Convective magnetic fields in a dispersive half-space

Convective magnetic fields in a dispersive half-spaceAuthor(s): Gabriel BartonSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 466, No. 2120 (8August 2010), pp. 2383-2400Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/25706351 .

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PROCEEDINGS -OF- A Proc. R. Soc. A (2010) 466, 2383-2400

THE ROYALwk doi:10.1098/rspa.2009.0551 SOCIETY A Ik Published online 10 March 2010

Convective magnetic fields in a dispersive half-space

By Gabriel Barton*

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

It is known that inside a material half-space the magnetic field B owing to the currents generated there by a slowly moving exterior charge (velocity u) is almost the same whether the material is a good Ohmic conductor or a highly refractive

non-dispersive/non-dissipative insulator. By contrast, the drag force experienced by the charge is completely different for conductors and insulators. To gain insight into the somewhat surprising coincidence regarding B fields, we study a microscopic model whose macroscopic Drude-type dielectric function e((o) can fit a fair variety of dispersion and dissipation. We look for B only to first order in w/c, but with otherwise arbitrary u.

Then, B is given by the Biot-Savart rule. The term linear in u follows directly from the

polarization produced as if electrostatically by the charge in its instantaneous position, and depends only on e(0), the strictly static (zero frequency) response function; only the corrections of higher order in u depend on just how e varies with w, and we determine the first such corrections.

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

1. Introduction, preview and summary

How the magnetic field B of a slowly moving exterior charge Q penetrates into certain special kinds of highly reflecting half-spaces z < 0 is described

by two results, each initially surprising in its own right. We write u for the velocity of the charge (with w/c?l), and ? for its distance from the surface. Long ago, Boyer (1974) and Furry (1974) calculated such fields inside a good, but not too good, Ohmic conductor characterized simply by its

conductivity a, finding an enhancement; more recently, the present author

(Barton 2009) calculated them inside a non-dispersive insulator characterized

by its real dielectric constant n2, finding that in the limit n->oo, they are practically the same as inside a good conductor, in spite of the drastic differences between the electromagnetic properties of the materials. This is in sharp contrast to the drag force experienced by a charge moving parallel to the surface, which is totally different in the two cases: proportional

*[email protected]

Received 16 October 2009

Accepted 9 February 2010 2383 This journal is ? 2010 The Royal Society



2384 G. Barton

to Q2u/d?? for conductors (Boyer 1974, 1999), but with an exponentially small factor exp(?2?cl>s/^) for insulators1 governed by the surface-plasmon frequency, oj^.

One reason to pursue such contrasts in some detail is the apparent importance of similar contrasts for understanding Casimir forces and the associated zero

point fields, long-standing quantum problems lucidly discussed, for example, by Ingold et al (2009). Here, however, we aim only to develop some insight into the basic physics underlying the coincidence regarding the B held, which is

purely classical; to that end, we keep the discussion as elementary and explicit as

possible. Equally, the end results could be viewed as contributions to the study of magnetic screening in general.

The problem becomes manageable if one works only to leading (first) order in u/c, exploiting the crucial observation spelled out in Barton (2009), that the answer may then be found by applying the Biot-Savart rule to the current density j generated in the material via electrostatics, i.e. neglecting retardation (as if c ?> oo). Here too we start, in unrationalized Gaussian units, by subdividing the total magnetic field into B + JBpoi, with B the field that would be generated by the charge in the absence of the half-space, and Bpo\ generated by the currents

j (conduction or polarization or both) that it induces in the material; and

approximate Bpo\ ^ B^/c, dropping contributions of higher order in 1/c. Finally,

to ease the typography, we now define = b. Thus,

Bto^ = B + Bpoh Spoi-^1 and B = b, (1.1)

the object being to determine b. Our reasoning and main results are laid out as follows. Section 2 sets up a Drude

model flexible enough to accommodate both insulators and conductors. Section 3 writes down the familiar electrostatic potentials, polarizations and surface charge densities 2 induced by a stationary charge; these serve, first, as crucial auxiliaries in deriving the exact2 solutions when the charge is moving (?4a), and then as the basis of expansions by powers of u (?46). The calculation is routed through the two-dimensional Fourier transforms of the potentials as functions of the surface

parallel position coordinates; the essential tools here are certain non-trivial factors oi these transforms, defined in equation (4.12), supplied in equations (4.13)

(4.15), and approximated in equation (4.19). On the other hand, it is central to our method that it avoids Fourier transforms with respect to time. In this

problem, they tend to obscure rather than illuminate the response, especially to

parallel motion, as originally observed by Boyer (1974), and as further witnessed

(in the author's opinion) by continuing uncertainties in the allied problems of

drag forces, both classical and quantum (e.g. Philbin & Leonhardt 2009 and references therein).

xIn notation from ?2, the exact drag force reads -(aQ2(jj^/4u2) Kq(2^cos/u). That it falls

exponentially as u/^cos -> 0 is typical given the frequency gap tog 2We call exact expressions true to first order in u/c, but to all orders in dimensionless parameters like u/[distancex(plasma frequency or conductivity)], i.e. without approximations based on the

dispersive and dissipative properties of the material.

