Matrix Transformation and Solutions of Wave Equation of Free Electromagnetic Field

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1 Matrix Transformation and Solutions of Wave Equation of Free Electromagnetic Field Xianzhao Zhong Meteorological College of Yunnan Province, Kunming, 6508, China Abstract In this paper, the generalized differential wave equation for free electromagnetic field is transformed and formulated by means of matrixes. Then Maxwell wave equation and the second form of wave equation are deduced by matrix transformation. The solutions of the wave equations are discussed. Finally, two differential equations of vibration are established and their solutions are discussed. Key words: matrix transformation, wave equation. PACS: De 1 Introduction By means of matrix establishment and transformation [1], the generalized wave equation for free electromagnetic field [] is transformed and formulated in terms of a diagonal matrix. Then both wave equations, Maxwell wave address: zhxzh46@163.com 1

2 equation [3] and the second form of wave equation are deduced from the the generalized wave equation in terms of matrixes. Solutions of the two kinds of wave equations are discussed. For Maxwell wave equation, the wave amplitude F 0 is independent of wave number k and wave frequency ω. For the generalized wave equation, it is a function of wave number k and wave frequency ω. Solving the second form of wave equation, two solutions, F 1 and F 1, are obtained. Multiplying F 1 by F 1 results in a function F that is identical to the solution of the generalized wave equation. Finally, a pair of differential vibration equations are established, whose solution product is identical to the solution of the generalized wave equation. Transforming the Generalized Wave Equation by Means of Matrix The generalized wave equation of free electromagnetic field is F 1 c C F t 1 F = 0, (1) c t where F is the electromagnetic field []. Now we transform Eq.(1) into diagonal matrix. Suppose C = p () and Thus Eq.(1) is rewritten as t = ε. (3) (p pε ε )F = 0. (4)

3 It is required from Eq.(4) that, for arbitrary F, p pε ε = 0. (5) Rewriting Eq.(5), we have p 1 pε 1 pε ε = 0, (6) which is then transformed into ( ) 1 1 p p ε 1 1 ε = 0. (7) Let 1 1 A = 1 (8) 1 whose eigenvalue equation is det(λe A) = 0. (9) Solving Eq.(9), two eigenvalues are obtained. They are λ 1 = 5 (10) and Solving the equation of eigenvector we obtain the eigenvectors as λ = 5. (11) (λe A) x 1 = 0, (1) x α 1 = ( 1 5 ) T (forλ 1 ) (13) 3

4 and α = (1 5 + ) T (forλ ). (14) Then two matrixes 1 1 Y = (15) and 1 5 Y T = (16) are built up in terms of the eigenvectors α 1 in Eq.(13) and α in Eq.(14). And the matrix A in Eq.(8) is transformed into a diagonal matrix Y T AY = = diag(λ 1 Λ ) (17) 5 in virtue of Eq.(15) and Eq.(16). Thus Eq.(4) is transformed into ( ) p p ε F 1 = 0. (18) 5 ε After transformation, the electromagnetic field has changed from the field F to the field F 1. Equation (18) is developed into 5 5(p ε )F 1 10(p + ε )F 1 = 0. (19) Substituting Eq.() and Eq.(3) into Eq.(19) and taking its first bracket, we have F 1 1 c F 1 t = 0, (0) which is the well-known Maxwell wave equation. Taking the second bracket of Eq.(19) results in (p + ε )F 1 = 0, (1) 4

5 which leads to the second form of wave equation. F c F 1 t = 0, () 3 Solutions of Generalized Wave Equation of Free Electromagnetic Field The solution of the free electromagnetic field formulated by Eq.(1) is F = F 0 exp(λkr λωt)exp(iλkr iλωt), (3) where k is the wave number and ω is the wave frequency. In the last section, we transform the the generalized wave equation and obtain the well-known Maxwell wave equation [3] and the the second form of wave equation by matrix transformation. From Maxwell wave equation we have the solution F 1 1 c F 1 t = 0 (4) F 1 = F 10 exp(ikr iωt), (5) where the wave amplitude F 10 is a constant quantity, independent of the wave number k and wave frequency ω. The other form of wave equation is F c F 1 t = 0. (6) By separation of variables of the spatial r and the temporal t [4], F 1 (r, t) = R(r)T (t), (7) 5

