15 Maxwell’s Equations for the Electromagnetic Field Maxwell’s equations describe the basic laws of the electromagnetic field Over the 40 years preceding Maxwell’s enunciation of his equations (1865) the four fundamental laws describing the electromagnetic field had been discovered These are known as Ampe`re’s law, Faraday’s law, Coulomb’s law, and the magnetic continuity law These four laws were cast by Maxwell, and further refined by his successors, into four differential equations: =ÂH¼jþ =ÂE¼À @D @t @B @t ð15-1aÞ ð15-1bÞ = ÁD ¼ ð15-1cÞ =ÁB¼0 ð15-1dÞ These are Maxwell’s famous equations for fields and sources in macroscopic media: E and H are the instantaneous electric and magnetic fields, D and B are the displacement vector and the magnetic induction vector, and j and are the current and the charge density, respectively We note that (15-1a) without the term @D=@t is Ampe`re’s law; the second term in (15-1a) was added by Maxwell and is called the displacement current A very thorough and elegant discussion of Maxwell’s equations is given in the text Classical Electrodynamics by J D Jackson, and the reader will find the required background to Maxwell’s equations there When Maxwell first arrived at his equations, the term ð@D=@tÞ was not present He added this term because he observed that (15-1a) did not satisfy the continuity Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved equation To see that the addition of this term leads to the continuity equation, we take the divergence =Á, of both sides of (15-1a) = Á ½=  H ¼ ð= Á jÞ þ @ ð= Á DÞ @t ð15-2aÞ The divergence of curl is zero, so the left-hand side is zero and we have ð= Á jÞ þ @ ð= Á DÞ ¼ @t Next, we substitute (15-1c) into (15-2b) and find that ! @ =Ájþ ¼0 @t ð15-2bÞ ð15-3aÞ or =Ájþ @ ¼0 @t ð15-3bÞ which is the continuity equation Equation (15-3b) states that the divergence of the current ð= Á jÞ is equal to the time rate of change of the creation of charge ðÀ@=@tÞ What Maxwell saw, as Jackson has pointed out, was that the continuity equation could be converted into a vanishing divergence by using Coulomb’s law, (15-1c) Thus, (15-1c) could only be satisfied if @ @D =Ájþ ¼=Á jþ ¼0 ð15-4Þ @t @t Maxwell replaced j in Ampe`re’s law by its generalization, and arrived at a new type of current for the electromagnetic field, namely, j!jþ @D @t ð15-5Þ for time-dependent fields The additional term @D=@t in (15-5) is called the displacement current Maxwell’s equations form the basis for describing all electromagnetic phenomena When combined with the Lorentz force equation (which shall be discussed shortly) and Newton’s second law of motion, these equations provide a complete description of the classical dynamics of interacting charged particles and electromagnetic fields For macroscopic media the dynamical response of the aggregates of atoms is summarized in the constitutive relations that connect D and j with E, and H with B; that is, D ¼ "E, j ¼ E, and B ¼ H, respectively, for an isotropic, permeable, conducting dielectric We can now solve Maxwell’s equations The result is remarkable and was the primary reason for Maxwell’s belief in the validity of his equations In order to this, we first use the constitutive relations: D ¼ "E ð15-6aÞ B ¼ H ð15-6bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Equations (15-6a) and (15-6b) are substituted into (15-1a) and (15-1b), respectively, to obtain =ÂH¼j þ" =  E ¼ À @E @t ð15-7aÞ @H @t ð15-7bÞ Next, we take the curl ð=ÂÞ of both sides of (15-7b): =  ð=  EÞ ¼ À @ ð=  HÞ @t ð15-8Þ We can eliminate =  H in (15-8) by using (15-7a), and find that À@ @E =  ð=  E Þ ¼ j þ" @t @t so =  ð=  EÞ ¼ À @j @2 E À " @t @t ð15-9Þ The left-hand side is known from vector analysis to reduce to =  =  E ¼ =ð= Á EÞ À =2 E ð15-10Þ Equation (15-9) then reduces to =ð= Á EÞ À =2 E ¼ À @j @2 E À " @t @t ð15-11Þ Finally, if there are no free charges then ¼ and (15-1c) becomes = Á D ¼ "= Á E ¼ or =ÁE¼0 ð15-12Þ Thus, (15-11) can be written as =2 E À " @2 E @j ¼ À @t @t2 ð15-13Þ Inspection of (15-13) quickly reveals the following If there are no currents, then j ¼ and (15-13) becomes =2 E ¼ " @2 E @t2 ð15-14Þ which is the wave equation of classical optics Thus, the electric field E propagates exactly according to the classical wave equation Furthermore, if we write (15-14) as =2 E ¼ @2 E 1=" @t2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð15-15Þ then we have =2 E ¼ @2 E v2 @t2 ð15-16Þ where v2 ¼ c2 The propagation of the electromagnetic field is not only governed by the wave equation but propagates at the speed of light It was this result that led Maxwell to the belief that the electromagnetic field and the optical field were one and the same Maxwell’s equations showed that the wave equation for optics, if his theory was correct, was no longer a hypothesis but rested on firm experimental and theoretical foundations The association of the electromagnetic field with light was only a speculation on Maxwell’s part In fact, there was only a single bit of evidence for its support, initially We saw that in a vacuum we have =ÁE¼0 ð15-12Þ Now it is easy to show that the solution of Maxwell’s equation gives rise to an