Proc. R. Soc. A (2010)



Convective magnetic fields 2385

Section 5a uses the Biot-Savart rule to derive the exact expressions (5.6) (5.9) for b = z x VQ in terms of a potential Q featuring only 2. From these

expressions, some remarkable symmetries of b follow at once. Moreover, they invite a subdivision Q = + L\Q and thence b= b^ + Ab, still exact, where

b^ (?56) can be written down in closed form directly from the electrostatics of

?3, and only Ab (?5c) depends non-trivially on dispersion and dissipation, i.e. on how the dielectric function s(co) varies with frequency. The rest of the paper concentrates mainly on sensible approximations to A6, via the A W, by ascending powers of u.

Section 6 discusses Ab_, induced by motion perpendicular to the surface; using cylindrical coordinates, this is relatively easy to handle. Parallel motion induces A6y, analysed and then approximated in ?7a. By virtue of the underlying restriction to order 1/c, it is purely convective, i.e. a function, laterally, only of distance from the instantaneous position of the charge. To leading and next

to-leading orders in u\\ (but not beyond), it separates naturally into two parts.

One part Ab^ (?76) is generated reversibly from the (fictitious) non-dissipative limit of e(a)); figure 1 illustrates the central portion of the pattern of field lines and figure 2 gives some indication of magnitudes. The other part Ab[^ (?7c) is generated irreversibly from the dissipative parts of e: figure 3 illustrates field lines and figure 4 indicates magnitudes.

The full expressions for by given in ?7 should perhaps be prefaced by warning the reader that they are complicated. They are spelled out mainly to demonstrate that our method can indeed deliver them straighforwardly though laboriously; but, unless wanted for some specific applications, they are best viewed as a basis for developing some overall insight into patterns and magnitudes, exemplified by the figures.

2. The model

To determine the currents j, we shall devise a Drude model3 entailing a

macroscopic dielectric function f(w) that, in different limits, can fit both insulators and conductors, namely,

(X)2T + (Ol ? (A)2 ? 1(JL>X

'

z<0: s=?_

co2T ? co2 ? icoX (2.1)

and z>0: f = l,

where wp is the (bulk) plasma frequency. Then, the conductivity <j, refractive index n, a corresponding polarizability a, all at zero frequency, and a surface

plasmon frequency cos (defined by hindsight) are

A"W Ws"Wr + i' (2-2)

3We make no attempt to introduce the temperature T explicitly, it would make itself felt mainly through the value of the damping constant A. To adapt the model to finite T in the quantum regime is a far more delicate problem, discussed at some length elsewhere (Barton 1997, 2000).




2386 G. Barton

and

w=o: e=i + 4^?2, ^rriry (2.3) (n2 + 1)

For good metallic conductors, X/cos is typically of order 10~3 to IO-2. The underlying microscopic model is strictly non-relativistic; it has been

discussed very fully elsewhere a propos of van der Waals forces (Barton 1997,

2000), and for non-dispersive insulators (as defined below) in Barton (2009). It features charge carriers with charge e, mass m and number density n, displaced from equilibrium by ?. Then, the squared plasma frequency and the basic equation of motion read4

<4 = ^ ud *-i>f-& + ?B. (2.4) p m at1 dt m

We introduce a microscopic displacement potential W] the macroscopic polarization P and surface charge density macroscopic potentials 0 and and connect the model to e through

f = -W, ? = -V0, P= -^E = heZ = -V\/;, W=?, (2.5) 47T ne

dP dxl/ j=

? = -V^

and 2 = P3(z = 0-). (2.6) ot ot

Integrating out the gradients, one finds

Insulators have oj2t 7^ 0. Non-dispersive insulators (considered in Barton

(2009)) would be realized in the limit (w^,w2)->oo at fixed finite frequency independent n2 and a < 1. They would become perfectly reflecting in the further limit n ?> 00, hence a ?> 1.

