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 ε(ω) can fit a fair variety of dispersion and dissipation. We look for B only to first order in u/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 ε(0), the strictly static (zero frequency) response function; only the corrections of higher order in u depend on just how ε varies with ω, and we determine the first such corrections.
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 u/c≪1), and ζ 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 , 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 , 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 to for conductors (Boyer 1974, 1999), but with an exponentially small factor for insulators1 governed by the surface-plasmon frequency, ωS.
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 field, 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 ). Here too we start, in unrationalized Gaussian units, by subdividing the total magnetic field into , with the field that would be generated by the charge in the absence of the half-space, and Bpol generated by the currents j (conduction or polarization or both) that it induces in the material; and approximate , dropping contributions of higher order in 1/c. Finally, to ease the typography, we now define . Thus, 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 Σ 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 (§4b). 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 ΔW of 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).
Section 5a uses the Biot–Savart rule to derive the exact expressions (5.6)–(5.9) for in terms of a potential Ω featuring only Σ. From these expressions, some remarkable symmetries of b follow at once. Moreover, they invite a subdivision Ω=Ω(0)+ΔΩ and thence b=b(0)+Δb, still exact, where b(0) (§5b) can be written down in closed form directly from the electrostatics of §3, and only Δb (§5c) depends non-trivially on dispersion and dissipation, i.e. on how the dielectric function ε(ω) varies with frequency. The rest of the paper concentrates mainly on sensible approximations to Δb, via the ΔW, by ascending powers of u.
Section 6 discusses Δb⊥, induced by motion perpendicular to the surface; using cylindrical coordinates, this is relatively easy to handle. Parallel motion induces Δb∥, 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 (§7b) is generated reversibly from the (fictitious) non-dissipative limit of ε(ω); figure 1 illustrates the central portion of the pattern of field lines and figure 2 gives some indication of magnitudes. The other part (§7c) is generated irreversibly from the dissipative parts of ε: figure 3 illustrates field lines and figure 4 indicates magnitudes.
The full expressions for b∥ 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 ε(ω) that, in different limits, can fit both insulators and conductors, namely, 2.1 where ωp is the (bulk) plasma frequency. Then, the conductivity , refractive index n, a corresponding polarizability α, all at zero frequency, and a surface-plasmon frequency ωS (defined by hindsight) are 2.2 and 2.3 For good metallic conductors, λ/ωS is typically of order 10−3 to 10−2.
The underlying microscopic model is strictly non-relativistic; it has been discussed very fully elsewhere à 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 , displaced from equilibrium by ξ. Then, the squared plasma frequency and the basic equation of motion read4 2.4 We introduce a microscopic displacement potential Ψ; the macroscopic polarization P and surface charge density Σ; macroscopic potentials ϕ and ψ; and connect the model to ε through 2.5 2.6 Integrating out the gradients, one finds 2.7
Insulators have . Non-dispersive insulators (considered in Barton (2009)) would be realized in the limit at fixed finite frequency-independent n2 and α≤1. They would become perfectly reflecting in the further limit , hence .
Conductors have . They are necessarily dispersive, 2.8 At zero frequency, , giving perfect reflection and α=1. Two important special cases are non-dissipative plasmas,5 2.9 and Ohmic conductors (Boyer 1974; Furry 1974) 2.10
We write the instantaneous position of the charge as ρ=(σ,ζ>0), and field points as r=(s,z). It will prove convenient to introduce 2.11 Notice that Z is a coordinate difference (namely z−ζ) only if z<0. Define also, for any function f(z), 2.12
The governing equations are 2.13 2.14 2.15
We shall rely heavily on two-dimensional Fourier representations of the Coulomb potential in various guises, e.g. 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 . 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.
For insulators, by equation (2.7), 3.1
The textbook solution features n and α from equation (2.3), and the image position , 3.2 where θ is the Heaviside step function; then, 3.3
4. Uniformly moving charge
(a) Exact solution
The crucial idea is to write ϕ=ϕ(0)+Δϕ, etc., where ϕ(0) is the function defined by equation (3.2) but now parametrically time-dependent via its dependence on 4.1 the x-axis being chosen along the component of u parallel to the surface. The fields B(0) are already known for the special case of Ohmic conductors from Boyer (1974, 1999), and for non-dispersive insulators from Barton (2009), but ψ(0) and B(0) are re-derived more succinctly below. New here is the derivation of Δϕ and thence of ΔB.
