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Electromagnetic Wave Propagation through Triangular Antenna Subject to Longitudinal Symmetric Conditions

S. Kumar P. G. Centre Department of Mathematics, Ram Dayalu Singh College, Muzaffarpur-842002, Bihar, India ABSTRACT: Electromagnetic (EM) field intensities happen to exist as solutions of Maxwell’s equations in a three dimensional space. In the present paper an attempt has been made to determine the components of EM field intensities belonging to a pair of groove regions adjacent to a convex triangular prism. Field intensities are supposed to be longitudinal symmetry and the triangular prism forms a part of an echellete grating of fixed period. Two existence theorem regarding longitudinal symmetric electric intensity vector and magnetic intensity vector associated with time dependent spherical wave have been established. Finally, the expression for the current density associated with the prism has been derived. KEYWORDS: Electromagnetic field intensities;Convex triangular prism; Maxwell’s equations

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Kumar S. Electromagnetic Wave Propagation through Triangular Antenna Subject to Longitudinal Symmetric Conditions. Mat.Sci.Res.India;10(2)


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Kumar S. Effect of Tin Additions on Microstructure and Mechanical Properties of Sand Casting of AZ92 Magnesium Base Alloy. Mat.Sci.Res.India;10(2). Available from: http://www.materialsciencejournal.org/?p=105


INTRODUCTION

A convex triangular obstacle forms a vital part of a periodic echellete grating. In recent years1-10 quite a good number of results have been reported pertaining to the reflection, grazing and the diffraction of a propagating EM wave through a smooth, conducting and convex triangular obstacle K (Fig. 1). The obstacle is supposed to be hollow with an open rectangular base having the flare angle β, the groove depth ‘h’ and the period ‘d’ associated with the said echellete grating. In the present paper a model M (Fig. 2), consisting of a convex triangular obstacle and its adjacent wedge regions Ri (i = 1, 2), has been considered for interacting with a propagating EM wave. EM field intensities  are generated due to propagating EM wave subject to governing Maxwell’s equations

Vol10_No2_Elect_Kumr_F0

where H , E and F=(H v E) being magnetic intensity vector, electric intensity vector and their combination respectively. The physical elements σ,Î, m, J and B stand for conductivity, permittivity, permeability, current density and magnetic flux density associated with the model. The object of present paper is to determine the vector field intensity F by using spherical polar coordinates (r, θ, f) subject to the assumption ∂F / ∂ø = 0 (Longitudinal symmetric condition), leading finally to a spherical wave. Two existence theorems have been established for finding the components of H and E  is associated with the Maxwell’s equations in H and E justifying thereby the transmission (σ ≠ 0) of the concerning EM waves through K. The result has been further utilized for computing the current density J.

 FORMULATION OF THE PROBLEM 

Consider the Maxwell’s equation

Vol10_No2_Elect_Kumr_F1

where  F=F(x1,x2,x3,t) stands for vector field intensity. Transforming (1) by using spherical polar coordinates x1=r sin θcos Φ , x2=r cos θ sin Φ , x3=r cos θ one can arrive at the equation

Vol10_No2_Elect_Kumr_F2

Now, using the variable separable method, one can arrive at the solution of the equation (2) in the form

and

Vol10_No2_Elect_Kumr_F3

Vol10_No2_Elect_Kumr_F4

and

Vol10_No2_Elect_Kumr_F5

where the constants k and x are independent of r, θ, Φ and t.

In particular, assuming  then equation (5) give arise to the surface harmonic function F2(θ,Φ) satisfying the equation

Vol10_No2_Elect_Kumr_F6

Again, separating ‘F2’ further in the form of the product

Vol10_No2_Elect_Kumr_F7

one can arrive at the equations

Vol10_No2_Elect_Kumr_F8

and

Vol10_No2_Elect_Kumr_F9

where x=cos θ,  m is independent of θ.

The equation (9) may be identified as associated Legendre’s equation which furnishes the associated Legendre’s function  as one of its solutions. Thus, combining (7), (8) and (9), the surface harmonic function F2(θ, f) may be expressed in the form

Vol10_No2_Elect_Kumr_F10

where C3 is an arbitrary constant.

