Electromagnetic Theory, Vol. 1 (1893) — part 18 of 31

The statement that there is a charge q at the origin means nothing more than that displacement diverges or emanates from that place, to the integral amount q. We also understand, of course, that the displacement has no divergence anywhere else. But the manner of emanation is, so far, left quite arbitrary. Certain general results may, however, be readily arrived at. To begin with, if the medium is isotropic, the emanation of the displacement must not favour one direction more than another, from which it follows that the displacement at distance r from the source is spread uniformly over the area 47rr2, so that q/lirr2 expresses its density. Now if we alter the permittivity to dis- placement parallel to k only, we favour displacement in that direction if the permittivity is increased, at the expense of the transverse displacement, because the total amount, which measures the strength of the source, is the same. Conversely, if we reduce the permittivity parallel to k, we favour the transverse displacement. The lines of displacement, originally spread equably, therefore separate themselves about the k axis (both ways), and concentrate themselves transversely, or about the equatorial plane, the plane passing through the charge which is cut by the k axis perpendicularly.

If we carry this process so far as to reduce the permittivity along k to zero, there can be no displacement at all parallel to k, so that the displacement must be entirely confined to the equatorial plane itself. In this plane it spreads equably from the source, because of the transverse isotropy assumed at the beginning. The amount q therefore spreads uniformly over the circle of circumference 2?rr in the displacement sheet, so

266 ELECTROMAGNETIC THEORY. CH. III.

that the density (linear) is now q/2irr. The law of the inverse square of the distance has been replaced by the law of the inverse distance, with confinement to a single plane, however.

Next, if we introducs planar eolotropy, say by reducing the permittivity in direction j, we cause the displacement in the sheet to concentrate itself about the i axis, at the expense of the displacement parallel to j. Finally, if we abolish altogether the permittivity parallel to j, the whole of the displacement will be confined to the i axis, half going one way and half the other.

We may go further by introducing heterogeneity. Thus, reduce the permittivity on one side of the equatorial plane. The effect will be to favour displacement on the other side j and, in the limit, when the permittivity is altogether abolished on the first side, the whole of the displacement goes unilaterally along the k axis on the other side.

In the case of the planar distribution of displacement, the surface-density is infinite, but the linear-density is finite, except at the source. We may, however, spread out the source along a finite straight line (part of the k axis) when the displacement will simultaneously spread out in parallel sheets, giving a finite surface-density. Similarly, in the case of the linear distribu- tion of displacement along the i axis, we may spread out the source upon a portion of the j, k plane, when the former line of displacement will become a tube, with finite surface-density of displacement inside it.

In the above manner, therefore, we obtain a general know- ledge of results without mathematics. Except, however, in the extreme cases of spherical isotropy, of planar isotropy with zero permittivity perpendicular to the plane, and of purely linear displacement due to the vanishing of the transverse permit- tivity, and other extreme cases that may be named, we do not obtain an exact knowledge of the results, except in one respect, that the total displacement must always be the same. Return, therefore, to the case of initial isotropy upset by changed per- mittivity parallel to the k axis, or, if need be, with three differ- ent principal permittivities, so that the displacement is

cF, . . . (32) if F be the electric force. We know that in the state of equi-

ELEMENTS OF VECTORIAL ALGEBKA AND ANALYSIS. 267

librium, there must be no voltage in any circuit, or the curl of P must be zero, so that F must be the slope - VP of a scalar P, the potential. Furthermore, the divergence of D is the electrification- density p. Uniting, then, these results, we obtain

div(-cVP) = />, ..... (33)

or (c1V12 + c2V22 + c3V32)P=-/0, . . . (34)

which is the characteristic of P when referred to the principal axes. Now when there is a point-source at the origin, and the three c's are equal, we know that the potential is q/^irrc at distance r. From this, remembering that r2 is the sum of the squares of the components of r, it is easy to see that the potential in the case of eolotropy is

where f is some constant, because this expression (35) satisfies (34) with p = 0, that is to say, away from the source.