Conductors have = 0. They are necessarily dispersive,

"I z<0: e = l- 2" (2.8)

At zero frequency, e((o -> 0) = n2 oo, giving perfect reflection and a = 1. Two

important special cases are non-dissipative plasmas,5 2 2

A = 0 = e = l-^, ?l = f; (2.9)

4It may be worth stressing (as in Barton 1997, p. 2466) that X = 0 makes the material non

dissipative in the sense that it can absorb energy only reversibly, through the creation of plasmons; but that nevertheless, it is dispersive, and perfectly causal in the sense that e satisfies the

Kramers-Kronig relation, by virtue of an imaginary part ttoj2 5(co'2t

? (J2).

5Then, bulk plasma waves ('bulk plasmons') have frequency wp for any wavevector. They produce no fields outside, hence as regards our strictly exterior charges, we can and shall ignore them

altogether (cf. Barton 1979, 1997, 2000).





and Ohmic conductors (Boyer 1974; Furry 1974)

?(Z<0)^1--JL = 1-. (2.10) 47tct lAOJ lb)

We write the instantaneous position of the charge as p = (<r, ? > 0), and held

points as r = (s, z). It will prove convenient to introduce

+ S = 8-<r = (S,X), S = Js* + Si,

cosX = ^ and R = ^S2 + Z2. (2.11)

Notice that Z is a coordinate difference (namely z ? C) only if z < 0. Define also, for any function f(z),

disc(/(z))=/(0+)-/(0-). (2.12)

The governing equations are

E = -V(f>, V24> = -4irQ8(r-p) + 4irV P, (2.13)

z<0: -V-P = VV = -4ttV20, (2.14)

disc(0) = O and disc(d30) = 4tt d3\(/\z=Q (2.15)

We shall rely heavily on two-dimensional Fourier representations of the

Coulomb potential in various guises, e.g.

1 f d2k ,m aQ f d2k ? = ?- exp[ife s] exp[?k\z\] and \f/K

' = ? ?- expfifc S ?

kZ]. \r\ J 2irk 2ir J 2irk

(2.16)

3. Stationary solutions

As already stated, our method for dealing with slow motion centres on the

response of the material half-space to a stationary charge at the instantaneous

position of the true charge; quantities governed directly by this response will be labelled by superscripts (0), leading to equations (2.13)-(2.15) with

(.E,P,0,^) -* (?(0\P(0\<^?\^0)). When the true charge is stationary (u = 0), all such quantities are independent of time t ; but in ?4 the same symbols6 are

retained for the same functions of position relative to a uniformly moving charge, when they do vary with t.

6As the least evil of possible choices, the significance of the supercript (0) in this paper differs from that in Barton (2009), where it was reserved for terms independent of c in expansions by powers of l/c. The superscripts '1' that appear in equation (1.1), signifying coefficients of 1/c, are also

relics from Barton (2009), i.e. they are not counterparts of our present (0).




2388 G Barton

For insulators, by equation (2.7),

<4#0: ̂ (0) = (^)^(0)^<0).

(3-1)

The textbook solution features n and a from equation (2.3), and the image position p,

^\+>i.z)mf?i>, ?.(,,_o, (3.2) [\r-p\ \r-p\\ \r-p\

where 6 is the Heaviside step function; then,

47r 27r [(s -

<r)2 + C2]^

(3.3)

For conductors, by equation (2.7),

w2r = 0 : 0(o)(z < 0) = 0 (regardless of A). (3.4)

In virtue of equation (2.3), we have equations (3.2) and (3.3) with n2 -> oo and a = 1,

0(O) = 0(Z) and 2J<?> = -?-^-(3.5)

4. Uniformly moving charge

(a) Exact solution

The crucial idea is to write 0 = + A0, etc., where (f>^ is the function defined

by equation (3.2) but now parametrically time-dependent via its dependence on

p(t) = p(0) + ut, u = u\\ + zu2, and u\\ = xu\\, (4.1)

the x-axis being chosen along the component of u parallel to the surface. The

fields B^ are already known for the special case of Ohmic conductors from Boyer

(1974, 1999), and for non-dispersive insulators from Barton (2009), but xf/^ and are re-derived more succinctly below. New here is the derivation of A0 and

thence of L\B. We substitute the expressions thus split into equations (2.14) and (2.15);

simplify by using equations (3.2) and (3.3); and note that these components by themselves satisfy the matching conditions (2.15). The governing equations





become

V2A<? = 0, V2A^ = 0, (4.2)

disc(A</>) = 0, disc(d3A<?) = 4ird3 A^=0 (4.3)

md 7/+^ir+w^-s4*=-{^+A-|r(-

(44)

We adopt an Ansatz satisfying equation (4.2) and equation (4.3i) by construction:

A0 = Re d2A;A0(k,?)exp[ifc. s- k\z\] (4.5)

and

z<0: Ai// = Re d2A:A^(fc,t)exp[ifc. s + kz]. (4.6)

Then, by equation (4.32), A0 = -2ttA^, (4.7)

reducing equation (4.4) to

The associated homogeneous equation is solved by

e"^2 exp ^fcius^l

- ^ j

= Ot, (4.9)

featuring the (hypothetical) undamped frequency ws, with subcritical damping when A < 2cos.

