On Magnetic Fields Generated in a Dielectric Half-Space by a Slowly Moving Point Charge Outside

On Magnetic Fields Generated in a Dielectric Half-Space by a Slowly Moving Point ChargeOutsideAuthor(s): Gabriel BartonSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 465, No. 2103 (Mar.8, 2009), pp. 809-822Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/30243328 .

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PROCEEDINGS OF

THE ROYAL SOCIETY A

Proc. R. Soc. A (2009) 465, 809-822 doi:10.1098/rspa.2008.0359

Published online 25 November 2008

On magnetic fields generated in a dielectric half-space by a slowly moving point

charge outside

BY GABRIEL BARTON*

Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK

A point charge moving with speed u<< c/n outside a non-dispersive dielectric half-space having refractive index n produces, inside the material, magnetic fields of the same order as and in fact larger than they would be in wholly empty space. The part of the field generated directly by the polarization currents is parallel to the surface, and has even parity with respect to it. For 1 << n c/u, these fields are practically independent of n, and, by a remarkable coincidence, the same as the (already known) fields that the same charge would produce in a half-space occupied by material having high (but not infinite) ohmic conductivity.

Keywords: B fields in dielectrics; velocity-field penetration; magnetic shielding

1. Introduction and summary

Long ago, but startlingly at the time, Furry (1974) observed that the magnetic fields B of steadily moving charges penetrate beyond an infinitesimally thin but perfectly reflecting because perfectly conducting plane; and, in a postscript, agreed with Boyer (1974), who meanwhile had pointed out independently, for parallel motion, that such fields penetrate also a half-space occupied by a good conductor.1 For a corrected update and a literature review about conducting half-spaces, see Boyer (1999). Roughly speaking, these so-called convective B fields diminish only like inverse distance squared, as they would in wholly empty space: the main reason for surprise was the contrast with the fields due to incident light waves having frequency (o, which diminish exponentially, on a scale set by the skin depth c/v2.rrao.

Our aim is to explore the same problem for an insulating half-space with real non-dispersive refractive index n, to first order in the velocity u, i.e. to O(u/c), provided only that nu/c<< 1. One must hedge such statements because, as S4 will show, limU/c-0o(c/u)lim,2 2 B vanishes, while limn2u limu/co(C/U)B is finite. Since perfect light reflection would ensue only at infinite n2, we shall

*[email protected]

1Appendix B explains just what we take the words good conductor to mean.

Received 2 September 2008 Accepted 27 October 2008 809 This journal is C 2008 The Royal Society



810 G. Barton

for brevity refer to the leading terms in the regime 1 << n2 << (c/u)2 as applicable to good refractors. Charges with u/c<< 1 we call slow. Remarkably, a given slow charge will turn out to generate exactly the same B field inside a good refractor as it would inside a good conductor; moreover, these fields are of the same order as the ones that the charge would generate in wholly empty space. The writer's interest in them stems from their bearing on image forces (Barton 2008; G. Barton 2008, unpublished data; to be cited as I, II), whose leading velocity- dependent terms are often of order u2, and purely classical: an indifference to Planck's constant that corresponds to the purely classical nature of the effects reported here.

The rest of this paper concerns itself only with the fields additional to those that the charge would generate in the absence of the material. Sections 2 and 3 review Maxwell's equations and specialize them to stationary charges, with equations (2.5) and (2.6) highlighting the important auxiliary position variable Z, which is even in the position of the field point relative to the surface. Section 4 introduces the convenient pseudo-Coulomb gauge; derives the exact equation (4.2) obeyed by the vector potential A in this gauge; and explains how the possibility of Cherenkov radiation forces approximations designed for slow charges to sharpen the obvious condition u/c<<l to the more demanding nu/c<< . Under this condition, S5 discusses the (merely asymptotic) expansion of A and thence of B=VXA by powers of u/c; notes that their leading terms, superfixed (1), are of first order; and derives their governing equation (5.2).