The constant / may be evaluated by calculating the displace- ment passing through the surface of any sphere r = constant, according to (35), and equating it to q. The result is /= (c^Cg)"*. The complete potential in the eolotropic medium due to the point-source q at the origin is therefore

g/47T ( }

--

The equipotential surfaces are therefore ellipsoids centred at the charge, the equation of any one being

. . . (37)

so that the lengths of the principal axes are proportional to the square roots of the principal permittivities. The lines of elec- tric force cut through the equipotential surfaces at right angles. But the lines of displacement do not do so, on account of the eolotropy. To find their nature, first derive the electric force from the potential by differentiation. Then (36) gives

-'

and the displacement is obtained by operating by c on this, which cancels the c~\ and gives

268 ELECTROMAGNETIC THEORY. OH. III.

-i

The displacement is therefore radial, or parallel to r itself. That is, the lines of displacement remain straight, only altering their distribution as the medium is made eolotropic.

Observe, in passing, that the scalar product FD varies as P4 ; that is, the density of the energy varies as the fourth power of the potential. Also note that Fr is proportional to P ; that is, the radial component of F, or the component parallel to the displacement, varies as P/r.

If we select any pair of equipotential surfaces between which the electric force and displacement are given by (38), (39), we may, if we please, do away with the rest of the electric field. That is, we may let the displacement and electric force terminate abruptly upon the two ellipsoidal surfaces. What is left will still be in equilibrium. For the voltage remains zero in every circuit possible. This is obviously true between the surfaces, because no change has been made there. It is also true in any circuit beyond the surfaces, because of the absence of electric force. And lastly, it is true for any circuit partly within and partly beyond the region of electric force, because the electric force is perpendicular to the surface. So the electric field is self-contained. The electrification is on the surfaces, where the normal component of the displacement measures the surface-density. That is, there is a total electrification q on the inner and - q on the outer surface. The elastance of the condenser, or " leyden," to use Lord Rayleigh's word, formed by the two surfaces, is the ratio of the voltage between them to q. That is, the coefficient of q in (36) is the elastance be- tween any equipotential surface and the one at infinity.

In thus abolishing the electric field beyond the limited region selected, we may at the same time change the nature of the medium, making it isotropic, for example,- or conducting, &c. Any change is permissible that does not introduce sources that will disturb the equilibrium of the electric field between the equipotential surfaces.

Starting from isotropy, when the equipotential surface is a sphere, if we keep Cj = c2 constant, and reduce c3, the sphere will become an oblate spheroid, like the earth (and other bodies) — flattened at the poles. Let the common value of ct

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 269

and c2 be denoted by c0, then the potential due to the central charge q is

P s __ Qfa _ , (40)

and the displacement and electric force are given by

As c3 is continuously reduced the oblate spheroid of equili- brium becomes flattened more and more, and is finally, when c3 = 0, reduced to a circular disc. The displacement is now entirely in the equatorial plane, so that if we terminate the displacement at distance r from the centre, we obtain a circular line of electrification (that is, on the edge of the disc). We may have the displacement going from the central charge to an equal negative charge spread over the circle, or from one circle to another in the same plane ; and so on.

Similarly, when cx and c2 are unequal, but c3 = 0, the circular disc is replaced by an elliptical disc. But, without introducing transverse eolotropy, we see that the abolition of the permittiv- ity in a certain direction has the effect of converting the usual tridimensional solutions relating to an isotropic medium to bidimensional solutions, in which the displacement due to sources situated in any plane perpendicular to the axis of symmetry is also in that plane. We may have any number of such planar distributions side by side, with the axis of symmetry running through them. But they are quite independent of one another. If we restore the permittivity parallel to the axis of symmetry, the displacement will usually spread out to suit tridimensional isotropy. The exception is when every plane perpendicular to the axis has identically similar sources, similarly situated. Then the restoration of the permittivity will produce no change.