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 4.2 4.3and 4.4
We adopt an Ansatz satisfying equation (4.2) and equation (4.31) by construction: 4.5 and 4.6 Then, by equation (4.32), 4.7 reducing equation (4.4) to 4.8 The associated homogeneous equation is solved by 4.9 featuring the (hypothetical) undamped frequency ωS, with subcritical damping when λ<2ωS.
On the right-hand side of equation (4.8), we use equations (3.32), (2.16) and the crucial identity 4.10 to extract an ordinary inhomogeneous differential equation for , 4.11 The associated homogeneous equation is the same as for Δψ, 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) that we require is the particular integral 4.12where 4.13 Notice that ΔW is independent of the surface-parallel wavenumber k2 normal to u∥. From here on we consider only motion that is either perpendicular to the surface (u∥=0), or parallel (u3=0), identified by subscripts ⊥ or ∥, respectively. Then, ΔW reduces to one or the other of 4.14and 4.15
Without dissipation (when λ=0), the denominator of ΔW⊥ 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 ΔW∥ vanishes when , 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 §4b that, for small u and small λ, they never enter the approximation governing the B field. Meanwhile, we note the exact reality condition .
The potential and from equation (4.7) are our central results; they allow one to find or to approximate Σ, and thence the B fields that are our main objective.
In particular, from equation (3.3), 4.16 whence 4.17 Because k2 occurs only in the exponential, ΔΣ, like Σ(0), is even in S2=y−σ2.
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 realized by good Ohmic conductors7 (cf. equation (2.10)), which on dimensional grounds translates into small λ in the sense that 4.18 As far as ΔW is concerned, this corresponds formally to high ωS, whence 4.19
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 4.20 possible (given finite u) only for a limited range of λ. Conversely, (merely) small-λ approximations, i.e. brute-force expansions by powers of λ , would make sense only if 4.21 possible (given finite λ) 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 λ=0) plus a dissipative term proportional to λ. 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 ψ, j and finally b(r) directly in terms of Σ(s). We will subdivide b=b(0)+Δb later.
(a) The Biot–Savart rule
Our non-retarded model has 5.1 This Neumann problem is solved by the image method (e.g. Barton 1989), 5.2 Hence, 5.3 with Σ and ∂Σ/∂t prescribed by equations (4.16) and (4.17). They vary with time because they depend on the time-dependent position of the charge, on ζ directly and on σ via S or ; and their time dependence in turn governs that of ψ and j. This said, we focus on the immediate agenda by abbreviating .
By the Biot–Savart rule, appropriate variants of equation (2.16) for each of the two Coulomb factors, and equation (4.10), 5.4 where ϵ is the signum function. Appeal to 5.5 and straightforward simplification,8 dropping the now redundant primes from , reduce this to 5.6 5.7 5.8and 5.9 with obvious analogues for and , featuring Σ(0) and ΔΣ, 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.
By symmetry, for perpendicular motion, the field lines are circles parallel to the surface and centred on S≡s−σ=0, with parities to match: b1 is even in S1 and odd in S2, and vice versa for b2. For parallel motion, we shall see presently that b is even in S2, but that parity with respect to S1 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 b(0)
This is the component studied in Barton (2009). To find it, one starts by replacing ∂Σ/∂t in equations (5.8) and (5.9) by ∂Σ(0)/∂t, with Σ(0) from equation (3.3) and ∂/∂t from equation (4.10). Then, it becomes obvious that b(0) depends on the material only through a prefactor α. The calculation is straightforward, and reproduces the results for non-dispersive insulators already known from Barton (2009) (with ), 5.10 It proves convenient to scale the lateral coordinates by Z. Writing the new dimensionless variables in sans serif, we define10 5.11 5.12 and have 5.13 5.14 From equation (4.19), one sees that the leading corrections to Ω(0), embodied in ΔΩ, are of relative order (ku/ωS)2∼(u/ZωS)2 and . 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 produce the same B fields as do Ohmic conductors, even though the latter are very dispersive indeed.
(c) The dispersive component Δb
By equations (4.17) and (5.8), whence,11 again in virtue of equation (5.5), 5.15 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 and therefore must vanish (given that is already known).
Since ΔΩ (like Ω(0)) features positions only through S and Z, the field at points r inside the material is a function only of the coordinate differences (S,Z)≡R=r−ρ, 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 b to be visualized very simply; and thence, given that b has even parity, that of the exterior field too.
From here on we settle for the nominally ‘high ωS’ 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 Δb∥ in §6, though straightforward in principle, requires so much more labour than Barton (2009) needed for b(0).