Section 6 establishes our central technical point, that the material-dependent contribution B1o can be found by applying the Biot-Savart Law to the zero- order polarization currents, identifiable directly via the familiar electrostatics already spelled out in S3. Though this makes the calculation straightforward in principle, it is anything but trivial in practice: the formalism from appendix A leads one to (6.3) and (6.4), representing Bo1 in terms of an auxiliary potential Q; and eventually yields for Q the elegant and convenient closed expressions (6.5)-(6.7). Remarkably, B(1) everywhere is parallel to the surface, and has even parity with respect to it. Section 7 spells out other consequences. Sections 7a-7c concern scaling properties and symmetries (especially parities) and introduce dimensionless form factors F allowing the field components to be displayed regardless of n2; these sections summarize themselves. Section 7d concerns B () as a function of time at a fixed point; it leans heavily on (and verifies) Boyer's observation that such pulses generated by motion parallel to the surface must have vanishing time integrals, essentially because appropriate linear com- binations of them reproduce the fields of steady currents, to which non-magnetic materials do not react at all. The last section, 7e, merely resumes the wholly unexpected coincidences between the results for non-dispersive insulators (with any n2 ~O(1)) and for good ohmic conductors, noting that they admit much quantitative information about the former through mere transcription of the extensive details given by Furry (1974) for the latter.

In principle, our predictions for insulators as opposed to conductors should be easy to check: all one needs is a small induction coil embedded in the material.

Proc. R. Soc. A (2009)



Convective B fields in dielectrics 811

2. Maxwell's equations

We use unrationalized Gaussian units. SI units would be perverse, because we shall expand by powers of 1/c.

Consider a non-dispersive dielectric, e-1 P - E, D- E + 4rP = eE. (2.1) 47r

It occupies the half-space z< 0, while z> 0 is vacuum. The dielectric function reads

e(z) = 0(-z)n2 + O(z), O(z < 0) = O, O(z > 0) = 1. (2.2)

The material is taken as non-magnetic, so that = 1 inside and out. In the vacuum outside there is a point charge Q at p=(a, moving

with velocity u dp/dt, ull da/dt, u3 = d (2.3)

We write field points as r, and define

r (s, ), R -p - (S, R3), S -a (X, Y), R3 z- . (2.4)

By hindsight we also define Z - |z + (. (2.5)

Notice that Z is an even function of z; and that inside the material

z < 0 = Z = -z + = -R3, (2.6) while for positive z, i.e. outside, Z= z+ ~ bears no useful relation to R3-

Maxwell's equations read

V-B = 0, VXE+B/c = 0, (2.7)

V.D = 4rrQb(r-p), V X B - D/c = (4irQ/c)ub(r-p). (2.8)

From them, we derive matching conditions that the fields satisfy across the surface z= 0. For any function F(z), define disc(F) = F(0 +) - F(0 -). By acting on the field equations with foyf dz..., one finds

disc(D3) = 0, disc(Ell) = 0, disc(B) = 0. (2.9)

Given these conditions, we need to, and shall, consider fields only at z*0. For the fields that the charge would generate in absence of the medium (e= 1

everywhere), we introduce the script capitals 8, 13.

3. Electrostatics

For a stationary charge, B vanishes, and without further loss of generality we choose A= 0 everywhere. Then, using superscripts (0) to indicate u= 0,

E() = -V(0),

V2(0) =

47r-Q 6(r-p),

disc = 0,

disc(ea3(gj)) -= 0.