Theory of the relative Motion of Electrification and the

Medium. The Solution for a Point- Source in steady

rectilinear motion. The Equilibrium Surfaces in General.

§ 164. Let us now pass to what is, at first sight, an entirely

different problem, but which involves essentially the same

mathematics. As before, let there be a point-source of displace-

270 ELECTROMAGNETIC THEORY. CH. TIT.

ment at the origin, but let the dielectric medium be homo- geneous and fully isotropic. The displacement will be radially distributed, without bias one way or another. Now, keeping the charge fixed in space, imagine the medium to move steadily from right to left past the charge. How will this affect the displacement ?

If we desire the solution expressing how the displacement changes from its initial distribution, as the medium is brought from rest into steady motion, we can obtain it by going the right way to work. But it is complex, and will, therefore, not be given here, especially as it is not intelligible without close study. But if we only wish to know the finally-assumed dis- tribution of displacement when the initial irregularities have subsided, we may obtain and express the result in a compara- tively simple manner. Thus, to begin from the foundation, we have the two fundamental circuital laws,

curl(H-h) = cpE + /ou, .... (42) curl(e-E) = fipH, (43)

where E and H are the electric and magnetic forces, /t and c the inductivity and permittivity, p the density of electrification, supposed to have velocity u ; and e, h are the motional electric and magnetic forces given by

e = VwB, h = VDw, . . . (44)

B and D being the induction and displacement, and w the velocity of the medium. (See equations (1), (2), or (3), (4), § 66, and equations (5), (6), § 44, for the motional forces. Also §§ 33 to 70 generally.)

Now in our present case, E and H are steady, and the elec- trification is at rest. The right sides of (42), (43) are therefore ze/o, which is a great simplification. By (42) and the second o~ (44) we obtain

curl H = curl h = curl VDw. . . . (45) But B is circuital, and therefore so is H. It follows that

H = VuD, (46)

if we substitute - u for w, for future convenience. This gives H explicitly in terms of the displacement, when that is known, so we have no further trouble with the magnetic force.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 271

To find the displacement, (43) gives, along with the first of

curl(E + VuB) = 0, .... (47) or say, curlf=0, if f=E + VuB. . , , (48)

Now put the value of B in f by (46), thus,

. . (49)

since /zc£2 = l, v being the speed of propagation of distur- bances, and u the tensor of u, supposed to be parallel to k.

Expanding by the fundamental formula (52), § 114, we obtain

. . (50)

Now let f «= (1 - tt2/v2)F. Insert in (50), and divide by (1 - tt2/^2). The result is

'

This vector F must also have no curl. Therefore, if P is the potential whose slope is F, we have

¥ d? „ dP „ n u?}dP %=-_-, J^2= - .b = -(!-)

ax ay v* dz

and the components of displacement are given by

*—•& »•—%' *—•&' <53>

if Ci = C2 = Cj an(j c3 = c(l-u2/v2). . . (54)

Now observe that (53) express the displacement in terms of the electric force in an eolotropic medium at rest, whose principal permittivities are given by (54), when P represents the electro- static potential. Our present problem of a moving isotropic medium is therefore reduced to one relating to a stationary eolotropic medium. Thus, to state the comparison fairly, when the isotropic medium moves bodily past a stationary charge, the displacement distribution becomes identically the same as if the medium were at rest, but had its permittivity in the direc-

272 ELECTROMAGNETIC THEORY. CIL III.

tion of motion reduced from c to c(l - w2/v2). The solution is therefore given by (40), (41) in § 163, by taking c0 = c, and c3 as in (54). The displacement concentrates itself about the equatorial plane, or plane through the charge perpendicular to the axis of motion in the one problem, or axis of reduced per- mittivity in the other. In the limit, when the speed of motion reaches v, the speed of light, the displacement is wholly planar, as in the eolotropic case when the permittivity c3 vanishes.

Observe, in (52), that the square of u occurs. It does not matter, therefore, which way the motion takes place, so far as the displacement is concerned. But it makes a great difference in the magnetic force which accompanies the lateral concen- tration of the displacement. Being given by (46), we see that H reverses itself when the direction of motion is reversed.

