sheets, although they are really portions of a large spherical sheet.

Now, imagine the motion of the two charges to be reversed ;. if we simultaneously reverse the induction in the spherical sheet, without altering the displacement, it will still be an electromagnetic sheet, but will contract instead of expanding. It will go on contracting to nothing when the charges meet. If they are then stopped nothing more happens. But if the charges can separate again, the result is an expanding spherical electromagnetic sheet as before.

(7.) Collision of Equal Charges of same Name.

§ 59. If, in the case of colliding plane sheets with charges, § 57, they be of the same name, then, on meeting, it is the- induction that vanishes, whilst the displacement is doubled. That is, we have momentarily a plane sheet of displacement.

If the charges be kept together thereafter, this plane sheet splits into two plane electromagnetic sheets joined by a spherical sheet. At the centre of the last is the (doubled) charge 2/5, which sends its displacement isotropically to the surface of the sphere, where it is picked up and turned round towards one pole or the other. The equator of the sphere is the line of division of the oppositely flowing displacements. The displacement gets greater and greater as the poles are neared, the total amount reaching each pole being p (half the central charge), which then diverges in the plane sheet touching the pole.

The final result, when the waves have gone out to infinity,. is, of course, merely the stationary field of the charge 2p.

(8.) Hemispherical Sheet. Plane, Conical and Cylindrical Boundaries.

§ 60. If a charge be initially in contact with a perfectly conducting plane, and be then suddenly jerked away from it at the speed v, the result is merely a hemispherical electromag- netic shell. The negative charge, corresponding to the moving

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ELECTROMAGNETIC THEORY.

CH. II.

point-charge, expands in a circular ring upon the conducting plane, this ring being the equator of the (complete) sphere.

This case, in fact, merely amounts to taking one-half of the solution in § 58, and then terminating the displacement normally upon a conductor.

In Fig. 4, A is the original position of p on the conducting plane CAE, and when the charge has reached B the displace- ment terminates upon the plane in the circle DF.

FIG. 4.

Instead of a plane conducting boundary, we may similarly have conical boundaries, internal and external (or one conical boundary alone), with portions of perfect spherical waves run- ning along them at the speed v.

If the two conical boundaries have nearly the same angle, and this angle be small, we have a sort of concentric cable (inner and outer conductor with dielectric between), of con- tinuously increasing thickness. The case of uniform thickness is included as an extreme case ; the (portion of the) spherical wave then becomes a plane wave.

OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 63

General Nature of Electrified Spherical Electromagnetic Sheet.