American Journal of Engineering Research (AJER)2015 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847p-ISSN : 2320-0936 Volume-4, Issue-8, pp-93-97 www.ajer.org Research PaperOpen Access w w w . a j e r . o r g Page 93 An Evaluation of the Plane Wave properties of Light using Maxwells Model Nwogu O. Uchenna 1, Emerole Kelechi 2, Osondu Ugochukwu 3,ImhomohE. Linus 4 1(Department of Electrical and Electronics Engineering, Federal Polytechnic Nekede ,Owerri, Imo State, Nigeria) ABSTRACT: The only two forces of nature that human beings can directly experience through their 5 senses aregravityandlight.Theothersensationssuchasasmell,heat,soundetcdetectmacroscopicpropertiesof mater and not its constituents. Electrostatic and electromagnetic charges are deeply likened with light. It is the acceleratedmovementofthesechargesthatgiveslight.Themovementisatthespeedoflight.Thispartof electromagneticradiation(light)istheonlyonethatisvisibletohumanbeing.Here,thelanguageof engineering electromagnetism; the second grand unification of science that unifies the 3 most powerful forces of light,magnetismsandelectricity(Maxwellsfourequations)andunderlyingvectorcalculusequationswere invokedtodecouplethecomponentpartsofatypicalelectromagneticradiationtobring outthisphenomenon called light, as a basic component part of an electromagnetic radiation. Keywords-current, displacement, field, magnitude, phasor, vector I.INTRODUCTIONLightisatransversewavehavinganoscillatingelectricandmagneticfield.Bothfieldsareperpendicularto eachotherandtothedirectionoftheirpropagation.Theratiooftheelectricfieldintensity()andmagnetic field intensity () is equal to a constant thus; EB = constant (c) Therefore an electromagnetic waves is defined as plane if at any given instant of time, the electric intensity() and the magnetic intensity(H ), remain the same ( in magnitude and phase ), over a plane normal to the direction of wave propagation and are completely propagating in the Z direction and the X-Y plane is called theplane of the wave or the wave front. Figure 1: Plane Wavefront. American J ournal of Engineering Research (AJ ER)2015 w w w . a j e r . o r g Page 94 From the above definition; BX = dBdy = 0 EX = dEdy = 0 :. dEXdBX = dEYdBy = Constant = field strength 377 (intrinsic impedance) According to Maxwell equationfor plane wave: E=ECosX;H =HCosy where X and Y are unit vector in agiven direction. II.METHODOLOGY Maxwells original equation where broken down into four equations. These four equations are a refined form of the 22 previous equations. It unifies the 3 most powerful forces of nature: Light, Electricity and Magnetisms.It consists of the electromagnetic language that binds these 3 universal forces [4]. Also Maxwells equations play a central role in the analysis in the transmission media [2]. Thefoursetsofequations satisfied everywhere by the electric field vectors( and D ),and magnetic field vectors (Hand B ) are; div D = div=0 curl = - dBdt= dHdt curl H = J + dDdt= + dEdt Where Dand Ddt are electric displacement and displacement current density respectively and J is the conduction current density. D= and H = BUrUo with = free charge density = permittivity of free space = 8.854 X 10-2fm -1 =dielectric constant or relative permittivity of the material = 1 in a vacuum, > 1 in homogenous, isotopic medium = permeability of free space = 1C2 where c speed of light. However, in the words of Maxwellians [4]; in the beginning God said; V X H= J + Ddt V X E = - Bdt V.B=0 American J ournal of Engineering Research (AJ ER)2015 w w w . a j e r . o r g Page 95 V.D= and there was light. ThisMaxwellsequationismadeupofelectromagnetismandlight.Theequationscanbedecoupledandby applyingconstitutiverelationsandothervectorcalculusconditions,thusasacomponentofelectromagnetic wave can be unveiled. III.PHYSICAL INTERPRETATIONS V X H= J + Ddt V X E = - Bdt V.B=0 V.D=Where, J = Conduction current density Ddt = time varying electric flux density B= magnetic flux density Bdt = time varying magnetic flux density = electric charge density The four equations describe the experimental al facts that [4]. Equation1:Magneticfield is not only produced by aconductor current but also by adisplacement