It should be carefully noted that P, which is the electrostatic potential in the eolotropic stationary medium, is not the electro- static potential in the moving medium, although it is the same function precisely. In the eolotropic medium its slope is the electric force F. In the moving medium its slope is the same force F, but this is no longer the electric force, which is E instead, obtained from F by (51) or (52). Again, whilst the displacement and electric force are not parallel in the stationary eolotropic medium ; on the other hand, in the moving medium they are parallel, because the medium is isotropic. To com- plete the differences, there is no magnetic force in the eolotropic case, but there is in the moving medium, and it is its existence which allows the parallel to exist as regards the identical distributions of displacement.

Instead of moving the medium past a fixed charge, we may fix the medium, and move the charge through t the reverse way at the same speed. This is why we put w = - u. The above results apply when the charge q moves with velocity u through the stationary medium. But it is necessary now to travel with the charge in order to see the same results, because the origin is taken at the charge, and it is now in motion. The preliminary mathematics is therefore less easy.

The above theory of convection- currents and the eolotropic comparison I have given before.* I have now to add a some- what important correction. For the opportunity of making

• "Electrical Papers," Vol. II., pp. 492-499, 504-516.

ELEMENTS OP VECTOUIAL ALGEBRA AND ANALYSIS. 273

this correction I am indebted to Mr. G. F. C. Searle, of Cam- bridge, who has been at the trouble of working his way through my former calculations. Whilst fully confirming my results for a point- charge, and therefore all results obtained there- from by integration, he has recently (in a private communica- tion) cast doubt upon the validity of the extension of the solution to the case of a moving charged conducting sphere> and has asked the plain question (in effect) : — What justification is there for taking distributions of displacement calculated in the above manner, and cutting them short by surfaces to which the displacement is perpendicular, which surfaces are then made equilibrium surfaces? For example, in the solution for a point- charge, I assumed that the solution was the same for a charged sphere, because the lines of electric force met it per- pendicularly. On examination, however, I find that there is no justification for this process. The electric force should not be normal to a surface of equilibrium, exceptions excepted.

The true boundary condition for equilibrium, however, needs no fresh investigation, being implicitly contained in the above. In the stationary case of eolotropy, the vector F must have no curl, and since it is the electric force, it is the same as saying that there must be no voltage in any circuit, so that F must be normal to an equilibrium surface. Now, in the problem of a moving medium with charge fixed the same condition for- mally obtains, viz., that F shall have no curl. But as it is not the electric force E, it is clear that the displacement (which is parallel to E) cannot be perpendicular to an equilibrium sur- face. The voltage calculated by the fictitious electric force F must come to zero in every circuit. The boundary condition is, therefore, that F is perpendicular to an equilibrium surface. That is, P = constant is the equation to a surface of equilibrium (no longer equipotential in the electrostatic sense), where P is, however, the same function as in the eolotropic stationary problem of electrostatics.

Therefore, by § 163, we see that the sphere of equilibrium when the medium (isotropic) is stationary, becomes an oblate spheroid when the medium is set into a state of steady motion, the shorter axis being parallel to the line of motion. In the limit, when the speed is v, the spheroid is flattened to a circular disc, on whose edge only is the electrification.

T

274 'ELECTROMAGNETIC THEORY. CH. III.

This applies when the whole dielectric medium is moving past a fixed spheroidal surface of electrification. It equally applies when the electrification moves the other way through a fixed medium. There will be no electric force in its interior. We may therefore fill it up with conducting matter, provided we do not interfere with the free motion of the external medium through it in the one case, or with the rest of the external medium when the conductor moves through it. But in the limiting case of motion at the speed of light, when only a single circular line of electrification is concerned, it would seem to be immaterial whether the conductor be a flat disc or a sphere, subject to the reservation of the last sentence.