Q(Z) (3.1)




812 G. Barton

The familiar solution features the image position p and a polarizability a,

S-,-), a= (n2 - 1)/(n2 + 1) (3.2)

2 Q' =-aOQ, Q" = Q + ' - Q. (3.4) n2+1

and reads

We shall retain these definitions for moving charges. Then 0(0), and likewise the vector potentials and fields to be introduced presently, depend on t parametrically, through the time dependence of p and p. Thus,

ao(o)/at = u~.O()/0ap = ua.(o)/ai. (3.5) In the absence of the medium, one would have just the Coulomb field

e(O) --VQ/Ir-pl = Q(r-p)/Ir-pl3. (3.6)

4. Wave equation

Though the substantive calculations in the present paper, designed for slow charges, will deal directly with B, a general view of the time dependence is best obtained from the wave equation for the vector potential in the pseudo-Coulomb gauge2 defined by

E = -Vo(o)- A /c, B=VXA and V-A=O forz:0. (4.1) Then Maxwell's equations entail

e(z) a2 4rrQ e(z) +E(o) C V+V2 A - u(r - p) - , (z 0). (4.2) c2 g2C c at

The matching conditions on A and on aA/az follow from (2.9) plus (3.1); we skip the details (given in II) because they are not needed if one requires only the B field to first order in u/c. For parallel motion, say for u= ui= (u, 0, 0), p= (f = ut,O, the inhomogeneity (the right-hand side) depends on x and t only through the combination x-ut; then it is either obvious from translation invariance, or follows via Fourier transforms, that the same is true of A. In this convective regime alat= u- a/ap = ua/a = - ua/axz, and (4.2) entails

(F nu)21 2 a 2i 2 2rrQ n2 aE(o) 1 -2 + 2 Ay2A uQ 6(r - p) (z<0). c a/ x 2 2(c c at

From (4.3) it is evident that the limits n-+ oo and u/c-+0 are incompatible, i.e. that they do not commute.3 In the prior limit n2-+ 00the material reflects perfectly. By virtue of (3.3) and (3.4), the interior E(O) then vanishes, but on the

2Pseudo because V -A need not vanish on the surface, where in fact it has a 6(z)-proportional singularity because disc(A3) 40. We stress that <p(o) enters not as an approximation, but to specify the gauge: (4.1) and (4.2) are exact. 3A similar incompatibility afflicts ohmic conductors: limu/c-o lim- -00# lim__, 0 limu/c--*o.





right of (4.2), as, of (4.3), the second term remains finite, while the presence of A / C2 shows that A must vanish. Hence A cannot change with time (with no externally applied constant E field there can be no non-zero time-independent A), whence no interior fields can be produced by uniformly moving charges that at t= - oo were infinitely far from the field point in question. In particular that is the case for any fixed u when nu/c- oo. Accordingly, n2 -+ generates the perfect reflection scenario already discussed in I. The intermediate regime where nu/c is of order unity is difficult to elucidate: for one thing, null/ c> 1 must elicit Cherenkov radiation (e.g. Schieber & Schiichter 1998). Here, we shall settle for the good refractor regime where n>> 1 yet nu/c<<1, i.e. u far below the light speed in the medium, which for small enough u/c still admits a wide range of values for n. Crucially, this allows expansion by powers of u/c; and, to first order in u/c, will allow us to find the fields directly, rather than via A. From here on, therefore, we can and shall dispense with A altogether.4

5. Approximations for u/c << 1

Bracketed superscripts will indicate orders in u/c, consistently with the notation already adopted for 0(0). In fact it proves easier to start by expanding formally in powers of 1/c, even though this is a dimensional parameter: for insulators, the proper expansions emerge automatically in due course.5 Thus, we try to write E= E(O) + E(1)/c-+ E(2)/c2 +..., and B similarly. Substituting into Maxwell's equations one readily sees that A(O)=0; and, from (4.2), that A expands by powers of 1/c2, whence A=A(1)/c+A(3)/c3+.... Accordingly we write

A A(1)/c +..,

E = -V(o) -A(1)/c2 +, B V X A(1)/c + O(1/c)3. (5.1)

However, exact solutions of Maxwell's equations or of (4.2) cannot be expanded convergently by powers of u/ c: the best one can find are asymptotic approximations to the fields up to and including O(u/c)2. The reasons, readily visible from the electromagnetic Green's functions for wholly empty space, are spelled out very explicitly by Landau & Lifshitz (1975; cited as LL), S65. Thus, there is no point in trying to continue (5.1) beyond the terms actually displayed there.