Now, introducing longitudinal symmetry condition ∂F / ∂ø = 0 for the EM field      F(r, θ, f, t) the equation (9) apparently reduces to the Legendre’s equation

Vol10_No2_Elect_Kumr_F11

furnishing thereby the Legendre polynomial Pn(x) as one of its solutions, and consequently the solution (10) reduces to the form

Vol10_No2_Elect_Kumr_F12

Now, recalling the equation (5) again, one can arrive at the ordinary differential equation

Vol10_No2_Elect_Kumr_F13

which possesses the only regular singular point at r = 0 and as such applying Frobenius method one can arrive at the series solution of (13) in the form

Vol10_No2_Elect_Kumr_F14

Vol10_No2_Elect_Kumr_F15

and the coefficients a2m  satisfy the recurrence relation

Vol10_No2_Elect_Kumr_F16

Again, referring to the equation (4), one can arrive at the equation

Vol10_No2_Elect_Kumr_F17

which furnishes the solution

Vol10_No2_Elect_Kumr_F18

where ‘A’ is an arbitrary constant,J√-1 , and

Vol10_No2_Elect_Kumr_F18b

and k is the wave number restricted by the inequality

Vol10_No2_Elect_Kumr_F19

Hence, combining (3), (12), (16) and (18) one can arrive at the solution

Vol10_No2_Elect_Kumr_F20

Spherical wave function and the components of electric and magnetic intensities vectors :

The expression (20) represent a spherical wave function

Vol10_No2_Elect_Kumr_F21

Vol10_No2_Elect_Kumr_F21b

stands for the free space spherical wave formed by superimposition of spherical waves of amplitudes An(F). The natures of these waves are similar to that given by (20)

Now, recalling the Maxwell’s equations

Vol10_No2_Elect_Kumr_F21M

and

Vol10_No2_Elect_Kumr_F21M2

one can arrive at the following relations :

Vol10_No2_Elect_Kumr_F22-23

Replacing F by H,  and E successively in (21) one can recast (22) and (23) in the forms :

Vol10_No2_Elect_Kumr_F24-29

Vol10_No2_Elect_Kumr_F30

Hence, one can arrive at the following theorems :

Theorem-1

A longitudinally symmetric electric intensity vector  is said to be associated with time dependent damped spherical wave ΦH(r,θ,t) of frequency ω with the damping factor (σ/2 ε) if the bounding surfaces  of ∂K are conducting   (σ≠0 ) and the components of magnetic intensity vector E are given by (27) to (29) and the frequency ω and the wave number k are mutually related as 4εk2 = μ(4 ε ω22)  subject to the restriction 2k√ ε > √ μ σ.

Determination of current density  

A current density J consists of displacement current and the conduction current according to Maxwell’s theory in electromagnetics11.

Hence, one can express J in the form

Vol10_No2_Elect_Kumr_F31

Now, combining the relations (21) and (31), J may be finally expressed in the following form

Vol10_No2_Elect_Kumr_F32

which represents a spherical wave with its modulus and the phase given by the following expressions

Vol10_No2_Elect_Kumr_F33

CONCLUSIONS

The present paper furnishes the existence of longitudinally symmetric EM waves. Such waves may be designated as longitudinally symmetric spherical waves. The concerning wave functions have been determined as solutions of the governing Maxwell’s equations in spherical coordinates . The governing Maxwell’s equations have been encountered for finding the magnetic field intensity and electric field intensity vectors. Finally the result has been used for computing the current density.

ACKNOWLEDGEMENTS

Thanks are due to UGC Minor Research Project for financial support [Grant no. PSB-004/11-12 dated 3 August-2011]

Fig.1: A convex triangular prism of dimension a, b, d and with flare angle β.              is perpendicular to the plane
Click on image to enlarge
A model ‘M’ consists of a triangular prism formed
Click on image to enlarge

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  3. Bhowmick K. N. and Kumar Sanjay, Indian J. Phys. 65 B (4): 329 (1991).
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  6. Kumar Sanjay, Proc. Math. Soc. B.H.U. 24: 29 (2008).
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