Whether it is possible to move matter through the ether without disturbing it forms an entirely different question, to which no definite answer can be given at present. We can, in any case, fall back upon the more abstract theory of electrifi- cation moving through the ether, without having conductors to interfere.

Theory of the relative Motion of Magnetification and the

Medium.

§ 165. From the preceding, relating to the magnetic effects produced by moving the medium bodily past stationary elec- trification ; or equivalently, by moving the electrification the other way through the medium ; or, more generally, by rela- tive motion of the sources of displacement and the medium supporting it, we may readily deduce the corresponding results when the sources are of the flux induction, instead of displace- ment. Thus, in (42), (43), do away with the convective electric current /ou, and substitute convective magnetic current cm, where o- is volume-density of magnetification (suppositional). Our circuital equations are then

curl(H-h) = cpE, . .... (55) curl (e-E) = /*pH + <ni, . . . (56)

with the auxiliary equations of motional electric and magnetic force

e = VwB, h = VDw, .... (57)

as before, where w is the velocity of the medium, and with the

ELEMENTS OP VECTOIUAL ALGEBRA AND ANALYSIS. 275

further auxiliary conditions that the divergence of the displace- ment is zero, whilst that of the induction is or. Now we really do not need to employ the formal mathematics. Knowing the results relating to the motion of electrification, we may infer the corresponding ones concerning the motion of magnetifica- tion, by making use of the analogies between the electric and magnetic sides of electromagnetism, translating results in the appropriate manner. But as our object is to illustrate the working of vectors as well as electromagnetic principles, it will not be desirable to leave out the mathematics entirely, especially as the safe carrying out of the analogies requires that they should be thoroughly understood first, which requires some practice.

First, if we convert the problem to one of stationary waves, by keeping the sources at rest, and letting only the medium move bodily past them, we have, when the stationary state of induction and displacement is reached, disappearance of the right members of (55) and (56). Then (56) gives, when united with the first of (57),

curl E = curl e = curl VwB. . . . (58)

In the result for a moving point-charge, curl E is perpendicular to w, so now we may write

..... (59)

where w = - u, because, similarly, curl H will be perpendicular to w, and so make VBu circuital. Here u is the equivalent velocity of the sources, when the medium is stationary. Thus E, and therefore D, are fully known in terms of B. Comparing with (46), we see that the electric force set up when a magnetic charge moves is related to the induction from the charge in the same way as the magnetic force set up when an electric charge moves is related to the displacement from it, with, however, a change of sign, or reversal of direction.

Using the second of (57) in the other circuital equation (55) produces

curl(H-VuD) = 0 = curlf, say, . . (60)

where, by (59),

. . (61)

T2

276 ELECTROMAGNETIC THEORY, CH. III.

or, if u is parallel to k,

f = H + VkVkH = H + (kH8 - H)

. . (62)

from which we see that the vector F derived from f by f=(l -u2/v2)T has no curl, or is derived from a scalar poten- tial fl thus, F = - vQ, and that the induction is given by

"'§ B=-"S' B3=-,d-«wf. (63;

That is, the induction distributes itself in the same way as if the medium were at rest, but had its inductivity reduced from p to /*(! -u?/vz) in the line of motion. In fact, the theory of the effects produced by relative motion of magnetic sources and the medium is essentially the same as that of moving electric sources ; translating displacement in the latter case to induction in the former, permittivity to inductivity, electric potential (real or fictitious, as the case may be) to magnetic potential, and the magnetic induction which accompanies moving electrification to electric displacement set up by moving magnetification, remembering, however, the change of sign.

We know, therefore, the result of the steady rectilinear motion of any distribution of magnetic sources. If the same motion is common to all the elementary sources, we reduce the problem to that of magnetic eolotropy without relative motion of the medium and sources, and there is a definite system of sur- faces of equilibrium (ft = constant) where we may, if we please, terminate the electric and magnetic fields, thereby substituting new arrangements of magnetic sources for the old. In this case it is most convenient to keep the sources at rest, and move the medium alone.

