## Abstract

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.

## 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 *ω*, which diminish exponentially, on a scale set by the skin depth .

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 (

*u*/

*c*), provided only that

*nu*/

*c*≪1. One must hedge such statements because, as §4 will show, vanishes, while is finite. Since perfect light reflection would ensue only at infinite

*n*

^{2}, we shall for brevity refer to the leading terms in the regime 1≪

*n*

^{2}≪(

*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 The writer's interest in them stems from their bearing on image forces (Barton 2008; G. Barton 2008, unpublished data; to be cited as I, II), whose leading velocity-dependent terms are often of order

*u*

^{2}, 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*≪1 to the more demanding

*nu*/

*c*≪1. Under this condition, §5 discusses the (merely asymptotic) expansion of

**and thence of**

*A***=∇×**

*B***by powers of**

*A**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 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 §3. 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 in terms of an auxiliary potential *Ω*; and eventually yields for *Ω* the elegant and convenient closed expressions (6.5)–,(6.7). Remarkably, everywhere is parallel to the surface, and has even parity with respect to it. Section 7 spells out other consequences. Sections 7*a*–,7*c* concern scaling properties and symmetries (especially parities) and introduce dimensionless form factors ** F** allowing the field components to be displayed regardless of

*n*

^{2}; these sections summarize themselves. Section 7

*d*concerns 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 combinations of them reproduce the fields of steady currents, to which non-magnetic materials do not react at all. The last section, 7

*e*, merely resumes the wholly unexpected coincidences between the results for non-dispersive insulators (with any

*n*

^{2}∼(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.

## 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,(2.1)It occupies the half-space *z*<0, while *z*>0 is vacuum. The dielectric function reads(2.2)The material is taken as non-magnetic, so that *μ*=1 inside and out.

In the vacuum outside there is a point charge *Q* at *ρ*=(

**,**

*σ**ζ*>0), moving with velocity(2.3)We write field points as

**, and define(2.4)By hindsight we also define(2.5)Notice that**

*r**Z*is an even function of

*z*; and that inside the material(2.6)while for positive

*z*, i.e. outside,

*Z*=

*z*+

*ζ*bears no useful relation to

*R*

_{3}.

Maxwell's equations read(2.7)(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 , one finds(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 (*ϵ*=1 everywhere), we introduce the script capitals , .

## 3. Electrostatics

For a stationary charge, ** B** vanishes, and without further loss of generality we choose

**=**

*A***everywhere. Then, using superscripts (0) to indicate**

*0**u*=0,(3.1)The familiar solution features the image position and a polarizability

*α*,(3.2)and reads(3.3)

(3.4)

*We shall retain these definitions for moving charges*. Then *ϕ*^{(0)}, and likewise the vector potentials and fields to be introduced presently, depend on *t* parametrically, through the time dependence of *ρ* and . Thus,(3.5)In the absence of the medium, one would have just the Coulomb field(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(4.1)Then Maxwell's equations entail(4.2)The matching conditions on

**and on ∂**

*A**/∂*

*A**z*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 ,

*=(*

*ρ**ξ*=

*ut*,0,

*ζ*), 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

**. In this convective regime ∂/∂**

*A**t*=

*.∂/∂*

*u***=**

*ρ**u*∂/∂

*ξ*=−

*u*∂/∂

*x*, and (4.2) entails(4.3)

From (4.3) it is evident that the limits *n*→∞ and *u*/*c*→0 are incompatible, i.e. that they do not commute.3 In the prior limit *n*^{2}→∞ the material reflects perfectly. By virtue of (3.3) and (3.4), the interior *E*^{(0)} then vanishes, but on the right of (4.2), as, of (4.3), the second term remains finite, while the presence of shows that must vanish. Hence ** A** cannot change with time (with no externally applied constant

*E*field there can be no non-zero time-independent ), whence no interior fields can be produced by uniformly moving charges that at

*t*=−∞ were infinitely far from the field point in question. In particular that is the case for any fixed

*u*when

*nu*/

*c*→∞. Accordingly,

*n*

^{2}→∞ 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,

*nu*

_{∥}/

*c*>1 must elicit Cherenkov radiation (e.g. Schieber & Schächter 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

**. From here on, therefore, we can and shall dispense with**

*A***altogether.4**

*A*## 5. Approximations for *u*/*c*≪1

Bracketed superscripts will indicate orders in *u*/*c*, consistently with the notation already adopted for *ϕ*^{(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*^{(0)}+

*E*^{(1)}/

*c*+

*E*^{(2)}/

*c*

^{2}+⋯, and

**similarly. Substituting into Maxwell's equations one readily sees that**

*B*

*A*^{(0)}=0; and, from (4.2), that

**expands by powers of 1/**

*A**c*

^{2}, whence

**=**

*A*

*A*^{(1)}/

*c*+

*A*^{(3)}/

*c*

^{3}+⋯. Accordingly we write(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 (*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), §65. Thus, there is no point in trying to continue (5.1) beyond the terms actually displayed there.