On the right-hand side of equation (4.8), we use equations (3.32), (2.16) and the crucial identity

<410)

to extract an ordinary inhomogeneous differential equation for A\/>,

x |(fa/3

+ ifc it||)2 - \{foi% + ifc u\\) J. (4.11)

The associated homogeneous equation is the same as for A^, with the same

solutions (4.9). It is the absence of k from this equation that makes our problem

relatively tractable. Since the complementary function is a linear combination of the C? from

equation (4.9), it decays exponentially in time, and the solution of equation (4.11)




2390 G. Barton

that we require is the particular integral

M> =

-j^jik exP ? ~ kK]?W(k, tig, u?) (4.12)

where

j (ku3 + ifc nj|)2 - A (ku$ + ik un) \

A^^-j-i-L, (4.13) | (A:^3 + ik ̂ n)

? A (fa^ + ifc u\\) + Wg J Notice that A W is independent of the surface-parallel wavenumber hi normal to

From here on we consider only motion that is either perpendicular to the surface (u\\

= 0), or parallel (^3

= 0), identified by subscripts _L or ||, respectively.

Then, A W reduces to one or the other of

{(ku3) -Xku3 + (Os\

and

I ? (fe u\\)2 ? iXk u\\ j

*wisrh?7-~7\' (4-15) j ?

[k ixy) ? iAfc u\\ +

Without dissipation (when A = 0), the denominator of AWl is manifestly positive, and it could vanish only if damping became supercritical, a scenario we do not consider further. On the other hand, and as equation (4.9) should

perhaps have led one to anticipate, the denominator of AWy vanishes when

k u\\ =

?(x)s-yjl

? A2/4wg

? iA/2, a resonance effect signalling that appropriately

moving charges can create physical rather than merely virtual surface plasmons. Such plasmons contribute crucially to the electric field responsible for the drag force experienced by the charge; but we shall see in ?46 that, for small u and small A, they never enter the approximation governing the B field. Meanwhile, we note the exact reality condition

AVF|*(fc) = A W|| (?fe).

The potential AV> and A0 = ? 27rAi/> from equation (4.7) are our central results;

they allow one to find or to approximate H, and thence the B fields that are our

main objective. In particular, from equation (3.3),

AU(S, 0 = -d3 Ai/zUo = -^Re [d2A;exp[ifc 5 - kQ A W, (4.16)

(2tt)2 j

whence

d*S[S>? =

-^%Re f d2A;exp[ifc S -

kQ [-ik u\\ -

ku3] (4.17) dt (27r)2 J

Because occurs only in the exponential, A?, like is even in S2 = y ?

o<i>





(b) Approximations

Approximations hinge on the denominators of L\W in equations (4.14) and

(4.15), call them V. There are obviously corresponding approximations to S, to

dS/dt, and eventually to B. Write u generically for any pertinent component of u, and note that in

the Fourier integrands, k is effectively of order 1/R. Least troublesome is the

regime where u 2

ku^ ? <?(Os and A<^cl>s P ?ws,

realized by good Ohmic conductors7 (cf. equation (2.10)), which on dimensional

grounds translates into small A in the sense that

col / x A =

-^z ? (wp

~ cos) =^cos<â. (4.18)

As far as A W is concerned, this corresponds formally to high cos, whence

{(ArM3)2 - XkuA \(k wii)2 + iAfc uA AW? ?

J 9--? and Aî--^-^?z-^. (4.19)

ws ws

Trying to improve on equation (4.19), (merely) small-u approximations, i.e. brute-force expansions of W by powers of u, would make sense only if

1 U 1 1 M û 2 ku~ ? A ana Aku ~ ? <JC cos =>

2p '

(4-2?) -?A?^- -?ws, Run )

possible (given finite only for a limited range of A. Conversely, (merely) small-X

approximations, i.e. brute-force expansions by powers of A , would make sense

only if

u 1 ,7 û 2 A<? ku^

? ana Aku ^ ? ? ws ==>

(4.21)

A?-?-^ => A?ws,

possible (given finite A) only for a limited range of u. We will not pursue such improvements, and settle for the approximations

(4.19), which give all results in the form of a non-dissipative term (as if for A = 0)

