An explicit representation for the axisymmetric solutions of the free Maxwell equations

  • 1 Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble, France
Mayeul Arminjon

Abstract

Garay-Avendaño and Zamboni-Rached defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields – and hence, by summation on frequencies, all propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modeled as an exact solution of the free Maxwell equations.

1 Introduction

Axially symmetric solutions of the Maxwell equations are quite important, at least as an often relevant approximation. For instance, axisymmetric magnetic fields occur naturally as produced by systems possessing an axis of revolution, such as disks or coils [1,2], or astrophysical systems like accretion disks [3] or disk galaxies [4]. Axisymmetric solutions are also used to model EM beams and their propagation (e.g., refs. [5,6]). In particular, nondiffracting beams are usually endowed with axial symmetry, see, e.g., refs. [7,8,9]. Naturally, one often considers time-harmonic solutions, since a general time dependence is obtained by summing such solutions. Two classes of time-harmonic axisymmetric solutions of the free Maxwell equations, mutually associated by EM duality, have been introduced recently [10]. The main aim is to “describe the propagation of nonparaxial scalar and electromagnetic beams in exact and analytic form.” However, as noted by the authors of ref. [10], the analytical expression for a totally propagating time-harmonic axisymmetric solution Ψ of the scalar wave equation, from which they start [equation (1) below], covers all such solutions [9] – thus not merely ones corresponding to nonparaxial scalar beams. The first class of EM fields defined in ref. [10] is obtained by associating with any such scalar solution Ψ a vector potential A by equation (10) below [10]. The second class is deduced from the first one by EM duality [10].

The aim of the present work is to show that, by combining these two classes, one is able to describe all totally propagating time-harmonic axisymmetric EM fields – and thus, by summing on frequencies, all totally propagating axisymmetric EM fields. To this aim, we shall prove the following theorem: Any time-harmonic axisymmetric EM field (whether totally propagating or not) is the sum of two EM fields, say (E1, B1) and (E2,B2), deduced from two time-harmonic axisymmetric solutions of the scalar wave equation, say Ψ1 and Ψ2. The first EM field is derived from the vector potential A1 = Ψ1ez, and the second one is deduced by EM duality from the EM field that is derived from the vector potential A2 = Ψ2ez. As this result is based on determining two vector potentials from merely two scalar fields, it was not necessarily expected.

This article is organized as follows. Section 2 presents and comments on the results of ref. [10], and somewhat extends them in particular by noting that equations (11)–(13) apply to any time-harmonic axisymmetric solution of the scalar wave equation. Also, equations (15)–(20) are new. Section 3 presents the main results of this work – it gives the proof of the theorem. That proof is not immediate but uses standard mathematics with which one is familiar from classical field theory. Section 4 summarizes main results and presents a method to obtain an explicit representation for all totally propagating axisymmetric solutions of the free Maxwell equations.

2 From scalar waves to Maxwell fields

2.1 Axially symmetric scalar waves

We adopt cylindrical coordinates ρ, ϕ and z about the symmetry axis, that is, the z axis. Any totally propagating, time-harmonic, axisymmetric solution of the scalar wave equation (of d’Alembert) can be written [9,10] as a sum of scalar Bessel beams:

ΨωS(t,ρ,z)=eiωtK+KJ0(ρK2k2)eikzS(k)dk,
where ω is the angular frequency, K := ω/c1 and J0 is the first-kind Bessel function of order 0. (Here, c is the velocity of light.) The Bessel beams were first introduced by Durnin [7]. A physical discussion of these beams and their “nondiffracting” property can be found in ref. [11]. On the other hand, a “totally propagating” solution of the wave equation is one that does not have any evanescent mode. In simple words, an evanescent mode can be described as a wave that behaves as a plane wave in some spatial direction, however with an imaginary wavenumber so that its amplitude decreases exponentially. See, e.g., refs. [12,13]. In the present case, the totally propagating character of the wave (1) means precisely that the axial wavenumber k = kz verifies −KkK, [10], so that the radial wavenumber kρ=K2k2 is real, as is also kz.

