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Two Systems of Maxwell’s Equations and Two Corresponding Systems of Wave Equations in a Rotating Dielectric Medium

By Evgeny A. Bondarenko

Submitted: October 2nd 2018Reviewed: November 19th 2018Published: March 4th 2019

DOI: 10.5772/intechopen.82595

Downloaded: 235

Abstract

In this chapter, on the base of two basic systems of Maxwell’s equations for electromagnetic field vectors E→ and B→ in a uniformly rotating dielectric medium, the two corresponding systems of wave equations have been derived (to the first order in an angular velocity Ω). From their comparative analysis, it can be seen that the structure of the wave equations for electromagnetic field vectors in the first system is asymmetrical with respect to Ω, while the structure of such equations in the second one is symmetrical. On this basis, it can be concluded that if the principle of symmetry is accepted as a criterion for selection, then second system of wave equations (and, therefore, corresponding the second set of Maxwell’s equations) for vectors E→ and B→ may be preferred.

Keywords

  • Maxwell’s equations
  • wave equations
  • Sagnac effect
  • fiber-optic gyro
  • ring laser gyro

1. Introduction

In order to develop the theory of fiber-optic gyro (or, e.g., ring laser gyro with resonator containing a dielectric medium with index of refraction n), one needs to have a system of Maxwell’s equations and corresponding system of wave equations for electromagnetic field vectors Eand Bwhich are written in a frame of reference which uniformly rotates in an inertial frame with angular velocity Ω(Ω=|Ω|). Since the module vof vector v=Ω×rof a tangential velocity of such rotating device is much smaller than the speed of light, it is sufficient for these systems to be linear in vor, equivalently, in Ω.

From the literature, it can be seen that there are mainly two basic systems of Maxwell’s equations for electromagnetic field vectors Eand Bderived from the first principles and written for the case of a uniformly rotating dielectric medium. These two systems of equations are in good agreement with the experiments conducted for ring optical interferometers, fiber-optic gyros, and ring laser gyros containing in their arms the gas discharge tubes with Brewster’s windows. Both systems are based on the Galilean description of rotation: x=x0cosΩt0+y0sinΩt0, y=x0sinΩt0+y0cosΩt0, z=z0, t=t0(superscript 0refers to the inertial frame).

In the absence of free charges and currents, the first system, which was first obtained in work [1] from the formalism of the theory of general relativity, may be written in the form (we keep the terms only up to first order in Ω):

×E+Bt=0,E1
B=0,E2
×B1c2v×E1c2tn2Ev×B=0,E3
E1n2v×B=0,E4

and the second one, which was first obtained in work [2] on the base of the use of the tetrad method in this theory, may be presented as

×E+tB+1c2v×E=0,E5
B+1c2v×E=0,E6
×B1c2tn2Ev×B=0,E7
E1n2v×B=0.E8

Systems of Maxwell’s Eqs. (14) and (58) are written here in units of the SI. They contain only two electromagnetic field vectors Eand B. In the named original works [1] (see expressions (5), (14b) therein) and [2] (see relations (6.11), (6.12) therein), these systems contain, besides vectors Eand B, two more electromagnetic field vectors Dand H(which must be excluded). Systems (1–4) and (5–8) can be also found, for example, in subsequent works [3, 4, 5, 6, 7, 8] and [9], respectively. In the case of vacuum (when n=1), sets (1–4) and (5–8) are transformed into the ones known correspondingly from works [10] and [11].

In the above systems, all the quantities are specified by the formulas

=x̂/x+ŷ/y+ẑ/z,E=Exx̂+Eyŷ+Ezẑ,B=Bxx̂+Byŷ+Bzẑ,Ω=Ωxx̂+Ωyŷ+Ωzẑ,r=xx̂+yŷ+zẑ,v=Ω×r=vxx̂+vyŷ+vzẑ,vx=ΩyzΩzy,vy=ΩzxΩxz,vz=ΩxyΩyxE9

where x̂, ŷ, and ẑare the unit vectors which form an orthogonal coordinate basis x̂ŷẑof a rotating frame; Ex, Ey, and Ezand Bx, By, and Bzare the components of vectors Eand Bin this basis; Ωis the vector of angular velocity with which the basis x̂ŷẑrotates in the inertial frame; Ωx, Ωy, and Ωzare the components of vector Ω; ris the radius vector of the given observation point in basis x̂ŷẑ; x, y, and zare the components of vector r; vis the vector of linear tangential velocity of the observation point calculated in the inertial frame; vx,vy, and vzare the components of vector v; n=εrμr1/2is the index of refraction of a dielectric medium; and εrand μrare the relative permittivity and permeability of a medium, respectively.

