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African Journal of Science and Research,2015,(4)3:13-15 ISSN: 2306-5877 Available Online: http://ajsr.rstpublishers.com/ LIGHT AS A FORM OF ELECTROMAGNETIC WAVE IN TIME HARMONIC FIELDS (THEMF). 1Nwogu 1, 2 Uchenna, 2Emerole Kelechi, 3Akujuobi Azubuike Department of Electrical Electronic Engineering Federal Polytechnic Nekede, Owerri, Imo State, Nigeria. 3 Department of Computer Science, University of Portharcourt, Portharcourt Rivers State, Nigeria. Email: [email protected] Received:13, Mar,2015 Accepted: 24,Apr,2015. Abstract Light can also be shown to exist in time harmonic fields, where complex phasor analysis, Maxwell’s equations and constituent relations from vector calculus are deployed to unearth the plane velocity embedded in electromagnetic radiations. A sinusoidal vector phasor radiation that contains this phenomenon in phase, direction and magnitude is x-rayed and using vector calculus equations and constitutive relations to bring out the phenomenon of light (plane velocity) in relation to the speed of light in a medium. Keywords: light, electromagnetic, vector, domain, frequency, velocity, harmonic. INTRODUCTION Fields for which the time variation is sinusoidal are called time harmonic fields. To further work in this domain, phasor analysis (complex quantities) are employed to get a simple frequency steady state response. ⃗ (x,y,ƶ) = Re[E(x,y,ƶ, t) ejɷt]………….(1) From (1) above, ⃗ (x,y,ƶ) is a vector phasor that contain the information in direction, magnitude and phase. Also in the time harmonic field, the rate of change of E can be written as; ɷ (x,y,ƶ,t) = Re[jɷE(x,y,ƶ, t) ej t]……….(2) We can deduce that if the electric field vector E(x,y,ƶ) is represented in phasor form as E(x,y,ƶ), then (x,y,ƶ,t) can be represented by the phasor, jɷ E(x,y,ƶ). Also the integral ∫ can be represented by the phasor (x,y,ƶ,) These time harmonic relationships can be applied to both magnetic and electric field quantities in addition with Maxwell equations and other constitutive relations to expose the underlying entity called light. LIGHT IN TIME HARMONIC FIELDS Consider the field phasor or vector (E , H) and source phasor (β, j) in a simple linear isotropic medium as shown in Fig 1. Fig. 1 Field phasor and Vector phasor diagrams When the H and E fields vary with time sinusoidally, the time derivations can be replaced by factors of jw and Maxwell’s equation can be deduced to [4]: V X H = E + jwϵE………(3) V X E = - jwH…………….(4) V. B = 0…………………..(5) V.D= ……………………(6) Where = jɷD, - jɷB, D = Є Ē, B = μH and from ohm’s law, J = Ē Consider a magnetic field density B represented in the harmonic field as B = Bcos(wt-BZ)y, it can be seen that the signal is an oscilloscope in a ;positive B direction (direction of wave propagation) and the magnetic field in the y direction. But by right hand rule, the electric field component will be in the x direction. Decomposing the equation 7 and assuming free space. Emerole Kelechi et.al. 14 B :. γ √ ⁄ )… .………….(15) The real and imaginary part of (15) gives the wave attenuation , and the waves propagation constant of radio signals in Air β H :. = attenuation = √ ⁄ ⁄ )2 -1 ………………………………..……(16) E ...... A A .... .... S ......... E S β = propagation constant = ɷ√ ⁄ D ⁄ )2 +1 ………..…………………..(17) . D2 I. Field vectors Fig. 2 Field vectors B = μH = B˳cos(ɷZ - βZ)y H= cos(ɷt - βZ)y In free space, β˳ = ⇒ H = H˳cos(ɷt –βƵ)y H = H˳cos(ɷt –βƵ)y Amp/meters From Max. equation, V X H = jɷЄ˳Ē :. E = = electric field component............………………………(9) From Equa. (9), V X H gives H˳cos (cot – βƵ) y = ɷ√ ⁄ =H˳sin (ɷt – βƵ) (since, Ē XH fields has a 900 or phase lead in time harmonic) :. ɷt – βƵ = 0 :. ɷt – βƵ = = = phase velocity = Where Ƶ = distance propagated T = displacement :. = = time harmonic phase velocity of light …………………..