Thus, the (axial) “wave vector spectrum” S is a (generally complex) function of the real variable k = kz (−KkK), that is, the projection of the wave vector on the z-axis. This function S determines the spatial dependency of the time-harmonic solution (1) in the two-dimensional space left by the axial symmetry, i.e., the half-plane (ρ ≥ 0, z ∈] −∞, +∞[). Thus, any totally propagating, time-harmonic, axisymmetric scalar wave Ψ can be put in the explicit form (1), in which no restriction has to be put on the “wave vector spectrum” S (except for a minimal regularity ensuring that the function Ψ is at least twice continuously differentiable: the integrability of S, S ∈ L1([−K, + K]), would be enough). The general totally propagating axisymmetric solution of the scalar wave equation can be obtained from (1) by an appropriate summation over a frequency spectrum: an integral (inverse Fourier transform) in the general case or a discrete sum if a discrete frequency spectrum (ωj)(j = 1,…,Nω) is considered for simplicity:

Ψ(t,ρ,z)=j=1NωΨωjSj(t,ρ,z),
where, for j = 1,…,Nω, ΨωjSj is the time-harmonic solution (1), corresponding with frequency ωj and wave vector spectrum Sj. The different weights wj, which may be applied at the different frequencies, can be included in the functions Sj, replacing Sj by wjSj.

2.2 Reminder on time-harmonic free Maxwell fields

In this section, we recall equations more briefly recalled in ref. [10]. The electric and magnetic fields in SI units are given in terms of the scalar and vector potentials V and A by

E=VAt,
B=rotA.
These equations imply that E and B obey the first group of Maxwell equations. If one imposes the Lorenz gauge condition
1c2Vt+divA=0,
then the validity of the second group of the Maxwell equations in free space for E and B is equivalent to asking that V and A verify d’Alembert’s wave equation [14,15]. Moreover, if one assumes a harmonic time-dependence for V and A:
V(t,x)=eiωtVˆ(x),A(t,x)=eiωtAˆ(x),
then the wave equation for A becomes the Helmholtz equation: 2
ΔA+ω2c2A=0,
and the Lorenz gauge condition (5) can be rewritten as follows:
V=ic2ωdivA.
If A is time-harmonic [equation (6)2] and obeys (7), then V given by (8) is time-harmonic and satisfies the wave equation. Then, the electric field (3) can be rewritten as follows:
E=iωA+ic2ω(divA).
Thus, the data of a time-harmonic vector potential A obeying the wave equation, or equivalently obeying equation (7), determine a unique solution of the free Maxwell equations, by equations (4) and (9), and that solution is time-harmonic with the same frequency ω as for A.

2.3 Time-harmonic axisymmetric fields: from a scalar wave to a Maxwell field

To any solution Ψ(t,ρ,z)=eiωtΨˆ(ρ,z) of the scalar wave equation having the form (1), the authors of ref. [10] associate a vector potential A as follows:

A:=Ψez,orAz:=Ψ,Aρ=Aϕ=0.
(We shall denote the standard point-dependent orthonormal basis associated with the cylindrical coordinates by (eρ, eϕ, ez).) Equation (1) is valid, as we mentioned earlier, for totally propagating, axisymmetric, time-harmonic solutions of the scalar wave equation. Thus, in the way recalled in the previous section, a unique solution of the free Maxwell equations is defined, which is time-harmonic. The equations for the different components of this solution (E, B) are as follows:
Bϕ=Azρ,Eϕ=0,
Eρ=ic2ω2Azρz,Bρ=0,
Ez=ic2ω2Azz2+iωAz,Bz=0.
These equations follow easily from equations (4), (9) and (10), and from the axisymmetry of Az = Ψ(t, ρ, z), by using the standard formulas for the curl and divergence in cylindrical coordinates. Equations (11)–(13) provide an axisymmetric EM field whose electric field is radially polarized (E = Eρeρ + Ezez).

An “azimuthally polarized” solution (E′, B′) (in the sense that E=Eϕeϕ) of the free Maxwell equations can alternatively be deduced from the data Ψ by transforming the solution (11)–(13) through the EM duality, that is:

E=cB,B=E/c.