7The criterion (4.18) for a conductor to be treated as 'good' at this stage does not, of course,

feature c, which disappeared from our arguments when they were restricted to order 1/c at the

outset. By contrast, starting from the equation of telegraphy that follows directly from Maxwell's

equations, low-velocity approximations (u/c <$C 1) apply even to conductors 'good' in the sense that

dR/c > 1 provided % = auR/c2 1 =>> c/R < o <3C c2/uR. Both conditions are readily satisfied for small enough u/c; it is in this sense that one requires '<r to be high but not too high'. The author is much indebted to an anonymous referee for pointing out that appendix B of Barton (2009) (where 'L' takes the place of our present 'i?') wrongly replaced the basic condition by its opposite x ̂ > 1




2392 G. Barton

plus a dissipative term proportional to X. This tallies with the remark in ?4a that, for our present purposes, one can disregard resonance effects associated with the creation of physical plasmons.

5. The B field

We streamline the vector identities used in Barton (2009) to express j and

finally b(r) directly in terms of S(s). We will subdivide b = b^ + A6 later.

(a) The Biot-Savart rule

Our non-retarded model has

z<0: i = -V^, VV = 0 and 2 = - d3x/s\z=0. (5.1) ot

This Neumann problem is solved by the image method (e.g. Barton 1989),

VKr) = 2\d2s"df{S"^=-2 [dV [3(5", O]?^: and SW - <r. J | r

? s" | J | r ? s" |

Hence,

i(r) = 2fdV^|^V-J?, (5.3)

with U and dS/dt prescribed by equations (4.16) and (4.17). They vary with time because they depend on the time-dependent position of the charge, on ? directly and on a via S or S"; and their time dependence in turn governs that of \j/ and j. This said, we focus on the immediate agenda by abbreviating E(S", ?)

= U(s7/).

By the Biot-Savart rule, appropriate variants of equation (2.16) for each of the two Coulomb factors, and equation (4.10),

f 1 b(r') = d3rj(r) x V?

J \r ?

r\

J dt L<0 V \r-s"\) \ \r'-r\) and b(r') = 2 d2s dz dV^p^ ^eik<s-^+kz[ik + zk] V ' J J-oo J a* J 27T&

f d2/ x ?eil<s'-s^z'-zl[-il

- zle(z

- z%

J 2irl

where e is the signum function. Appeal to

d2s exp[i(fc -s-l-s)] = {2it)26(k -

I) (5.5)





and straightforward simplification,8 dropping the now redundant primes from r', reduce this to

b(r) = zxVQ(r) = V x (-zQ), (5.6) dQ , dQ fl\dQ dQ

Q{r) = -2Re dV^-^ ^ exp[ifc (a - ?") - jfe|z|] (5.8) J dt J kz

and 6fr)=4irfdV^- , * x (? - ?")_

J a< - *")2 + z2

y(s-s")2 + z2 + \z\\

with obvious analogues for ft'?'(r) = ix Vfl^(r) and A5(r) = zx VAfl(r),

featuring 2J(0) and AH, respectively. It might bear stressing that, although we have expressed b wholly in terms of the surface-charge density, it is generated by currents flowing inside and not on the surface of the material.

Equation (5.6) shows that b(r) is everywhere parallel to the surface, and

equation (5.8) that it is even with respect to distance from the surface. The field lines are the level curves of Q.

By symmetry, for perpendicular motion, the field lines are circles parallel to the surface and centred on S = s ? (7 = 0, with parities to match: b\ is even in Si and odd in S2, and vice versa for 62- For parallel motion, we shall see presently that b is even in S2? but that parity with respect to Si is a somewhat deceptive notion, depending on behaviour not under reflections but under time reversal, and sharp only if the material is non-dissipative.9

(b) The quasi-stationary component 6(0)

This is the component studied in Barton (2009). To find it, one starts by

replacing dE/dt in equations (5.8) and (5.9) by dE^/dt, with E^ from equation (3.3) and d/dt from equation (4.10). Then, it becomes obvious that b^ depends on the material only through a prefactor a. The calculation is straightforward, and

reproduces the results for non-dispersive insulators already known from Barton

(2009) (with (X, Y)^(SUS2)),

q(Q)= -aquz and g(Q) = a q-UW__\_

js*+si+" ys?+si+z* z+y^2+si+z* (5.10)

8To sidestep J dz... one can simplify the volume integral in equation (5.4) through the vector identities (V/) x v = V x (fv)

? /V x v, V x V# = 0 and Jz<0 d3rV x w = J d2sz x itf|z=o? with

appropriate /, v, p and it;. Then, equation (5.8) is easily recovered by evaluating the surface integral, also using the two-dimensional Fourier representations (2.16).