Remark about systems (1–4) and (5–8)

1. According to the textbook [12], in a uniformly rotating frame of reference with spatial rectangular coordinates x, y, z and time coordinate t, the quadratic form ds2may be presented as ds2=gikdxidxkwhere x0=ct, x1=x,x2=y,and x3=z. In general, when Ω=Ωxx̂+Ωyŷ+Ωzẑ, the nonzero components of a space–time metric tensor (with determinant g=1) of such rotating frame may be written as g00=1v2/c2, g11=g22=g33=1, g01=g10=vx/c, g02=g20=vy/c, and g03=g30=vz/c. Then, components καβ=gαβ+g0αg0β/g00of a spatial metric tensor [with determinant κ=1v2/c21] of such frame will be κ11=1+κvx2/c2, κ22=1+κvy2/c2, κ33=1+κvz2/c2, κ12=κ21=κvxvy/c2, κ13=κ31=κvxvz/c2, and κ23=κ32=κvyvz/c2. From these formulas for καβand κ, it follows that the spatial metric tensor of a rotating frame of reference, in a linear with respect to Ωapproximation, has a diagonal form with nonzero elements κ11=κ22=κ33=1and its determinant κ=1. Therefore, in such approximation, geometry of space in a rotating frame of reference remains Euclidean (flat space). So, spatial rectangular coordinates x, y, and z(or, e.g., cylindrical coordinates ρ, ϕ, z) of the given observation point in this frame have their usual sense, and the operator in systems (1–4) and (5–8) may be used in usual way. .

As one can see, the above two systems of Maxwell’s Eqs. (14) and (58) for electromagnetic field vectors Eand Bare not identical: system (1–4) has asymmetrical structure with respect to Ωin a sense that rotation manifests itself only in the third and fourth equations but not in the first and second ones; system (5–8) has symmetrical structure with respect to Ωbecause rotation manifests itself in all four equations. The reason of such difference between these two systems is that they were obtained in works [1, 2] with the help of two qualitatively different theoretical approaches.

In this situation, one may ask the following questions: (1) what will the form of the two corresponding systems of wave equations for the named vectors in first and second cases be? (2) Which system of such wave equations (first or second) is preferred? The answers to these questions are not given in the literature.

So, the task of this research is to derive the wave equations for vectors Eand B—at first on the base of system of Maxwell’s Eqs. (14) and then on the base of system (5–8). The results obtained in both cases must be compared. All calculations must be performed with accuracy approximated to the first order in v(v=v) or, equivalently, Ω(Ω=Ω).

2. Auxiliary relations

In this section, we are going to present some useful formulas for the quantities ××G, ×v×G, v×G, vG, and 2v×Gwhich involve vectors v=Ω×rand G(G=E, B).

A. Consider the term ××G. It is known (see, e.g., handbook [13]) that

××G=2G+G.E10

B. Consider the identity ×v×G=GvvG+vGGv. In the case v=Ω×r, we have Gv=Ω×G, v=0, and, to the first order in Ω, vG=0. Therefore

×v×G=vG+Ω×G.E11

C. Consider the identity v×G=G×vv×G. Since ×v=2Ω,

v×G=2ΩGv×G.E12

D. Consider the identity vG=vG+Gv+v××G+G××v. Since Gv=Ω×G, ×v=2Ω, and G××v=2Ω×G,

vG=vGΩ×G+v××G.E13

E. Consider the vector 2v×G. There is the formula

2v×G=2v×Gxx̂+2v×Gyŷ+2v×Gzẑ.E14

Let us first calculate the projection 2v×Gxof this vector onto the axis x̂. Taking into account (9), we have

2v×Gx=2x2+2y2+2z2vyGzvzGy,E15

or

2v×Gx=2x2+2y2+2z2ΩzxΩxzGzΩxyΩyxGy.E16

After the calculation, we get

2(v×G)x=vy(2Gzx2+2Gzy2+2Gzz2)vz(2Gyx2+2Gyy2+2Gyz2)2Ωx(Gyy+Gzz)+2(ΩyGyx+ΩzGzx).E17

Let us add to the right-hand side of (17) the following terms: 2ΩxGx/xand +2ΩxGx/x. Then

2(v×G)x=vy(2Gzx2+2Gzy2+2Gzz2)vz(2Gyx2+2Gyy2+2Gyz2)2Ωx(Gxx+Gyy+Gzz)+2(ΩxGxx+ΩyGyx+ΩzGzx),E18

or

2v×Gx=v×2Gx2ΩxG+2xΩG.E19

Similarly, we may obtain

2v×Gy=v×2Gy2ΩyG+2yΩG,E20
2v×Gz=v×2Gz2ΩzG+2zΩG.E21

Therefore

2v×G=v×2G2ΩG+2ΩG.E22

Finally, taking into account that, to the first order in Ω, ΩG=0, we get

2v×G=v×2G+2ΩG.E23

Formulas (10)(13) and (23) will be used in the next sections.