Now we observe this: the fact that the function Ψ have the form (1) plays no role in the derivation of the exact solution (11)–(13) to the free Maxwell equations. The only relevant fact is that Ψ = Ψ(t, ρ, z) is a time-harmonic axisymmetric solution of the scalar wave equation. Thus, with any axisymmetric time-harmonic solution Ψ of the scalar wave equation, we can associate two axisymmetric time-harmonic solutions of the free Maxwell equations: the solution (11)–(13) and the one deduced from it by the duality (14). These two solutions will be called here the “GAZR1 solution” and the “GAZR2 solution,” respectively, because both were derived in ref. [10] – although this was then for a totally propagating solution having the form (1), and we have just noted that this is not necessary.

It is implicit that, in equations (11)–(13), Bϕ, Eρ and Ez are the real parts of the respective right hand side [as are also E and B in equations (3), (4) and (9)]. Thus, if Az is totally propagating and hence may be written in the form (1), then by dJ0/dx = −J1(x), we obtain the following equations:

BϕωS=e[eiωtK+KK2k2J1(ρK2k2)S(k)eikzdk],
EρωS=e[ic2ωeiωtK+KK2k2J1(ρK2k2)ikS(k)eikzdk],
EzωS=e[ieiωtK+KJ0(ρK2k2)(ωc2ωk2)S(k)eikzdk],
where K := ω/c. In the case with a (discrete) frequency spectrum, one just has to sum each component: (15), (16) or (17), over the different frequencies ωj, with the corresponding values Kj = ωj/c and spectra Sj = Sj(k)(−Kjk ≤ + Kj) – as with a scalar wave (2):
Bϕ=j=1NωBϕωjSj,
Eρ=j=1NωEρωjSj,
Ez=j=1NωEzωjSj.

3 From Maxwell fields to scalar waves

Now an important question arises: Do the GAZR solutions generate all axisymmetric time-harmonic solutions of the Maxwell equations (in which case, by summation on frequencies, they would generate all axisymmetric solutions of the Maxwell equations)? Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. Can one find a GAZR1 solution and a GAZR2 solution, whose sum gives just that starting solution?

Note from equations (11)–(14) that the GAZR1 solution and the GAZR2 solution are complementary: in cylindrical coordinates, the GAZR1 solution provides non-zero components Bϕ, Eρ and Ez, the other components Eϕ, Bρ and Bz being zero – and the exact opposite is true for the GAZR2 solution. In view of this complementarity, we can consider separately the two sets of components: Bϕ, Eρ and Ez on one side and Eϕ, Bρ and Bz on the other side.

3.1 Sufficient conditions for the existence of the decomposition

For the “GAZR1” solution, which gives nonzero values to the first among the two sets of components just mentioned, we have the following result:

Proposition 1

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic axisymmetric solution (A1z, B1ϕ, E1ρ, E1z; E1ϕ = B1ρ = B1z = 0), of the form (11)–(13), and having the same frequency ω as the starting solution (A, E, B), be such that B1ϕ = Bϕ, E1ρ = , E1z = Ez, it is sufficient that we have just

B1ϕ=Bϕ.

Proof

Let A1z(t, ρ, z) be a time-harmonic axisymmetric solution of the wave equation, with frequency ω, and assume that B1ϕ as defined by equation (11) [with A1z in the place of Az] is equal to Bϕ, where B is defined by equation (4). That is, assume that