To zero order one has *ϕ*^{(0)} and *E*^{(0)}=−∇*ϕ*^{(0)} from §3, while *B*^{(0)}=0. The only term to first order is *B*^{(1)}/*c*≡∇×*A*^{(1)}/*c*. To second order one would need . Unfortunately, calculating *A*^{(1)} turns out to be quite difficult; it is governed by (4.2) without the first term on the left, i.e. by(5.2)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.

It proves convenient to split(5.3)into the contributions , generated directly by the uniformly moving point charge (the familiar Liénard–Wiechert solutions, cf. LL §38 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(5.4)

(5.5)

## 6. from the polarization currents

Given the difficulty just explained of securing *A*^{(1)} in full, it is lucky that follows directly from the Biot–Savart Law applied to the polarization current density in the medium, i.e. to(6.1)The writer suspects that this is obvious. A formal proof starts by noting that in principle 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

**be the vector from**

*R**this*charge to the field point

**. Then eqn (65.5) of LL shows that, in some gauge we need not specify, its vector potential to order**

*r**v*/

*c*may be written as

*e*

**/**

*v**cR*, and its

*B*field therefore as (

*e*/

*c*)∇

_{r}×(

**/**

*v**R*)=(

*e*/

*c*)

**×**

*v***/**

*R**R*

^{3}. But this is precisely the Biot–Savart Law, which therefore applies by linearity to

*in toto*.

In (6.1), the time dependence resides in *ρ*(

*t*), with ∂

*/∂*

*ρ**t*=−∂

**/∂**

*R**t*=

**. Accordingly,(6.2)where the last step has used [∇×**

*u***=0]⇒[**

*a***×∇**

*a**ψ*=−∇×(

*ψ*

**)], and then the vector identity7 .**

*a**We see that*

*is parallel to the surface*. This is surprising, because the polarization currents are by no means perpendicular: away from some exceptional points, .

Next, one Fourier transforms the two coulombic terms, using (A 3). Then yields a delta function, reducing the two two-dimensional Fourier integrals to just one; ∂/∂*t* is trivial because *ρ* occurs only in an exponent. The result reads(6.3)(6.4)Then tallies with ∇×

*A*^{(1)}found in II; but , whence is not the entire vector potential in the pseudo-Coulomb gauge. Regarding

*Z*≡|

*z*|+

*ζ*we recall (2.5) and (2.6).

To evaluate *Ω* one replaces i(*l*_{∥}.*u*_{∥})→*u*_{∥}.∂/∂** S**, integrates over the azimuthal angle of

*l*_{∥}, and uses (A 7)–(A 14). Taking the

*x*-axis along

*u*_{∥}, and introducing conveniently scaled coordinates (

*Σ*

_{1},

*Σ*

_{2}), one finds(6.5)(6.6)When

*u*

_{3}=0 these expressions can be rationalized to(6.7)

## 7. Properties of

Section 7*e* will comment on the coincidences between 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 *Ω* and thereby are functions only of ** R**. In particular, they do not depend on

*z*and

*ζ*separately, but only through

*Z*≡|

*z*|+

*ζ*. (The same is true of

*P*^{(0)}.) Thus, for a given position

*of the charge and at a given point*

*ρ***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**

*r**Z*, the potential

*Ω*is a function only of the scaled lateral coordinates

*Σ*

_{1,2}. Equations (7.1)–,(7.3) below show the same for .

### (b) Symmetries

*For any direction of* ** u**, one notes from (6.6) and (6.3) that

*Ω*(

*z*)=

*Ω*(−

*z*), whence . This is unexpected because the polarization currents are, necessarily, far from having any definite parity. Trying for some insight into the apparent paradox we risk one observation. Since

*P*^{(0)}is a function only of

**, the pattern of polarization currents is the same at all depths: all layers contribute constructively to the field at any exterior point**

*R**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 |

*z*| 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 *Ω* is even under *Y*→−*Y*, we see from (6.3) that is odd while is even.

Under *X*→−*X* on the other hand *Ω* has no definite parity unless either *u*_{∥} or *u*_{3} is zero.

*For perpendicular motion* (*u*_{∥}=0, *u*_{3}=*u*), we see that *Ω* is even in *X*, whence is even while is odd.

*For parallel motion* (*u*_{∥}=*u*, *u*_{3}=0) the parities are opposite: *Ω* is odd in *X*, whence is odd while is even.

### (c) Magnitudes

Equation (6.3) shows that , whence the field lines are the level curves of *Ω* in the (*X*,*Y*) or in the (*Σ*_{1},*Σ*_{2}) planes. We consider only interior points (*z*<0,*Z*=−*R*_{3}), which suffices because is even in *z*.

It proves convenient to introduce strength-independent form factors by defining and ^{(1)}(** r**)=Q

**×**

*u***/**

*R**R*

^{3}≡(

*uQ*/

*Z*

^{2})(

*) from (5.5). We shall compare*

*R***with the surface-parallel components of ; note that for good refractors, i.e. near**

*F**α*=1, the strength factors become practically the same.