But should the relative motion of source and medium not be the same for every source, then, since we cannot move the medium bodily more than one way at a time, we may imagine it to be stationary, and obtain the effect due to the motion of the magnetification by superimposing the separate effects of the elementary sources, which are known by the above.

ELEMENTS OF VECTOKIAL ALGEBRA AND ANALYSIS. 277

Theory of the relative Motion of Magnetisation and the Medium. Increased Induction as well as Eolotropic Disturbance.

§ 166. There is, however, no such thing as magnetifi cation, the magnetic analogue of electrification. The induction is always circuital. If it were not circuital, we should have unipolar magnets. We must, therefore, somewhat modify the conditions assumed to prevail in the above, in order to come closer to reality. Consider, therefore, instead of magnetifica- tion, a distribution of intrinsic magnetisation. This quantity is I = /xh0, where h0 is the equivalent intrinsic magnetic force. Abolish o- in (56), and introduce h0 in (55). The circuital equations are now

curl (H - VDw - h0) = cpE, . . . (64)

curl (VwB - E) = /^H, .... (65)

where we have introduced the motional forces (57). In the steady state, the right members vanish as before. Now assume that

E = VwB, (66)

so that the theory is unchanged so far, the displacement depending on the induction and velocity of the medium in the same way as when the source was magnetification. We shall see the limitation of application later. Next put (66) in (64) and we find, if the fraction 1 — u2/v2 be denoted by s,

curl f= curl h0, (67)

curl F = curl h0/s, .... (68)

where f and F are the same vectors (in terms of H) as in tho last case, such that sF = f. We see that the vector F has now definite curl, and that the induction is derived from F by

B^/iFp B2 = ^F2, B3 = MsF3; . (69)

that is to say B = AF, where A is the inductivity operator whose scalar principals are /x, p, and ps. We therefore have to find the steady state from these complete connections,

divB = 0, B = AF, curl F = curl h0/s . (70)

That is, the induction is the same as in a stationary eolotropic medium (according to A), provided the intensity of the intrinsic

278 ELECTROMAGNETIC THEORY. CH. III.

magnetic force be increased from h0 to h0/s. This is an important point.

We know that there are certain respects in which the theory of induction due to intrinsic magnetic force is identical with that due to magnetification. For instance, the steady induction outside a magnet due to h0 is the same as that due to a dis- tribution of cr, measured by the convergence of the intrinsic magnetisation. From this we might hastily conclude that the disturbance from the steady distribution produced by moving the medium would be the same for h0 as for the equivalent o-, provided we keep outside the region of magnetisation. But the above investigation only partly confirms this conclusion. It shows a likeness and a difference. The likeness is in the eolotropic peculiarity brought in by the motion. In both cases we may do away with the motion provided we simultaneously reduce the inductivity parallel to the (abolished) motion from fj. to s/x, so as to cause the induction to retreat from this direc- tion and concentrate itself transversely. The difference is in the reckoning of the strength of sources. In the case of magnetification (as of electrification) we do not alter the strength of the source. But in the case of magnetisation we do, or, at any rate, produce an equivalent result. For, along with the reduction of inductivity in the direction of motion, we require to increase the intensity of the intrinsic magnetic force from h0 to h0/s. Remember that s is a proper fraction, going from unity to zero as the speed of motion increases fron. 0 to vt the speed of propagation of disturbances.

To exemplify this, put

..... (71)

as we see we may do, by the third of (70). Using this in the second and first of (70) we obtain

.... (72) the equation of the potential 12. Or, in terms of Cartesians,

XV + v22 + «v88)fl -/*-Woi + v2A02 + *VAS) . (73)

The equivalent magnetification is the convergence of Ah0/s, not of fihQ the real intensity of magnetisation, nor yet of /*h0/s, the same increased in a constant ratio.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 27 T

There is, therefore, a remarkable difference between the two cases of motion of the medium parallel to and transverse to the lines of real magnetisation. Thus, first let h0 be parallel to w or k. Then the s"1 outside the brackets in (73) cancels the s inside, so that the right member becomes simply /xV3^03, where A03 is now the tensor of h0. That is, the effective magnetification is the convergence of the real magnetisation, just as when the medium is at rest. But if h0 is perpendicular to k, say parallel to i, the right member of (73) becomes /xs-'V^ou or t*16 effective magnetification is s~l times the con vergence of the magnetisation, and is therefore increased.