A1zρ=AρzAzρ.
Denoting by A1 := A1zez the vector potential that provides the GAZR1 solution (B1ϕ, E1ρ, E1z; E1ϕ = B1ρ = B1z = 0), let us compute EρE1ρ and EzE1z. We obtain the following equation by equation (9):
ωic2(EE1)=(divA)+ω2c2A,
where
A:=AA1:=AA1zez.
In order that the vector potential A of the a priori given solution (A, E, B) be axisymmetric, its components Aρ, Aϕ and Az must depend only on t, ρ and z, i.e., be independent of ϕ. Therefore:
divA=1ρ(ρAρ)ρ+Azz.
By using this and (24), we obtain
divA=1ρ(ρAρ)ρ+Azz.
Hence, in (23), we have
(divA)=(Aρρ+Aρρ+Azz)=(2Aρρ2+1ρAρρ1ρ2Aρ+2Azρz)eρ+(2Aρzρ+1ρAρz+2Azz2)ez.
The radial component of the vector (23) is thus:
ωic2(EE1)ρ=2Aρρ2+1ρAρρAρρ2+2Azρz2A1zρz+ω2c2Aρ.
However, the vector potential A obeys the Helmholtz equation (7), that is, for the radial component (using the fact that Aρϕ=Aϕϕ0):
(ΔA)ρ+ω2c2AρΔAρAρρ22ρ2Aϕϕ+ω2c2Aρ2Aρρ2+1ρAρρ+2Aρz2Aρρ2+ω2c2Aρ=0.
Inserting (29) into (28) gives:
ωic2(EE1)ρ=2Aρz2+2Azρz2A1zρz=z(AρzAzρ+A1zρ).
Therefore, if equation (22) is satisfied, then we have E1ρ = Eρ.

Similarly, from (27), the axial component of the vector (23) is expressed as follows:

ωic2(EE1)z=2Aρzρ+1ρAρz+2Azz22A1zz2+ω2c2(AzA1z).
The axial component of the Helmholtz equation (7) is expressed as follows:
(ΔA)z+ω2c2AzΔAz+ω2c2Az2Azρ2+1ρAzρ+2Azz2+ω2c2Az=0.
If equation (22) is satisfied, we have
2Azρ2=2Aρρz+2A1zρ2.
In equation (32), we replace 2Azρ2 by its value given on the right hand side in equation (33), and we replace Azρ by its value given in equation (22). Thus, we obtain
2Aρρz+2A1zρ2+1ρAρz+1ρA1zρ+2Azz2+ω2c2Az=0.
By substituting this equation in equation (31), we rewrite the latter as follows:
ωic2(EE1)z=2A1zρ21ρA1zρ2A1zz2ω2c2A1z.
We recognize the right-hand side as that of equation (32), although with the minus sign, and with A1z in the place of Az. That is, equation (35) is just
ωic2(EE1)z=ΔA1zω2c2A1z.
But this is zero, since A1z is by assumption a time-harmonic solution of the wave equation, with frequency ω. Therefore, if equation (22) is satisfied, then we have E1z = Ez too. This completes the proof of Proposition 1.□

It follows for the dual (“GAZR2”) solution:

Corollary 1

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic solution (A2z, E2ϕ, B2ρ, B2z; B2ϕ=E2ρ=E2z=0) with the same frequency, deduced from equations (11)–(13) by the duality (14), be such thatE2ϕ=Eϕ,B2ρ=Bρ,B2z=Bz, it is sufficient that we have just

E2ϕ=Eϕ.

Proof

The GAZR2 solution (A2z,E2ϕ,B2ρ,B2z;B2ϕ=E2ρ=E2z=0) is deduced from the GAZR1 solution (A2z,B2ϕ,E2ρ,E2z;E2ϕ=B2ρ=B2z=0), associated with the same potential A2z, by the duality relation (14). Suppose that equation (37) is satisfied. With the starting solution (A, E, B) of the free Maxwell equations, we may associate another solution by the inverse duality:

B˜=1cE,E˜=cB.
The assumed relation (37) means that
B2ϕ=B˜ϕ.
Indeed, by applying successively (14)1, (37) and (38)1, we obtain:
B2ϕ=1cE2ϕ=1cEϕ=B˜ϕ.
In turn, the relation (39) means that we may apply Proposition 1 to the GAZR1 solution (A2z, B2ϕ,…) and the solution (A˜,E˜,B˜). 3 Thus, Proposition 1 reveals that, since B2ϕ=B˜ϕ, we have also
E2ρ=E˜ρandE2z=E˜z,
i.e., in view of (14)2 and (38)2:
cB2ρ=cBρandcB2z=cBz.
This proves Corollary 1.□

3.2 Generality of the decomposition

Proposition 2

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A2zof the wave equation, with the same frequency, such that the associated GAZR2 solution(E2,B2), deduced from A2zby equations (11)–(13) followed by the duality transformation (14), satisfy

E2ϕ=Eϕ,B2ρ=Bρ,B2z=Bz.