current or time varying electric flux density (Amperes law) Equation 2: Electric field can be produced by a time varying magnetic field (Faraday and Lenzs law) Equation 3: There is no isolated magnetic change in nature i.e. the existence of a monopole. Equation 4: Establishes the occurrence of isolated electric charges in nature (Gauss law). IV.MAXWELLS EQUATION IN FREE SPACE(CHARGE FREE ZONE) A free space or charge free zone is space with nothing in it at all. It doesnt exist in the known universe, but interstellar space is a good approximation [6] Featuresoffreeinclude:Uniformityeverywhere,noelectricchargeinandaroundit.Itcarriesno electriccurrentandnothingthatelectromagneticradiationsarepropagatedinfreespacebyelectromagnetic theorem, we use Maxwells equation as well as a constitutive tool to understand and decipher plane waves into their constituent parts. Using Maxwells equations(M.E) (1)and decoupling it by multiplying it by a vector V we have that [4]: V X [V X H ] =V X J + V x Ddt (1) Invoking constitute relations to decouple the L.H.S. of the equation (1), we have, assuming D= E American J ournal of Engineering Research (AJ ER)2015 w w w . a j e r . o r g Page 96 :. R.H.S = V X J + V X[ dEdt ]= V X J +ddt [ V X E]..(1a) To decouple L.H.S of the equation, we use the vector identity theorem, which state the curl of the curl of a vector is equal to the gradient of divergence minus the laplacian. Thus:L.H.S = V [V.] V2 Where V [V.] = gradient of the divergence of the vector = 0 V2H= Lapalcian Bringing the L.H.S and R.H.S of the equations together, we have that0-V2H =VXJ+ddt[VXE] ..(2) Form the L.H.S,VXE=Maxwells equation (M.E) (2), applying theconstitutiverelations to further decouple, we have; [ V X E] = dBdt B=H :.[VXE]=Ht..(3) Substituting equation (3) back in equation (2) we have, ...(4) The above equation(4) is called the general wave equation for the H field[4]. We can further deduce the wave equation on the E field by decoupling Maxwells equation 2 by taking the curl of the equation: VX [M.E (2)] V X V X E =V X -Bdt V X V X E = - ddt [V X B]... .(5) By applying vector identity to the L.H.S and the constitutive relations on the R.H.S, we have that; V (V X E) - V2 X E = ddt (V X H ) = ddt (J J+ dEdt) The L.H.S. can also be decoupled further with M.E (4) V.(V X E) where V .D= , substituting we have that V () Bringing the two sides of the equations, together we have; ...(6) Equation (6) is called the general wav equation for the E field. But for a source free region0 - V2H=- d2H2= - VX J V2 E- d22 = d

+ V(

) American J ournal of Engineering Research (AJ ER)2015 w w w . a j e r . o r g Page 97 J = 0 and = 0 Substituting equation (4) and (6) we can deduce the general vector wave equation for a source freeregion in the Hand E field respectively: Also, if Hand E field vary with t time sinusoidally , the time derivative can be replaced by factors of jw to obtain; 2dt 2 = (jw)2 = -w2 substituting inthe wave equations, we have [4]: Comparing the general equations with the source velocity equation V 2w-12co2t2 =So (free space) where =source velocityWe have that :. sp =12= = 1 Assume free space, = = 136 X 10-9 = 8.854X10-6C2/nm2, = = 4 X10-7Tm/A = sp = 14X107X136 X 109 = 3X108m/s = speed of light. This shows that light is an integral part of both the electric field E and the magnetic field H . V.CONCLUSIONThe above derived equation shows that light can behave like a projecting magnetic or electric field. That is, wehavelightembeddedinboththeelectricandmagneticfields.Electromagneticwavesareproducedthat propagates through a vacuum at the speed of light. REFERENCES [1] George Kennedy and Bernard Davis, Electronic Communication systems (McGraw-Hill Education, 1992).[2] Johnson Ejimanya, Communication Electronics(Print Konsult, 2005).[3] Jeff Hecht, Understanding Fiber optics (Prentice Hall, 2006).[4] S.Adekola,A.Ayorinde,I.MoeteOntheRadiationfieldsoftheHelicalHyperbolicAntenna,PIERSProceedingsKuala Lumpur,MALAYSIA, March 27-30, 201, 1644-1647 [5] J. Parryhill, Basic Electromagnetic wave properties ofligh t(2009).[6] www.icbse.org/education: Introduction to light waves (2012). V2 H - d2H2 = 0 V2 E - d2E2 = 0 V2H+ w2 H= 0 V2E + w2 E = 0