To zero order one has 0(0) and E(O) -Vq(O) from S3, while B(O)-= 0. The only term to first order is B(')/c-VXA(1)/c. To second order one would need

E(2)/C2 - _A(1)/c2. Unfortunately, calculating A(~) turns out to be quite difficult; it is governed by (4.2) without the first term on the left, i.e. by

8E(O) V2A(1)

= -4. uQb(r - p) - ce (z I 0), (5.2) at plus appropriate matching conditions, and requires a complementary function in addition to separate particular integrals for the two terms on the right. The full calculation is given in II. 4 Of course A reappears centre stage in any Hamiltonian version of the theory, such as is developed in II. 5For conductors they would not: unlike the dimensionless parameter n, the conductivity has dimensions of inverse time, which complicates the problem appreciably.




814 G. Barton

It proves convenient to split E = 8 + Epol, B = B + Bpo1 (5.3)

into the contributions 8, B generated directly by the uniformly moving point charge (the familiar Lienard-Wiechert solutions, cf. LL S38 and Feynman et al. 1964), and those generated by the (surface) polarization charges and by the polarization current density inside the medium.6 For comparison, we expand

QR (1 - U2/C2) R3 {1-[u2 -(u.R)2/R21/C23/2

-QR 1 ~ + [u2-3(u-R)2/R2]/c2 ... + (0) +(2)I/c2 +, (5.4)

13 = (u/c)X8=B( /c + O(1/c3)", 3(1)/C =- uX S(O)c Qu XR (5.5) cR3

6. B(1) from the polarization currents

Given the difficulty just explained of securing A(1) in full, it is lucky that

B(1)= VXA(1 follows directly from the Biot-Savart Law applied to the polarization current density in the medium, i.e. to

8P(o)t(r) _ (n _ 1) -p t -p (6.1) at at 4, r-p| it 27rr-p| The writer suspects that this is obvious. A formal proof starts by noting that in principle P(0) too can be ascribed to a combination of moving point charges. Call a typical one e, its velocity v, and (just in this paragraph) let R be the vector from this charge to the field point r. Then eqn (65.5) of LL shows that, in some gauge we need not specify, its vector potential to order v/c may be written as ev/cR, and its B field therefore as (e/c)VrX (v/R) = (e/c)vX R/R3. But this is precisely the Biot-Savart Law, which therefore applies by linearity to P(0) in toto.

In (6.1), the time dependence resides in p(t), with ap/at= -aR/at=u. Accordingly,

B o)(rl') = a

d3 )(r) X V/ = - aQa po at

27ir at J

raX Vi aQ IddaX(621 1 - 2r atj xdyz X

az=o, a =Ir-p (6.2)

where the last step has used [VXa=O]>[aXV4= -VX(4la)], and then the vector identity fvolumed VV X (Via) = fsurfacedA X (/a). We see that )i parallel to the surface. This is surprising, because the polarization currents are by no means perpendicular: away from some exceptional points, aP2)lat o0. 6 There are no true surface currents even on perfectly reflecting insulators: they exist only on perfect conductors. 7 Here, dA stands for a surface element, not to be confused with the vector potentials that A symbolizes everywhere else in this paper.





Next, one Fourier transforms the two coulombic terms, using (A 3). Then ff dx dy ... yields a delta function, reducing the two two-dimensional Fourier integrals to just one; a/Ot is trivial because p occurs only in an exponent. The result reads

B( =X Vp2, B(1 -- 2/By,

B (1), = 2a/ax, (6.3) polI Xpo0,1 pol,2

aQ d121i11 T2= - - [i(/ll ull) +l1u3] exp{il, - S -/1 Z}. (6.4) 27r 1 11

Then Blo)=VX(- 2) tallies with VXA(1) found in II; but V-(--i)= -a2/az 0, whence -2 is not the entire vector potential in the pseudo- Coulomb gauge. Regarding Z- ++ we recall (2.5) and (2.6).