Proof

In view of Corollary 1, we merely have to prove that there exists a time-harmonic axisymmetric solution A2z of the wave equation, such that equation (37) is satisfied. From equations (9) and (25), we have

Eϕ=iωAϕ.
By using this with equations (11) and (14)1, we may rewrite the sought-for relation (37) as follows:
A2zρ=iωcAϕ.
This equation can be solved by a quadrature:
A2z(t,ρ,z)=h(t,z)+ρ0ρiωcAϕ(t,ρ,z)dρ.
We thus have to find out if it is possible to determine the function h, so that A2z given by (46) obeys the wave equation. Moreover, the unknown function A2z must have a harmonic time dependence with frequency ω as has Aϕ, i.e.,
A2z(t,ρ,z)=ψ(ρ,z)eiωt,Aϕ(t,ρ,z)=Aˆϕ(ρ,z)eiωt,
so we must have h(t, z) = eiωtg(z), too. Hence, we may rewrite (46) as follows:
ψ(ρ,z)=g(z)+ρ0ρiωcAˆϕ(ρ,z)dρ,
and now the question is to know if g can be determined so that ψ obeys the scalar Helmholtz equation, i.e., [cf. equation (32)]:
ψΔψ+ω2c2ψ2ψρ2+1ρψρ+2ψz2+ω2c2ψ=0
knowing that Aϕ or Aˆϕ does obey the ϕ component of the vector Helmholtz equation (7), i.e.,
(ΔAˆ)ϕ+ω2c2AˆϕΔAˆϕAˆϕρ2+2ρ2Aˆρϕ+ω2c2Aˆϕ2Aˆϕρ2+1ρAˆϕρ+2Aˆϕz2Aˆϕρ2+ω2c2Aˆϕ=0.
We have from equations (45) and (47):
ψρ=iωcAˆϕ,
hence,
2ψρ2=iωcAˆϕρ.
And we obtain from (48):
ψz=dgdzρ0ρiωcAˆϕz(ρ,z)dρ,
whence,
2ψz2=d2gdz2ρ0ρiωc2Aˆϕz2(ρ,z)dρ.
Entering equations (51), (52) and (54) into (49)1, we obtain:
ψ=iωcAˆϕρiωcAˆϕρ+d2gdz2ρ0ρiωc2Aˆϕz2(ρ,z)dρ+ω2c2(giωcρ0ρAˆϕ(ρ,z)dρ).
Therefore, the scalar Helmholtz equation (49)2 can be rewritten as follows:
d2gdz2+ω2c2g=iωc[Aˆϕρ+Aˆϕρ+ρ0ρ(2Aˆϕz2+ω2c2Aˆϕ)dρ],
or, using (50):
d2gdz2+ω2c2g=iωc[Aˆϕρ+Aˆϕρ+ρ0ρ(2Aˆϕρ21ρAˆϕρ+Aˆϕρ2)dρ].
An integration by parts gives us:
ρ0ρ(2Aˆϕρ2+Aˆϕρ2)dρ=[Aˆϕρ+Aˆϕρ]ρ0ρ+ρ0ρ1ρAˆϕρdρ,
so equation (57) can be rewritten as follows:
d2gdz2+ω2c2g=iωc[Aˆϕρ(ρ0,z)+Aˆϕ(ρ0,z)ρ0].
As is well known and easy to check, this very ordinary differential equation can be solved explicitly by the method of variation of constants. (The general solution g of (59) depends linearly on two arbitrary constants.) And by construction, any among the solutions g of (59) is such that, with that g, the function ψ in equation (48) obeys the scalar Helmholtz equation (49). This proves Proposition 2.□

Corollary 2

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A1zof the wave equation, with the same frequency, such that the associated GAZR1 solution (E1, B1), deduced from A1zby equations (11)–(13), satisfy

B1ϕ=Bϕ,E1ρ=Eρ,E1z=Ez.