……………..(10) GENERAL WAVE HARMONIC FIELDS EQUATION IN Electromagnetic wave propagation and physical interpretations. In free space, the conductivity, δ 0. This shows that the magnitude of both E and H alternates as it is being propagated in any direction in a given medium. Applying the complex wave propagation equation to decouple the propagation components of E and H, we have: Consider a case of perfect dielectric, where δ 0 (free space) Solving for and β in equation (16) and (17) and substituting δ 0 TIME From Maxwell equations in the time harmonic fields, Equ (1) = V X H = (δ + jɷϵ) Equ (2) = V X E = -jɷμH Taking the curl of (1) & (2) X ( X H) = (δ + jɷϵ)( X Ē) ………………………………………….(11) X ( X E) = jɷμ ( X H) …….………………………………..….(12) Note that the X ( X A) ( .A) – 2 A2 ubstituting the values of X E and X H in (11) and (12) and taking note of the Laplacian of a vector in the Cartesian co-ordinate yields the vector wave equation where γ complex wave propagation constant [4]: V2.H = jɷμ (δ + jɷЄ)H γ2H ......…… (13) V2.E = jɷμ (δ + jɷЄ) E = γ2E……….…(14) :. γ2 jɷμ (δ + jɷЄ) = 0 (attenuation) For β, substituting the value of δ in (17) = ɷ√ ⁄ =ɷ Β ɷ ....................………………(18) Equation (18) shows why signal propagates in a dielectric medium :. Β ɷ In free space, μ μ˳, ϵ = ϵ˳ :. = ...........................…………(19) Comparing equation (10) and (19), we have that. phase velocity of light in an electromagnetic field Substituting the values of for free space, we have 3X108m/s = velocity of light (Q.E.D) This shows that light is a form or an integral part of both the electric E and magnetic field H. This also shows that as a result of speed component of light, signal (electromagnetic) is able to propagate in a medium even when the attenuation is zero as in a perfect dielectric or when you rapid alternation ( as in a perfect conductor. CONCLUSION It has been shown that Maxwell’s equations leads directly to Electrical and Magnetic fields satisfying the wave equation for which the solution are linear combinations of plane waves travelling with the speed of light C = . The African Journal of Science and Research,2015,(4)3:13-15 R.H.S (Right Hand Side) of the above equation is a quantity relating to the equation governing electrical and magnetic fields. It has units of velocity, but can be seen that its derivatives components (E and B) forces had no physical velocity components initially [5]. The fact is, Maxwell’s equations explain how these waves can physically propagate through space: the changing magnetic field creates a changing electric field from Faraday’s law. In turn, the Electric field creates a changing Magnetic field through Maxwell’s correction to Ampere laws. The perpetual cycle allows these waves, now known as electromagnetic radiation to move through free space at a velocity C, which is the speed of light. From the decoupling procedures, it could deduce this component speed of light (C), in the form of plane velocity vsp embedded in the electromagnetic radiation is related to the speed of light. (The ratio of the absolute electromagnetic unit charge to the absolute electrostatic unit charge, which is the ratio ). References 1)Electronic Communication Systems by Kennedy and Davis (5th Edition, 2000) 2)Communications Electronic by Johnson I. Ejimanya (2005) 3)Understanding Fiber Optics by Jeff Hedit (2006) 4)Radiation Fields of Helical Antenna by S.A Adekola, Ike Moete,A Ayorinde(2010) 5)Basic Electromagnetic Wave Properties of Light by Matter J. Pary Hill (2009) 6)www.icbse.org/education: introduction to light waves (2012). 7)www.jewave.com (2013) 8)www.nptel.acrom/course117.