To evaluate Y2 one replaces i(11 .u

1) -* u, alaS, integrates over the azimuthal angle of il1, and uses (A 7)-(A 14). Taking the x-axis along u1j, and introducing conveniently scaled coordinates (Z:1,Z2), one finds

aQ= a Q ullX3 (6.5) /X2+ 2 + Z2 Z+/'X2 y2 + a2 1 1 _ (X, Y)

ZZ /+E + U3 Z (66)

When U3=0 these expressions can be rationalized to

lla QX [ Z (ullaQ/Z) ,l

1 X2 2 + y2 1- + Z( 1 + 2

(6.7) 7. Properties of B(1)

Section 7e will comment on the coincidences between B (1)z) and the results of Furry (1974) for the interior of a well-conducting half-space. He gives so much quantitative information that we restrict our own illustrations to a minimum.

(a) Scaling

Equation (6.5) shows that inside the material Y2 and thereby B(o1 are functions only of R. In particular, they do not depend on z and gjg separately, but only through Z- Iz| + . (The same is true of P(0).) Thus, for a given position p of the charge and at a given point r the field does not depend on the position of the surface, provided only that the surface lies between the two. More specifically, (6.6) makes it explicit that, apart from the overall scale that is set by Z, the potential 2 is a function only of the scaled lateral coordinates 21,2. Equations (7.1)-(7.3) below show the same for B(1)P

(b) Symmetries For any direction of u, one notes from (6.6) and (6.3) that 2(z)=2(-z),

whence Bo (z) = Bpo(-z). This is unexpected because the polarization currents

are, necessarily, far from having any definite parity. Trying for some insight into




816 G. Barton

the apparent paradox we risk one observation. Since P(O) is a function only of R, the pattern of polarization currents is the same at all depths: all layers contribute constructively to the field at any exterior point z>0, the contributions diminishing uniformly with increasing depth of the layer. By contrast, at a field point z<0 inside, layers at depths greater and less than Izl contribute with opposite signs, a destructive effect that is seen, a posteriori, to be exactly counteracted by enhancement from the fact than now there are currents arbitrarily close to the field point in question.

Since Q is even under Y-* - Y, we see from (6.3) that B(1) is odd while Bpo)2 is even.

Under X-- - X on the other hand Q has no definite parity unless either u1l or U3 is zero.

For perpendicular motion (ua 0 =0, u3= u), we see that Q is even in X, whence B(1) is even while B(1) is odd.

pol,1 pol,2

For parallel motion (u = , u, u3=0) the parities are opposite: Q is odd in X, whence B(1) is odd while B(1) is even.

(c) Magnitudes

Equation (6.3) shows that B).VQ = 0, whence the field lines are the level curves of 0 in the (X, Y) or in the (21, Z2) planes. We consider only interior points (z<0 ,Z= - R3), which suffices because B (1) is even in z.

It proves convenient to introduce strength-independent form factors by defining B)(r) (au/Z2) QF(R) and B 1)(r)=QuXR/R3 (uQ/Z2)F(R) from (5.5). We shall compare F with the surface-parallel components of .F; note that for good refractors, i.e. near a=l1, the strength factors become practically the same.

(i) Perpendicular motion

Given u= (0,0,u), one has F11 = X S/(X2 + Y2 + Z2)3/2 = .F; in other words

B(1 = aB(), (u

= 0). (7.1) pol I I

Thus, inside good refractors the total field is double what it would be in the absence of the material.

(ii) Parallel motion

Given u= (u,0,0), the form factors read

F1] _

1 S1 2(1

+ 2-/1 + E +

,

(7.2) F I +22+ I)+ F2 -D 1+IJ2+(1-

+ 1+2+/++

D+2 D = I + + Z22 (1 + 2: + 2:2)3/2. (7.3) 1 2 Proc. R. Soc. A (2009)




4

2

-E

ClD

si ial

-2

-4-

Figure 1. Some field lines of Bl as given by equations (7.2) and (7.3).