9By an oversight, the parities for are described wrongly in Barton (2009) (at the end of ?7(c)). The correct conclusions follow readily from equations (5.6) and (5.10) here.




2394 G. Barton

It proves convenient to scale the lateral coordinates by Z. Writing the new dimensionless variables in sans serif, we define10

(Sl,s2)=^, s=y^=|, (5.H)

cosxs!i = ^ and rWl + s2 = |, (5.12) and have

lP-(^)=^ and

6<??=(^)coS(x)^. (5.14)

From equation (4.19), one sees that the leading corrections to Q(?\ embodied in

A?, are of relative order (ku/cos)2 ~

(uf Zcos)2 and kXu/to\~ Xu/Zio\. It is this that explicates the mechanism whereby, as found in I, to first order in u non

dispersive insulators in the perfect-reflector limit n -> oo produce the same B fields as do Ohmic conductors, even though the latter are very dispersive indeed.

(c) The dispersive component A b

By equations (4.17) and (5.8),

&Q(r) = -2Re [ d2s"-^-

\d2k" {exp [ik" (*" -

a) -

k'X) [-ifc" u? -

f d2k 1 x A W(fc")

? exp [ik (s -

s") -

k\z\] ,

whence,11 again in virtue of equation (5.5),

It might be found reassuring to verify that equation (5.15) ensures compliance with the constraint emphasized by Boyer (1974) and pursued in Barton (2009), that for parallel motion, the integral J^dSife and therefore J^dSiAft must vanish (given that J^?00d5i6^ =0 is already known).

Since Ai2 (like Q^) features positions only through S and the field at points r inside the material is a function only of the coordinate differences (S,Z)

=

R = r ? p, regardless of the position of the surface, i.e. regardless of field point and source point, separately. This startling property, stressed originally by Furry (1974), and following him in Barton (2009), we call the Furry pattern. Presently, it will allow the time variation of the interior field 6 to be visualized very simply; and thence, given that b has even parity, that of the exterior field too.

10In Barton (2009), Si52 were called Eip. We are running short of congenial symbols. 11 The prescription Re will be dropped from expressions that are manifestly real as they stand.





From here on we settle for the nominally 'high co^ approximations (4.19): the more readily (i) because they should serve for most scenarios likely to be met in

practice and (ii) because our main motivation is to help illustrate the physics that so startlingly reduces the effects of dispersion. It is sheer bad luck that actually evaluating A6y in ?6, though straightforward in principle, requires so much more

labour than Barton (2009) needed for 6(0).

6. Perpendicular motion

The held lines of b_?^ and i\b? alike are circles parallel to the surface and centred on S = 0. Thus, A6? =

?A^ =

<pdL\Q__/dS,

* d2 k f00

L\Q? = 2aQu3 ?eikS-kZkkW__ = -47raQus dke~kz L\W__ J0(JfcS), (6.1) J & Jo f?? AtzolQu2 f??

A&i? = 4iraQm dk<TkZ ^W^kJ^kS) ~-dke~kz J^kS^m -

Xk2] Jo k>s Jo

(6.2)

UTraQujS \, ? u3(S2 -

3Z2)) and -bx^-?^{xZ- R2 (6.3)

where the approximation implements equation (4.19). For conductors, this reduces to

^_6oQ4gfz_ ^(S2-3Z2)]

6a g?/2 sJl_47TU3 (s2 -

3) 1 " Z* 'r*\a Zu*' r2 }'

( j

to be compared with the conclusion from Barton (2009) that bf^

= cx(cB__cp) =

(aQu3/Z2)s/r3, where B is the free field defined in ?1, and quoted in equation (7.1) below.

Viewing these expressions one must bear in mind that, like all our results, they are restricted to exterior charges, i.e. to ? > 0. For motion towards the half-space, their warranty expires when the charge crosses the surface; in particular, they say

nothing about the fields owing to any transition radiation (of surface plasmons, in our model) that might then be emitted. Until then, or for a receding charge, the interior field fits the Furry pattern. In other words, it changes (in magnitude and direction) as if it were attached rigidly to the charge; roughly speaking, at any fixed point, it falls or rises as ? rises or falls. By the same token, Q and b are

well defined and finite everywhere, as long as ? is strictly positive, only when ? vanishes (when the image charge and the real charge coincide) could they diverge, if the field point r too approached (<x,0).




2396 G. Barton

7. Parallel motion

Here, the variation with time is purely convective and in that sense trivial:

laterally the Furry pattern depends only on S = s ? a = s ? U\\t, whence it moves in step with the charge, in whose rest frame it remains stationary.

To ease the typography, this section omits the subscripts || from 6y.