3. First system of wave equations for vectors Eand Bin a rotating medium

In this section, we are going to derive the first system of wave equations for electromagnetic field vectors Eand Bwhich will correspond to the original system of Maxwell’s Eqs. (14).

3.1 Equation for vector E

To derive the wave equation for vector E, we apply the operator ×to expression (1):

××E+t×B=0.E24

Taking into account (10) and (3), we rewrite (24) in the form

2En2c22Et2+1c2tv×Bt1c2t×v×EE=0,E25

or, using (1),

2En2c22Et21c2t×v×E+v××EE=0.E26

With the help of (11), we get

2En2c22Et2+1c2tvEΩ×Ev××EE=0.E27

Consider the term Ein (27). According to (4), E=1n2v×B. Taking into account (12) and (3), we have v×B=2ΩBn2/c2/tvE, so

E=2n2ΩB1c2tvE.E28

Consider the last term in (28). In accordance with (13),

vE=vEΩ×E+v××E,E29

therefore

E=2n2ΩB1c2tvEΩ×E+v××E.E30

Finally, substituting (30) into (27), we obtain the desired wave equation for vector E:

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0.E31

3.2 Equation for vector B

In order to derive the wave equation for vector B, we apply the operator ×to expression (3):

××B1c2××v×En2c2t×E+1c2t×v×B=0.E32

Taking into account (10), (1), and (2), we rewrite (32) as

2Bn2c22Bt21c2t×v×B+1c2××v×E=0.E33

Since ××v×E=2v×E+v×E,

2Bn2c22Bt21c2t×v×B1c22v×E+1c2v×E=0.E34

Using (11), we have

2Bn2c22Bt2+1c2tvBΩ×B1c22v×E+1c2v×E=0.E35

Consider the term v×Ein (35). In accordance with (12), v×E=2ΩEv×E, or, with (1), v×E=2ΩE+/tvB. Then

v×E=2ΩE+tvB,E36

or, taking into account (13),

v×E=2ΩE+tvBΩ×B+tv××B.E37

With the help of (3), we find

v×E=2ΩE+tvBΩ×B+v×n2c22Et2.E38

Substituting (38) into (35), we obtain

2Bn2c22Bt2+2c2t[(v)BΩ×B]+2c2(ΩE)1c2[2(v×E)v×n2c22Et2]=0.E39

Consider the last term in (39). In accordance with (23), 2v×E=v×2E+2ΩE; hence

2v×Ev×n2c22Et2=v×2En2c22Et2+2ΩE.E40

It is clear that, to the first order in Ω,

v×2En2c22Et2=0.E41

So, expression (40) may be rewritten as

2v×Ev×n2c22Et2=2ΩE.E42

After inserting (42) into (39), we obtain the desired wave equation for vector B:

2Bn2c22Bt2+2c2tvBΩ×B=0.E43

3.3 Result of Section 3

According to (31) and (43), the first system of wave equations for electromagnetic field vectors Eand Bin a rotating medium has the following form:

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0,2Bn2c22Bt2+2c2tvBΩ×B=0.E44

4. Second system of wave equations for vectors Eand Bin a rotating medium

In this section, we are going to obtain the second system of wave equations for electromagnetic field vectors Eand Bwhich will correspond to the original system of Maxwell’s Eqs. (58).

4.1 Equation for vector E

To derive the wave equation for vector E, we apply the operator ×to expression (5):

××E+t×B+1c2t×v×E=0.E45

Taking into consideration (7) and (10), we rewrite (45) in the form

2En2c22Et2+1c2tv×Bt1c2t×v×EE=0,E46

or, using (5),

2En2c22Et21c2t×v×E+v××EE=0.E47

Taking into account (11), we have

2En2c22Et2+1c2tvEΩ×Ev××EE=0.E48

Consider the term Ein (48). According to (8), E=1n2v×B, where, taking into account (12),

v×B=2ΩBv×B.E49

With the help of (7), we find v×B=2ΩBn2/c2/tvE; hence

E=2n2ΩB1c2tvE.E50

Consider the term vEin (50). In accordance with (13),

vE=vEΩ×E+v××E.E51

Then

E=2n2ΩB1c2tvEΩ×E+v××E.E52

Substituting (52) into (48), we obtain the desired wave equation for vector E:

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0.E53

4.2 Equation for vector B

In order to derive the wave equation for vector B, we apply the operator ×to expression (7):