To rationalize one would set

1/D= 2 + ,

+ 2 -2 + + /(1+ f 2+ )3/2( + 2 y)2. (7.4) 1/=[ C1 2 1 2 1 2

Figure 1 shows some field lines. The field vanishes at i1= + (1 + v/5) /2=

+ 1.272, 12= 0. We illustrate magnitudes with two examples. At r-=(0,0,z), i.e. along the perpendicular from charge to surface,

F1 = F1 = 0; = 1, F2 = 1/2. (7.5)

Again there is anti-shielding: inside good refractors (near a= 1), the total field along this line is larger by a factor 3/2 than it would be in the absence of the material.

At r= (0, y,z) at fixed z, i.e. along a line at fixed depth, level with the charge, and at right angles to the velocity, again F1 = Fl = 0, but now

1 1 2 F2 = 1 (7.6)

(1 + E2)3/2 ' + 2] (1 + 1/2

Both functions are plotted8 in figure 2. At 2-= 0 one recovers (7.5); by contrast, as 1L2 -+00 one finds 2 1/ ~ while F2 ~ 1,/Z.

These examples have sampled variation with y and z. The variation with x is effectively sampled by (7.7) below, albeit from a slightly different point of view.

8They cross at 12 = (1 + 1v5)/2, the same as the value of 1 where F vanishes when 12= 0. No deep reason for the coincidence is visible.




818 G. Barton

1.0

0.8

0.6

0.4

0.2

0 1 2 3 4

sigma2

Figure 2. The form factors Y2 and F2 as functions of y2 = Y/Z at fixed X= 0, from equation (7.6).

Note F2(0)= 1 and F2(0)= 1/2.

(d) Pulses

Given a charge in uniform parallel motion, p=( = ut-a,O we follow

Boyer's lead (1974, 1999) and determine the pulse B(1) - auQF as a function of time at fixed r= (0,0,z<0). In this geometry ,I= huh and Z2= O, whence FI= 0. Boyer's point, made originally

' propos of ohmic conductors but equally

applicable to insulators, is that by taking an appropriate combination of such pulses differently phased, i.e. with different values of a, we can construct a steady line current, which magnetically speaking does not see the material at all. In other words, such a combination produces identically zero Bpol, and therefore zero B(1), a constraint that must be shown to be satisfied by our form factor F= fyF2.

To adapt Q to our present scenario, we define T'-ut/Z, start with a=0 (whence Zl= 7), and observe9 that

1 + (--T2)(1 + T2)1/2 aG2 2 (1 + 2)3/2[1 + (1 + T2)1/2]2 - (7.7)

G2 (7) = G2(+) =

0. (7.8) (1 + 72)1/2[1 + (1 + 2)1/2] 2(+o) = 0(78)

Figure 3 plots F2 and G2 against 7. The plot of F2 evidently serves equally to sample its variation with x at fixed t. The polarization-generated field at a fixed interior point is proportional to G2: it starts from zero, reverses when the charge

9Although our expressions are warranted only to first order in u/c, we must and do work to all orders (i.e. exactly) in the dimensionless variable Tr=ut/Z, which is linked to times and to distances, but knows nothing about c.





0.

0-

-8 - - - 8

z

-0.