6 for the electromagnetic field F 1 in Eq.(6), there is the first form c R(r) = 1 T (t) = λ ω (8) R(r) r T (t) t where ω is supposed to be a constant and the λ a variable. For Eq.(8), there is a solution or rewritten For Eq.(6), there is the second form For Eq.(31), there is a solution or rewritten F 1 = R 0 T 0 exp(±λkr)exp(±iλωt) (9) F 1 = F 0exp(±λkr)exp(±iλωt). (30) c R(r) = 1 T (t) = λ ω. (31) R(r) r T (t) t F 1 = R 0 T 0 exp(±λωt)exp(±iλkr) (3) F 1 = F 0exp(±λωt)exp(±iλkr). (33) Now multiplying Eq.(30) by Eq.(33) results in F = F 1F 1 = F 0 exp(λkr λωt)exp(iλkr iλωt), (34) where k is the wave number and ω is the wave frequency ω. Equation (34) is identical to Eq.(3). By matrix transformation, both wave equations, Eq.(4) and Eq.(6), are derived from the generalized wave equation Eq.(1), thus the general quadratic form becomes a standard quadratic one. The generalized wave equation conforms perfectly with Maxwell wave equation and the second form of wave equation. 6

7 4 Two Vibration Equations and Their Solutions Now let us establish a pair of differential vibration equations for the electromagnetic field. The first vibration equation in terms of the temporal variable t is d F dt + (λω)df dt + λ ω F = 0. (35) In terms of the spatial variable, the second wave equation is F λk F + λ k F = 0. (36) The solution of Eq.(35) is F = F 0 exp( λωt)exp( iλωt). (37) Rewriting Eq.(37) as a function of t, we have T (t) = F 0 exp( λωt)exp( iλωt). (38) In the spheroidal coordinates, Eq.(36) of spherically symmetric becomes For Eq.(39) there is the solution d F dr (λk)df dr + λ k F = 0. (39) F = F 0 exp(λkr)exp(iλkr) (40) or rewritten R(r) = F 0 exp(λkr)exp(iλkr). (41) Evidently, for the free electromagnetic field F, both equations, Eq.(35) and Eq.(36), are vibration equations. Equation (35) is one of the function 7

8 of t and Eq.(36) is of the function of r. Multiplying Eq.(38) by Eq.(41), we arrive at F = R(r)T (t) = F 0 exp(λkr λωt)exp(iλkr iλωt). (4) Equation (4) is identical to Eq.(3). Subtracting Eq.(35) from Eq.(36), we get c F λωc F + λ ω F ( d F dt + λω df dt + λ ω F ) = 0. (43) Equation (43) needs to be rearranged because there are two variables, t and r, included in the equation. Then Eq.(43) is changed to c F F t λω(c F + F t ) + λ ω (F F ) = 0. (44) Considering the continuity equation for passive field, Eq.(44) becomes which is Maxwell wave equation. cf + F t = 0 (45) F 1 c F t = 0, (46) References [1] Gene. H. Golub, Charles F.Ven loan. MATRIX COMPUTATIONS. ISBN rd ed c Hopkins University press. [] X.Z. Zhong. Electromagnetic Field Equation and Field Wave Equation. vixra. org: [84] ,

9 [3] J.D. Jackson. CLASSICAL ELECTRODYNAMICS, 3rd ed. John-Wiley & Sons, Inc., [4] M.D. Weir, R.L. Finney and F.R. Giordano. THOMAS CALCULUS. 10th ed. Publishing as Pearson Education. Inc.,

ROTATIONAL STABILITY OF A CHARGED DIELEC- TRIC RIGID BODY IN A UNIFORM MAGNETIC FIELD

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