××Bn2c2t×E+1c2t×v×B=0.E54

Taking into account (5) and (10), we rewrite (54) in the form

2Bn2c22Bt21c2tv×n2c2Et1c2t×v×BB=0,E55

or, using (7),

2Bn2c22Bt21c2t×v×B+v××BB=0.E56

Consider the term ×v×Bin (56). According to (11), ×v×B=vB+Ω×B. Thus,

2Bn2c22Bt2+1c2tvBΩ×Bv××BB=0.E57

Consider the term Bin (57). In accordance with (6),

B=1/c2v×E.E58

Taking into account (12), we have

v×E=2ΩEv×E,E59

or, using (5),

v×E=2ΩE+tvB.E60

So

B=2c2ΩE1c2tvB.E61

Hence

B=2c2ΩE1c2tvB.E62

Consider the term vBin (62). In accordance with (13),

vB=vBΩ×B+v××B.E63

Then

B=2c2ΩE1c2tvBΩ×B+v××B.E64

Substituting (64) into (57), we obtain the desired wave equation for vector B:

2Bn2c22Bt2+2c2tvBΩ×B+2c2ΩE=0.E65

4.3 Result of Section 4

According to (53) and (65), the second system of wave equations for electromagnetic field vectors Eand Bin a rotating medium has the following form:

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0,2Bn2c22Bt2+2c2tvBΩ×B+2c2ΩE=0.E66

5. Two systems of wave equations for vectors Eand Bin a rotating medium: a comparative analysis

The first system of wave equations for electromagnetic field vectors Eand B[which was derived from the corresponding original system of Maxwell’s Eqs. (14)] has the form

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0,E67
2Bn2c22Bt2+2c2tvBΩ×B=0.E68

The second system of wave equations [which was obtained from the corresponding original system of Maxwell’s Eqs. (58)] is

2En2c22Et2+2c2tvEΩ×E2n2ΩB=0,E69
2Bn2c22Bt2+2c2tvBΩ×B+2c2ΩE=0.E70

Systems (6770) have been derived in preprint [14] on the base of the procedure developed (for the case of vacuum) in work [15]. It should be noted that Eqs. (67) and (69) for vector Ein both systems, after dropping the terms Ω×Eand ΩB, take the approximate form

2En2c22Et2+2c2tvE=0,E71

known, for example, from work [8] (see relation (33) therein).

Expressions (67, 68) and (69, 70) represent two qualitatively different systems of wave equations for vectors Eand Bin a uniformly rotating dielectric medium. From a comparative analysis of these systems, it follows:

  1. The structure of equations for vectors Eand Bin the first system (67, 68) is asymmetrical with respect to Ω. Therefore, the propagation of E- and B-components of electromagnetic waves in a rotating medium in general case Ω=Ωxx̂+Ωyŷ+Ωzẑwill be governed by qualitatively different laws.

  2. The structure of the wave equations in the second system (69, 70) is symmetrical with respect to Ω. Hence, the propagation of the named field components in such case will be governed by similar laws.

Systems of wave Eqs. (67, 68) and (69, 70) [in conjunction with corresponding original sets of Maxwell’s Eqs. (14) and (58)] may serve as a theoretical basis for a detailed study of the process of electromagnetic wave propagation in a rotating dielectric medium. But before the beginning of such study, the researcher must first solve the problem of choosing between them (because the final results will be different).

[In the author’s opinion, from the position of the principle of symmetry, the system of wave Eqs. (69, 70) is preferable to system (67, 68), and the corresponding original system of Maxwell’s Eqs. (58) is preferable to the original system (14).]

6. Conclusion

In this chapter, on the base of two basic systems of Maxwell’s Eqs. (14) and (58) for electromagnetic field vectors Eand Bin a uniformly rotating dielectric medium, the two corresponding systems of wave Eqs. (67, 68) and (69, 70) have been derived (to the first order in Ω).

From their comparative analysis, it can be seen that the structure of the wave equations for electromagnetic field vectors in system (67, 68) is asymmetrical with respect to Ω, while the structure of such equations for these vectors in system (69, 70) is symmetrical.

On this basis, it can be concluded that if the principle of symmetry is accepted as a criterion for selection, then system of wave Eqs. (69, 70) for electromagnetic field vectors Eand Bmay be preferred. The same conclusion may be also made about the original system of Maxwell’s Eqs. (58) on the base of which set of wave Eqs. (69, 70) has been obtained.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Evgeny A. Bondarenko (March 4th 2019). Two Systems of Maxwell’s Equations and Two Corresponding Systems of Wave Equations in a Rotating Dielectric Medium, Electromagnetic Fields and Waves, Kim Ho Yeap and Kazuhiro Hirasawa, IntechOpen, DOI: 10.5772/intechopen.82595. Available from:

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