Figure 3. The form factors F2 and G2 at fixed x= 0 and Y= 0, as functions of the scaled time variable r= ut/Z, from equations (7.7) and (7.8).

is level with the point in question, and eventually vanishes, as it should. In the absence of the material G2(7) would be replaced by g2(7),

F2 1/(1 +2)3/2, 2 = 1 I+-/(1 +2)1/2, (7.9)

with 2(00)= 2 as dictated by Ampere's Law. Finally, to verify Boyer's constraint we switch on an ultimately steady line

current, considering to this end a semi-infinite line charge, with charge A per unit length, velocity (u,0,O) and leading point at ut. Then there are charge elements A dx' at ( ', 0, ag with all x' < ut. Accordingly, labelling the fields of such currents with overbars, we have

1Bpo() = aua) Bo(T Bpol(T) dT'F2(r') = - G2(r). (7.10) c cZJ- cZ The constraint is simply

Bpol(T --+ o) = 0 dtBpo(r, t) = 0. (7.11)

It is exact, i.e. it holds for all ull/c; and (7.10) and (7.8) show that our expressions satisfy it to O(u/c).

(e) Coincidences

Our conclusions for insulators and those of Furry (1974) and of Boyer (1974, 1999) for conductors manifest coincidences that are the most surprising in the writer's experience of electromagnetism, notorious for unintuitive end results though it is. Recall that we are considering a point charge in the half-space z> 0 with velocity u, and fields only to order u/c. Furry's B' is the field generated directly by the conduction currents produced by the charge in an infinitesimally thin but perfectly conducting (and perfectly reflecting) sheet occupying the xy plane, with vacuum half-spaces on both sides. We write B" for the field generated directly by the conduction currents produced by the same charge when




820 G. Barton

there is no such sheet, but the half-space z<0 is occupied by a good but not perfect conductor, with vacuum for z>0. We keep B for the field generated directly by the charge; Furry writes it as B0.

(i) Furry finds B"(z > 0) = B'(z > 0), but B"(z < 0)= -B'(z < 0). Regard- ing B" for parallel motion he agrees with Boyer.

(ii) Comparison with our results then shows, a posteriori, that Bo1) = aB" Thus,

(l/a)Bpol(z<0) = +B"(z<0) = -B'(z <0), (7.12)

(1/a)o(z>0) = +B"(z> 0) = +B'(z > 0). (7.13)

Since B(1) is even, B" is even while B' is odd.

(iii) Furry illustrates B'. In view of (7.12) and (7.13), these illustrations adapt trivially to B(1. One notes that on its far side the sheet would screen

perfectly, because B'(z < 0) = -B(z < 0). (iv) His fig. 3 helps to visualize the scaling property discussed in S7a. (v) The field lines B (1) correctly drawn but not labelled in our figure 1 are

mapped quantitatively in his fig. 4, with more detail in his table I. Up to a prefactor, our Q is his very differently obtained potential -41.

(vi) For parallel and only for parallel motion, the total field has just one set of field lines confined to a plane, namely those in the YZ plane through the

charge (i.e. normal both to the trajectory and to the surface). Equation (7.6) and figure 2 have sampled B1 along a section through this plane inside the material. Since B(01 is even in z while B is not, the lines of Btotai= B B(o)1 have a kink where they cross the surface. We do not attempt to draw them (not even for a= 1). Furry's fig. 5 and his tables II and III display quantitative information about B + B'; in view of (7.12) and (7.13), for a= 1 this applies to our Btotal outside but not inside.

Appendix A. Formulary

s - (x, y); r - (s, z); (A 1)

d3k exp(ik. r)- d2 k exp(ik s)

00

dk3 exp(ik3 z) r 2r2k 2 2kK2 (k2 + k ) (A 2)

1 d2k1l exp(- k1 z + ikl, s). (A 3) 2r k1l

Throughout, we define

k1l > 0.





Bessels:

J d4 cos(x cos 0){1, cos 0, sin , } = {2rJo(x),0, O, (A 4) 7r "

J dd sin(x cos ) {1,cos 0, sin , } = {0, 27rJ(x), 0}, (A 5) --7T J do cos(x cos

){cos 2, sin 2, sinl cos } = {27rJ(x),27r[Jo(x) - J(x)], 0}.