(a) The potential kQ\\

According to equation (5.6), and in view of dx^y =

ds^, one has A&i =

?dkQ\\/dS2 and A&2 = dkQ\\/dS\. For comparisons, we cite the correspondingly normed free fields in the half-space z < 0,

c? = C x ^

= ? HZ + iS.) = ? (* + *,) (7.1)

With u ? xu\\, our approximation (4.19) yields

2a 0 r d2k Afl|,

~ ?f-Re

? exp (ifc S

- A;Z) (-i) [(fc u{f + iA(fc u,,)2]. (7.2)

CJg J AC

Let be the (two-dimensional) polar angles of k and 5, and define t? =

X ?

<p, so that cos x = S'i/5', while k - S = kS cos t? and txy =

A^y [cos % cos t? + sin % sin t?]. Changing J^d^...

to J^dt?... and invoking symmetry to drop

from the integrand terms odd in 1?, one obtains

2a Q f?? dA: f71" Afi||

~ ?^ ~Te~kZ dlM (ku\\)3 sin(^cos[cos3 X cos3 # + 3cos x sin2 x

JO ^ J?TT x cos # sin2 t?] + A(^)2 cos(kS cos t?)[cos2 % cos2 + sin2 x sin2 t?]}. (7.3)

It proves convenient to subdivide, in an obvious notation,

Afiy ~

Afijr) + Afij0 and A 6 ~ A6(r) + A6(i), (7.4) where '(r)' identifies the reversibly generated (non-dissipative) component free of

A, and '(i)' the irreversibly generated (dissipative) component proportional to A and thereby to l/a. (One could equally well take '(r)' and c(i)' to refer to the real and imaginary terms inside the square brackets in equation (7.2).)

Evaluating Jd#... and then J dA; ... is painfully laborious, though relatively straightforward with the aid of various integrals derived in appendix A of Barton

(2009). The results are easiest to reach in Cartesians, and read

^ ^H*-f-^+**;",'-M,i

?a) Z2 \ S2 (2R2-RZ-Z2) , x and G(l) = ?- \ 1-l- ?- . 7.7) R{R + Z)\ R2 S2 J

y '





At this point, one can see that both and are even in 52, as Q must

be on general grounds, but that Afij^ (like Q^)

is odd in S\, whereas A?2^ is

even. No mystery is involved, the positive direction along the Si-axis is denned

by the direction of motion, and L\Q^ (like @\^) is governed by that part of the

dielectric response that is even under time reversal, whereas is governed by the part that is odd.

In other respects, the potentials G and the corresponding fields are somewhat less opaque in terms of the scaled two-dimensional polar coordinates (5.11) and (5.12),

G<r> = cos(X)5l(r)(s) + cos3(rf(s), (7.8) -(r)_3s(l + 2r) 91 r3 (1 + r)2

' [ '

M s (8r3 + r2 - 6r - 3) to =-5-, (7.10) y3

r5(l + r)2 v '

G? = g?(s) + cos2(x)9?)(s), (7.H)

$ = and ?) = -2Jl~r~1. (7.12) y? r(l + r) y2 r3(l + r)

v '

Recall (say from equation (5.7)) 3G L dG l /AdG j, , dG

Recall also that S = 0 is the point level with and opposite to the charge. We

proceed to illustrate a few simple field patterns. (The held lines of in the

(Si,S2) plane were illustrated in fig. 1 of Barton (2009), and could be treated as a basis for comparison. They form two sets of loops, centred on stagnation points

on the Si-axis, at Si = ?^(1

+ V^5)/2 ? ?1.27, separated by a field line running

along the S2-axis.)

(b) The reversible component A6^r^

By equation (7.5) and equations (7.8)-(7.10),

(jj^Z4 '

as2i sax * ds

(7.14) The field lines form four sets of loops, centred on stagnation points on the

Si-axis, at Si ?0.462 and ?3.189. The two nearest the origin are illustrated in

figure 1, which qualitatively (but not at all quantitatively) is somewhat similar to the (entire) pattern for b^. It cannot show the loops centred on the two outer

stagnation points because the field in those regions is too weak to be visible on

the same scale. The rapid decrease of with Si, superimposed on the oscillation dictated by the stagnation-point zeros, is shown in figure 2; clearly, without the

change of scale, the eye could not detect the outer zero at all.




2398 G. Barton

-,-.-,-r?.-,-1-.---- -r--n- ?,-'-1-1-'-1-'-'-S

Figure 1. Field lines of the reversible dispersive component A/?(r) (parallel motion), given by level curves of G^T\ equation (7.8); showing the two stagnation points nearest the origin, at Si

~ ?0.462.

