In fact the equation satisfied by φ(kx,ky,kz,r) is ∇2φ = -4πρ (kx,ky,kz,r), where the charge density ρ is smeared in the whole space. See Ref. 5.
Note that each component φ(kx,ky,kz,r) of the Landau and Lifshitz expansion satisfies ∇2φ(kx,ky,kz,r)= -4πρ (kx,ky,kz,r) = -(q/2 π2)ei(kxx+kyy+kzz).Since this charge density ρ(kx,ky,kz,r) is smeared in the whole space, each term φ(kx,ky,kz,r) doesn't satisfy ∇2φ = 0 at any point.
The considerations following Eq. (16) give the physical reason by which the Toraldo di Francia expansion "is valid in any half-space with no points in common with the path of the particle" (cf. Ref. 3). Let us see the charge q at rest in S from a system S′, which is moving with respect to S with a velocity v = (Ʋ,0,0). The initial static δ surface charge density ρ(kx,ky,r) transforms into a δ surface charge density[Equation]plus a δ surface current density[Equation]Each term ε′(k′x,k′y,r′,t′) of the development in evanescent waves of the field of the moving particle proposed in S′ will not be the solution of □′ε′ = 0 but of □′ε′ =4π∇′ρ′+(4π/c2)(σJ′/σt′)iverges at the plane z′ = 0.
L. Landau and E. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, New York, 1979), pp. 124–125.
This development is not valid in an arbitrary plane that contains the charge. See the considerations that follow Eq. (16).
This result is rather surprising, since it shows that the static evanescent waves are "transmitted" without refraction. It can be readily shown that this stems from Snell's law for evanescent waves in the case in which the real part of k is in the plane of incidence. In this case we obtain[Equation]For K → 0 we obtain sinθi = n sinθt, which is the usual Snell's law of refraction for plane waves. For ω → 0 we obtain sinθi = sinθt which shows that the static evanescent waves are "transmitted" without refraction.
J. Stratton, Electromagnetic Theory (McGraw-Hill, New York and London, 1941), p. 573.
See Ref. 9, p. 577.
p. C. Clemmow, The Plane Wave Representation of Electromagnetic Fields (Pergamon, New York, 1966), p. 36.
In order to put in evidence the (partial) angular character of this representation it must be taken into account that each wave in Eq. (12′) has Rek in the x-y plane and is attenuated in the z direction. Thus by making the replacement β = tan-1ky/kx = tan-1q/p, K [Equation], one readily obtains[Equation]
In fact these authors point out that their Eqs. (1.36) and (1.37) "are true mode expansions of E and H in each of the two half spaces z ≥ 0 on either side of the plane in which the particle moves," but they don't give any reason why these two expansions don't match at z = 0 [see the discussion following Eqs. (1.36) and (1.37) in Ref. 12].
For different types of spherical charge distributions, the numerical factor in Eq. (35) varies, but always remains in the order of unity. Hence, the classical electron radius is usually defined as r0 = q2/m0c2. [See, for example, W. T. Grandy, Introduction to Electrodynamics and Radiation (Academic, New York, 1970), p. 96]. Notwithstanding this fact, we retain for r0 the value given by Eq. (35).
See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 107–108.
See, for example, Ref. 16, p. 118, Eq. (3.150). It is worth noting here that Jackson's book gives the integral representation given by Eq. (45) only as a "mathematical result" and not as an expansion of the Coulomb field in cylindrical evanescent waves of zero frequency. In our opinion this author overlooks the fact that the K0(|k|ρ) is also the solution of the inhomogeneous equation, as we have shown in Appendix C [cf. Ref. 16, Eqs. (3.98)–(3.101) and the discussion following Eq. (3.141)]. We think that from a physical point of view the expansion of the Coulomb field in cylindrical evanescent waves given by Eq. (45) is as important as the expansion in evanescent plane waves given by Eq. (12).
It is easily seen that the fields given by Eq. (50) satisfy Maxwell equations in vacuum for all ρ ≠ 0. This TM circularly cylindrical evanescent wave can also be derived from the "circularly cylindrical wave function" given by J. A. Stratton (see Ref. 9, p. 360) by making the argument of the Hankel function H0(1) in Eq. (29) to be imaginary.
See Ref. 16, pp. 110–113.
It is worth noting here that already in the case of a point charge at rest in front of a moving medium, the boundary conditions cannot be simulated by image charges. This problem can nowever be managed within our formalism, by applying to each evanescent wave of the two-dimensional expansion the boundary conditions for moving media.
W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1968), p. 205.
We obtain, for example, that the charge density ρ(t) which in an infinite medium of dielectric constant ε would give an electric field equal to E(t) is given by[Equation]where T(|k|d) is given by Eq. (66)!
See, for example, M. Abramowitz and I. Segun, Handbook of Mathematical Functions (Dover, New York, 1968), p. 360, Eq. (9.1.18).
See, for example, W. Gröbner and N. Hofreiter, Integraltafel (Springer, Wien, 1950), Vol. II, p. 59, Eq. (18a). Here the validity of the expression (B1) is restricted to z > 0. But taking into account that∫∞0J0(x)dx=1[same Ref., p. 196, Eq. (la)] expression (B1) is valid also for z = 0.
Here we call x(k) what Smythe calls Φ(k).
See Ref. 23, p. 375, Eq. (9.6.15) and p. 376, Eq. (9.6.27).