(A 6)

Though we shall make no direct use of it as an alternative to (A 2), we do need (Gradshteyn & Ryzhik 1980)

V Jdk exp(-kZ)Jo(kS) 12_ 1 (Z, S > 0 R vZ2 + S2).

(A 7) Under the same conditions, and with primes always denoting derivatives with respect to the argument,

" S8) W dk exp(-kZ)J(kS) = R(R + Z) '(A8)

dkexp(-kZ)kJ,(kS) Z+RZRS(A9) p as (Z + R)2R3 Remarkably,

a W/az = -S/R3 V/aS. (A 10) From (A 7)-(A 10) plus J= --J1 one readily derives

Jdk exp(-kZ)Jo(kS)k = -aV/dZ = Z/R3, (A 11)

{ dk exp(-kZ)Jd(kS){1, k} {-W, a V/S}, (A 12) 0I 21 aW_2W S dk

exp(-kZ)JoI(kS){k, k2} iS W 2W (A 13)

o as ' asaz a2 W a2 V 3S2 - R2 asaz as2 R5

Appendix B. Good conductors

Given the time dependence discussed in S4, the equation of telegraphy, (V2+ [4ra/c2]uV- [uV]2/2)... =0, shows that as regards the effects of the

conductivity1o a one must balance auV/c2 against V2. Estimating V 1/L, with L the characteristic length scale of the system (for example L~ ), one

10 Not to be confused with the position coordinate a used elsewhere in this paper.




822 G. Barton

sees that the critical parameter is Xu-uL/c2, whence the limits u/c-*O and aL/c-+,

~ are incompatible. The results for what the literature usually calls good

conductors are derived assuming X >> 1; this regime covers a wide plateau, where the B fields are independent of a, analogous to the n-independent plateau for good refractors. Various illustrations are worked out by Schaich (2001a,b).

Such expressions apply when c/aL <<u/c<<1: the velocity though low must not be too low.11 By contrast with our insulators one notes that this criterion depends on L. The complication appears to be a concomitant of dispersion; for instance, it recurs, with different detail, for media having a=0 but finite plasma frequency.

References

Barton, G. 2008 On the velocity-dependence of image forces. J. Phys. A 41, 164 027. (doi:10.1088/ 1751-8113/41/16/164027)

Boyer, T. H. 1974 Penetration of the electric and magnetic velocity fields of a nonrelativistic point charge into a conducting plane. Phys. Rev. A 9, 68. (doi:10.1103/PhysRevA.9.68)

Boyer, T. H. 1999 Understanding the penetration of electromagnetic velocity fields into conductors. Am. J. Phys. 67, 954. (doi:10.1119/1.19171)

Feynman, R. P., Leighton, R. B. & Sands, M. 1964 The Feynman lectures on physics, vol II, section 26.2. Reading, MA: Addison-Wesley.

Furry, W. H. 1974 Shielding of the magnetic field of a slowly moving point charge by a conducting surface. Am. J. Phys. 42, 649. (doi:10.1119/1.1987802)

Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of integrals, series, and products, equation 6.611.1. New York, NY: Academic Press.

Landau, L. D. & Lifshitz, E. M. 1975 The classical theory offields, 2nd edn. Oxford, UK: Pergamon. Schaich, W. L. 2001a Electromagnetic velocity fields near a conducting slab. Phys. Rev. E 64,

046 605. (doi:10.1103/PhysRevE.64.046605) Schaich, W. L. 2001b Surface response of a conductor: static and dynamic, electric and magnetic.

Am. J. Phys. 69, 1267. (doi:10.1119/1.1407253) Schieber, D. & Schichter, L. 1998 Reaction forces on a relativistic point charge moving above a

dielectric or a metallic half-space. Phys. Rev. E 57, 6008. (doi:10.1103/PhysRevE.57.6008)

" High velocities, i.e. u/c~ 0(1), are discussed by Schieber & Schichter (1998).




On Magnetic Fields Generated in a Dielectric Half-Space by a Slowly Moving Point Charge Outside

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