#### (i) Perpendicular motion

Given ** u**=(0,0,

*u*), one has ; in other words(7.1)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(7.2)(7.3)To rationalize one would set(7.4)Figure 1 shows some field lines. The field vanishes at ,

*Σ*

_{2}=0. We illustrate magnitudes with two examples.

At ** r**=(0,0,

*z*), i.e. along the perpendicular from charge to surface,(7.5)Again there is anti-shielding: inside good refractors (near

*α*=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

_{1}=

*F*

_{1}=0, but now(7.6)Both functions are plotted8 in figure 2. At

*Σ*

_{2}=0 one recovers (7.5); by contrast, as |

*Σ*

_{2}|→∞ one finds while .

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.

### (d) Pulses

Given a charge in uniform parallel motion, *ρ*=(

*ξ*=

*ut*−

*a*,0,

*ζ*), we follow Boyer's lead (1974, 1999) and determine the pulse as a function of time at fixed

**=(0,0,**

*r**z*<0). In this geometry

*Σ*

_{1}=

*ξ*/

*Z*and

*Σ*

_{2}=0, whence

*F*

_{1}=0. Boyer's point, made originally à 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

*B*_{pol}, and therefore zero , a constraint that must be shown to be satisfied by our form factor .

To adapt *Ω* to our present scenario, we define *τ*≡*ut*/*Z*, start with *a*=0 (whence *Σ*_{1}=*τ*), and observe9 that(7.7)(7.8)Figure 3 plots *F*_{2} and *G*_{2} against *τ*. The plot of *F*_{2} evidently serves equally to sample its variation with *x* at fixed *t*. The polarization-generated field at a fixed interior point is proportional to *G*_{2}: it starts from zero, reverses when the charge is level with the point in question, and eventually vanishes, as it should. In the absence of the material *G*_{2}(*τ*) would be replaced by _{2}(*τ*),(7.9)with _{2}(∞)=2 as dictated by Ampère'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 *λ* per unit length, velocity (*u*,0,0) and leading point at *ut*. Then there are charge elements at with all . Accordingly, labelling the fields of such currents with overbars, we have(7.10)The constraint is simply(7.11)It is exact, i.e. it holds for all *u*_{∥}/*c*; and (7.10) and (7.8) show that our expressions satisfy it to (*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 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 for the field generated directly by the conduction currents produced by the same charge when 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 for the field generated directly by the charge; Furry writes it as

*B*_{0}.

Furry finds , but . Regarding for parallel motion he agrees with Boyer.

Comparison with our results then shows,

*a**posteriori*, that . Thus,(7.12)(7.13)Since is even,″ is even while*B*′ is odd.*B*Furry illustrates

′. In view of (7.12) and (7.13), these illustrations adapt trivially to . One notes that on its far side the sheet would screen perfectly, because .*B*His fig. 3 helps to visualize the scaling property discussed in §7

*a*.The field lines 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

*Ω*is his very differently obtained potential −*ψ*.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 along a section through this plane inside the material. Since is even in*z*while is not, the lines of have a kink where they cross the surface. We do not attempt to draw them (not even for*α*=1). Furry's fig. 5 and his tables II and III display quantitative information about ; in view of (7.12) and (7.13), for*α*=1 this applies to our*B*_{total}outside but not inside.

## Footnotes

↵Appendix B explains just what we take the words

*good conductor*to mean.↵

*Pseudo*because ∇.need not vanish on the surface, where in fact it has a*A**δ*(*z*)-proportional singularity because disc(*A*_{3})≠0. We stress that*φ*^{(0)}enters not as an approximation, but to specify the gauge: (4.1) and (4.2) are exact.↵A similar incompatibility afflicts ohmic conductors: lim

_{u/c→0}lim_{σ→∞}≠lim_{σ→∞}lim_{u/c→0}.↵Of course

reappears centre stage in any Hamiltonian version of the theory, such as is developed in II.*A*↵For conductors they would not: unlike the dimensionless parameter

*n*, the conductivity has dimensions of inverse time, which complicates the problem appreciably.↵There are no true surface currents even on perfectly reflecting insulators: they exist only on perfect conductors.

↵Here, d

stands for a surface element, not to be confused with the vector potentials that*A*symbolizes everywhere else in this paper.*A*↵They cross at , the same as the value of Σ

_{1}wherevanishes when Σ*F*_{2}=0. No deep reason for the coincidence is visible.↵Although 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*τ*=*ut*/*Z*, which is linked to times and to distances, but knows nothing about*c*.↵Not to be confused with the position coordinate

used elsewhere in this paper.*σ*↵High velocities, i.e.

*u*/*c*∼*O*(1), are discussed by Schieber & Schächter (1998).- Received September 2, 2008.
- Accepted October 27, 2008.

- © 2008 The Royal Society