On the Si-axis (parallel to the trajectory at fixed depth), /3\ vanishes; figure 2

plots /?;> for 0 < Si < 2.5 and 100(32 for 2.5 < Si. Notice the change in scale, and

the zeros at the stagnation points. Asymptotically, /^(Si <?C 1) ~9/4 ?

75s2/4, and /^r)(Si?l)~6/S^

On the S2-axis (perpendicular to the trajectory at fixed depth), (3\ again

vanishes, while f$ ̂s monotonically as S2 rises, with /^(s2 <?C 1) ? 9/4 ?

15s2/4 and j(jW(s|?l)~6/s|.

(c) The irreversible component A6^

By equation (4.11) and equations (7.11)?(7.13),

r, AitaQXu2 r,

with (3^ given by the derivatives in equation (7.14) on changing -> G^\ The held lines form three sets of loops, with stagnation points on the Si-axis, at

Si =0 and Si ~ ?2.252. The central loops are illustrated in figure 3. Again, it is

impossible to show the outer ones on the same scale. One sees that loops centred on the origin are peculiar to fields generated irreversibly.

On the Si-axis (parallel to the trajectory at fixed depth), (3^ = 0; figure 4 plots

/?2 for 0 < Si < 2 and 10/32 for 2 < Si. The zero locates the right-hand stagnation

point. Asymptotically, /^(Si ? 1) ~

-9Si/4 +25s3/4, and /^(Si ? 1) ~

2/s3. On the perpendicular (s

= 0) from charge to surface, A6^ vanishes.





2-\

c/T \ v \ ?o \ (N \ tS \

XT' \

f : \ ^ n \ (-\| U- -r+?'- -1- - -1-' I^__J-1 1-

-1?7^- - -1-1-1- - - -1- -'- -'-1

v \ 1 ^-^2

y si4 5 6

o \ / I M / ?

1 /

-2-1

(r) Figure 2. The reversible dispersive component A/Jg along the line S2 = 0 (at fixed depth and

parallel to the trajectory, given by equation (7.14)).

S2

-xs" -cliT -ii4 0!:, 0^. "as "as Sl

Figure 3. Field lines of the irreversible dispersive component A/?^ (parallel motion), given by the

level curves of equations (7.11)?(7.13); showing the stagnation point at the origin.




2400 G Barton

0.1: "

-?

% : / CN I / V . 1 / ?T -0.3

- / v ; I / o .1 /

o I / ^ -0.4- 1 / XT" I / ^ \ / -0.5 - \ /

-0.6 \ V/

Figure 4. The irreversible dispersive component A/?^ along the line S2 = 0 (at fixed depth and

parallel to the trajectory, given by equation (7.11)).

On the S2-axis (perpendicular to the trajectory at fixed depth), (3^ again vanishes, while (3^ rises from 0 at S2 = 0 to a maximum of 0.27 at 0.68 and

eventually falls to zero like 2/S2.

References

Barton, G. 1979 Some surface effects in the hydrodynamic model of metals. Rep. Prog. Phys. 42, 963-1016. (doi:10.1088/0034-4885/42/6/001)

Barton, G. 1989 Elements of Green's functions and propagation, ch. 6. Oxford, UK: Oxford

University Press.

Barton, G. 1997 Van der Waals shifts in an atom near absorptive dielectric mirrors. Proc. R. Soc.

Lond. A 453, 2461-2495. (doi: 10.1098/rspa. 1997.0132) Barton, G. 2000 Atomic shifts near absorptive mirrors. Comments Mod. Phys. D: At. Mol. Phys.

2, 301-307.

Barton, G. 2009 On magnetic fields generated in a dielectric half-space by a slowly moving point charge outside. Proc. R. Soc. Lond. A 465, 809-822. (doi:10.1098/rspa.2008.0359)

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, 68-82. (doi:10.1103/PhysRevA.9.68)

Boyer, T. H. 1999 Understanding the penetration of electromagnetic velocity fields into conductors.

Am. J. Phys. A 67, 954-958. (doi:10.1119/l.19171) Furry, W. H. 1974 Shielding of the magnetic field of a slowly moving point charge by a conducting

surface. Am. J. Phys. 42, 649-667. (doi:10.1119/l.1987802)

Ingold, G.-L., Lambrecht A. & Reynaud S 2009 Quantum dissipative Brownian motion and the

Casimir effect. Phys. Rev. E 80, 041113.

Philbin, T. G. Sz Leonhardt, U. 2009 No quantum friction between uniformly moving plates. New

J. Phys. 11, 033035.




Convective magnetic fields in a dispersive half-space

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