Proof

Let A1z be a time-harmonic axisymmetric solution of the wave equation, and consider:

  1. (i)the GAZR1 solution defined from A1z by equations (11)–(13) (thus with A1z, B1ϕ, E1ρ, E1z,… instead of Az, Bϕ, Eρ, Ez,…, respectively).
  2. (ii)the GAZR2 solution defined from the same A1z by applying the duality (14) to the said GAZR1 solution:
E1ϕ:=cB1ϕ,B1ρ:=1cE1ρ,B1z:=1cE1z,
E1ρ:=cB1ρ=0,E1z:=cB1z=0,B1ϕ:=1cE1ϕ=0.
On the other hand, consider the free Maxwell field (E′, B′) deduced from the given time-harmonic axisymmetric solution (E, B) of the free Maxwell equations by the same duality relation:
E:=cB,B:=E/c.
Just in the same way as it was shown in Note 1, we know that a vector potential A′ such that B′ = rot A′ does exist and can be chosen to be axisymmetric (and is indeed chosen so) – as are E and B, and hence E′ and B′. The sought-for relation (60) is equivalent to
E1ϕ=Eϕ,B1ρ=Bρ,B1z=Bz.
Therefore, the existence of A1z as in the statement of Corollary 2 is ensured by Proposition 2.□

Accounting for Proposition 2 and Corollary 2, and remembering the “complementarity” of the GAZR1 and GAZR2 solutions, we thus can answer positively to the question asked at the beginning of this section.

Theorem

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exist a unique GAZR1 solution (E1, B1) and a unique GAZR2 solution(E2,B2), both with the same frequency as has (A, E, B), and whose sum gives just that solution:

E=E1+E2,B=B1+B2.

Remark

Thus, the uniqueness of the representation concerns the electric and magnetic fields. It does not concern the potentials A1 = A1zez and A2 = A2zez that generate (E1, B1) and (E2,B2), respectively.

4 Discussion and conclusion

The authors of ref. [10] introduced two classes of axisymmetric solutions of the free Maxwell equations, and they showed that these two classes allow one to obtain nonparaxial EM beams in explicit form. It has been proved that, by combining these two classes, one can define a method that allows one to get all totally propagating, time-harmonic, axisymmetric free Maxwell fields – and thus, by the appropriate summation on frequencies, all totally propagating axisymmetric free Maxwell fields. This method results immediately from the aforementioned theorem and from the general form (1) of a totally propagating, time-harmonic, axisymmetric solution of the scalar wave equation. However, that theorem is not an obviously expected result, and its proof is not immediate. We thus have now a constructive method to obtain all totally propagating axisymmetric free Maxwell fields. Namely, considering a discrete frequency spectrum (ωj)j=1,...,Nω for simplicity: there are 2Nω functions, kSj(k) and kSj(k)(j=1,...,Nω), such that the components Bϕ, Eρ, Ez of the field are given by equations (18), (19) and (20), respectively – while the components Eϕ, Bρ and Bz are given by these same equations applied with the primed spectra Sj, followed by the duality transformation (14).

In the forthcoming work, we shall apply this to model the interstellar radiation field in a disc galaxy as an (axisymmetric) exact solution of the free Maxwell equations. In this application, it is very important that, due to the present work, one knows that any (totally propagating) axisymmetric free Maxwell field can be obtained in this way.

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Footnotes

1

Beware that instead K := 2ω/c in ref. [10]. Our notation seems as natural and gives more condensed formulas.

2

Of course, just the same equation (7) applies to the “amplitude field” Aˆ.

3

The explicit expression of the corresponding vector potential A˜ as function of (A, E, B) is not needed: only the existence of an axisymmetric A˜, such that B˜=rotA˜, is needed. Precisely, A˜ is got as a solution of the PDE rotA˜=B˜. (Such a solution always exists in a topologically trivial domain, thus in particular if the domain is the whole space.) Hence, A˜ can indeed be chosen axisymmetric, i.e., with its components in cylindrical coordinates being independent of ϕ, because with this choice, the independent variables of the PDE rotA˜=B˜ simply do not include ϕ – since B˜=1cE is axisymmetric, similar to E by assumption.

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  • [1]

    Garrett MW. Axially symmetric systems for generating and measuring magnetic fields. Part I. J Appl Phys. 1951;22:1091–107.

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