Suppose, for example, our magnet is a straight filamentary magnet. It has two poles, of strength m and - m say, so that induction to the amount m diverges from one and converges to the other pole, whilst continuity is made between them in the filament itself. Now let the medium move past the filament, the direction of motion being parallel to it. Then the strength of the poles is unchanged, but the induction outside is diverted laterally by the effective reduced inductivity parallel to the filament. This case, then, resembles that of a pair of oppositely signed point-charges, if we keep outside the filament. In the limit, therefore, when the speed is raised to v, we have a pair of parallel plane induction-sheets whose cores are joined by the straight filament of induction, to make continuity.

For (66) to be valid, B should be such as to make £ circuital. This requires that w and curl B should be perpen- dicular to one another. They are perpendicular in the example just mentioned, on account of the symmetry of B with respect to the axis or line of motion. But if the same filament be held transversely across the line of motion the property stated is no longer true, so that (66) is not true, and this will, to an unknown extent, upset the later equations. We can, however, still employ them provisionally to obtain a first approximation to the results. On this understanding, then, the strength of the poles (effective) is made m/s and - m/s, with unimpaired effective inductivity parallel to the filament, whilst that per- pendicular to the filament is reduced, as before. So there is now an increased total induction as well as concentration about the plane through the filament perpendicular to the motion.

280 ELECTROMAGNETIC THEORY. CH. III.

We may, however, arrange matters in such a way that the equation (66) shall be still valid when the line of motion is perpendicular to the magnetisation, and so obtain an exact solution showing the increased induction, that is, exact in the absence of working errors. Let the region of magnetisation be confined between two infinite parallel planes, and the magneti- sation I = fih0 be uniformly distributed, parallel to the boundaries. We thus do away with the poles. Now when the medium is at rest the induction is B = I within the plate (which may be of any thickness), and zero outside, whilst there is no displacement. The intrinsic magnetic force h0 pro- duces the greatest effect that it can produce unaided. But if the medium be made to move steadily straight across the magnetised region, then, after certain transient effects have passed away (which may be readily calculated, because they form simply plane electromagnetic waves), although there will still be no induction (or displacement) outside the plate, that within it will be increased (without change of direction) from I to I/s. That is, the motional magnetic force comes in to assist the intrinsic magnetic force. Here we have the explana- tion of the previous result relating to the increased effective strength of poles.

Along with this increased induction, there will be electric force, according to (66). It is entirely within the magnetised region. As the speed increases up to vt the induction and the accompanying displacement in the plate go up to infinity.

The theory of an electrised plate is similar. The displace- ment due to intrinsic electrisation J = ce0, where e0 is the equivalent electric force, the electrisation being uniform and parallel to the sides of the plate, will be increased from D = J to D = J/s by setting the medium moving straight through it, it being now the motional electric force that assists the in- trinsic. The accompanying magnetic force is accordingly given

(74) Outside the plate there is no disturbance in the steady state.

It will, of course, be understood that if, instead of supposi- tional impressed or intrinsic forces in a uniform medium, we employ actual material plates, magnetised or electrised as the case may be, they must, in the first place, be non-conductors,

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 281

and next, they must not interfere with the supposed uniform motion of the medium. In short, the reservation is similar to that mentioned in connection with moving charged conductors. Of course, in our present case, the plates may be moved, whilst the medium is supposed to be at rest.

It will be readily seen that the full investigation of the effects of moving practicable electromagnetic arrangements of conductors presents considerable difficulties. We can, how- ever, get some information relating to the motion of a linear circuit.