6. Perpendicular motion
The field lines of and Δb⊥ alike are circles parallel to the surface and centred on S=0. Thus, , 6.1 6.2and 6.3 where the approximation implements equation (4.19). For conductors, this reduces to 6.4 6.5 to be compared with the conclusion from Barton (2009) that , where 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 ζ>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 ζ rises or falls. By the same token, Ω and b are well defined and finite everywhere, as long as ζ is strictly positive, only when ζ vanishes (when the image charge and the real charge coincide) could they diverge, if the field point r too approached (σ,0).
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−σ=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 b∥.
(a) The potential ΔΩ∥
According to equation (5.6), and in view of ∂x,y=∂S1,S2, one has Δb1=−∂ΔΩ∥/∂S2 and Δb2=∂ΔΩ∥/∂S1. For comparisons, we cite the correspondingly normed free fields in the half-space z≤0, 7.1
With , our approximation (4.19) yields 7.2
Let φ,χ be the (two-dimensional) polar angles of k and S, and define ϑ≡χ−φ, so that , while and . Changing to and invoking symmetry to drop from the integrand terms odd in ϑ, one obtains 7.3 It proves convenient to subdivide, in an obvious notation, 7.4 where ‘(r)’ identifies the reversibly generated (non-dissipative) component free of λ, and ‘(i)’ the irreversibly generated (dissipative) component proportional to λ and thereby to . (One could equally well take ‘(r)’ and ‘(i)’ to refer to the real and imaginary terms inside the square brackets in equation (7.2).)
Evaluating and then … 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 7.5 7.6and 7.7
At this point, one can see that both and are even in S2, as Ω must be on general grounds, but that (like ) is odd in S1, whereas is even. No mystery is involved, the positive direction along the S1-axis is defined by the direction of motion, and (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.
Recall (say from equation (5.7)) 7.13 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 field lines of b(0) in the (s1,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 s1-axis, at , separated by a field line running along the s2-axis.)
(b) The reversible component Δb(r)
The field lines form four sets of loops, centred on stagnation points on the s1-axis, at s1≃±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(0). 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 β2 with s1, 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.
On the s1-axis (parallel to the trajectory at fixed depth), vanishes; figure 2 plots for 0<s1<2.5 and for 2.5<s1. Notice the change in scale, and the zeros at the stagnation points. Asymptotically, , and .
On the s2-axis (perpendicular to the trajectory at fixed depth), again vanishes, while falls monotonically as rises, with and .
(c) The irreversible component Δb(i)
By equation (4.11) and equations (7.11)–(7.13), 7.15 with β(i) given by the derivatives in equation (7.14) on changing . The field lines form three sets of loops, with stagnation points on the s1-axis, at s1=0 and s1≃±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 s1-axis (parallel to the trajectory at fixed depth), ; figure 4 plots for 0<s1<2 and for 2<s1. The zero locates the right-hand stagnation point. Asymptotically, β(i)(s1≪1)≃−9s1/4 , and .
On the perpendicular (s=0) from charge to surface, Δb(i) vanishes.
On the s2-axis (perpendicular to the trajectory at fixed depth), again vanishes, while rises from 0 at s2=0 to a maximum of 0.27 at 0.68 and eventually falls to zero like .
↵1 In notation from §2, the exact drag force reads . That it falls exponentially as is typical given the frequency gap ωS.
↵2 We call exact expressions true to first order in u/c, but to all orders in dimensionless parameters like u/[distance×(plasma frequency or conductivity)], i.e. without approximations based on the dispersive and dissipative properties of the material.
↵3 We make no attempt to introduce the temperature T explicitly, it would make itself felt mainly through the value of the damping constant λ. 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).
↵4 It may be worth stressing (as in Barton 1997, p. 2466) that λ=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 ε satisfies the Kramers–Kronig relation, by virtue of an imaginary part πω2p.
↵5 Then, bulk plasma waves (‘bulk plasmons’) have frequency ωp 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).
↵6 As 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 1/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).
↵7 The 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≪1) apply even to conductors ‘good’ in the sense that provided . Both conditions are readily satisfied for small enough u/c; it is in this sense that one requires ‘σ 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 ‘R’) wrongly replaced the basic condition by its opposite χ≫1.
↵8 To sidestep one can simplify the volume integral in equation (5.4) through the vector identities (∇f)×v=∇×(fv)−f∇×v, ∇×∇g=0 and , with appropriate f,v,g and w. Then, equation (5.8) is easily recovered by evaluating the surface integral, also using the two-dimensional Fourier representations (2.16).
↵11 The prescription Re will be dropped from expressions that are manifestly real as they stand.
- Received October 16, 2009.
- Accepted February 9, 2010.
- © 2010 The Royal Society