electric field whose form is E ¼ E0 eiðkÁrÀ!tÞ ð15-17aÞ where E ¼ Ex ux þ Ey uy þ Ez uz ð15-17bÞ E0 ¼ E0x ux þ E0y uy þ E0z uz ð15-17cÞ k ¼ kx ux þ ky uy þ kz uz ð15-17dÞ r ¼ xux þ yuy þ zuz ð15-17eÞ k Á r ¼ kx x þ ky y þ kz z ð15-17fÞ Substituting (15-17a) into (15-12) quickly leads to the relation: kÁE¼0 ð15-18Þ where we have used the remaining equations in (15-17) to obtain (15-18) The wave vector is k and is in the direction of propagation of the field, E Equation (15-18) is the condition for orthogonality between k and E Thus, if the direction of propagation is taken along the z axis, we can only have field components along the x and y axes; that is, the field in free space is transverse This is exactly what is observed in the Fresnel–Arago interference equations Thus, in Maxwell’s theory this result is an immediate consequence of his equations, whereas in Fresnel’s theory it is a defect This fact was the only known difference between Maxwell’s theory and Fresnel’s theory when Maxwell’s theory appeared in 1865 For most of the scientific community and, especially, the optics community this was not a sufficient reason to overthrow the highly successful Fresnel theory Much more evidence would be needed to this Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Maxwell’s equations differ from the classical wave equation in another very important respect, however Returning to (15-13), Maxwell’s equations lead to =2 E À " @2 E @j ¼ À @t @t2 ð15-13Þ The right-hand term in (15-13) is something very new It describes the source of the electromagnetic field or the optical field Maxwell’s theory now describes not only the propagation of the field but also enables one to say something about the source of these fields, something which no one had been able to say with certainty before Maxwell According to (15-13) the field E arises from a term @j/@t More specifically the field arises not from j, the current per se, but from the time rate of change of the current Now this can be interpreted, as follows, by noting that the current can be written as j ¼ ev ð15-19Þ where e is the charge and v is the velocity of the charge Substituting (15-19) into (15-13), we have =2 E À " @2 E @v ¼ e ¼ e_v @t @t2 ð15-20Þ The term @v/@t is obviously an acceleration Thus, the field arises from accelerating charges In 1865 no one knew of the existence of actual charges, let alone accelerating charges, and certainly no one knew how to generate or control accelerating charges In other words, the term (e)@v/@t in 1865 was ‘‘superfluous,’’ and so we are left just with the classical wave equation in optics: =2 E À " @2 E ¼0 @t2 ð15-21Þ Thus, we arrive at the same result from Maxwell’s equations after a considerable amount of effort, as we by introducing (15-21) as an hypothesis or deriving it from mechanics This difference is especially sharp when we recall that it takes only a page to obtain the identical result from classical mechanics! Aside from the existence of the transverse waves and the source term in (15-13), there was very little motivation to replace the highly successful Fresnel theory with Maxwell’s theory The only difference between the two theories was that in Fresnel’s theory the wave equation was the starting point, whereas Maxwell’s theory led up to it Gradually, however, the nature of the source term began to become clearer These investigations, e.g., Lorentz’s theory of the electron, led physicists to search for the source of the optical field Thus, (15-13) became a fundamental equation of interest Because it plays such an important role in the discussion of the optical field, (15-13) is also known as the radiation equation, a name that will soon be justified In general, (15-13) has the form of the inhomogeneous wave equation The solution of the radiation equation can be obtained by a technique called Green’s function method This is a very elegant and powerful method for solving differential equations, in general However, it is quite involved and requires a considerable amount of mathematical background Consequently, in order not to detract from our discussions on polarized light, we refer the reader to its solution Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 15-1 Radiating field coordinates arising from an accelerating charge; P is the observation point (From Jackson) by Jackson (Classical Electrodynamics) Here, we merely state the result Using Green’s function method, the solution of the radiation equation in the form given by (15-20) is found to be ! È É e n Eðr, tÞ ¼  ðn À vÞ Â v_ 4"0 c2 3 R ð15-22aÞ where ¼1ÀnÁv ð15-22bÞ and n ¼ R/R is a unit vector directed from the position of the charge to the observation The geometry of the moving charge is shown in Fig 15-1 In the following chapter we determine the field components of the radiated field for (15-22) in terms of the accelerating charges REFERENCES Jackson, J D., Classical Electrodynamics, Wiley, New York, 1962 Sommerfeld, A., Lectures on Theoretical Physics, Vols I–V, Academic Press, New York, 1952 Born, M and Wolf, E., Principles of Optics, 3rd ed., Pergamon Press, New York, 1965 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved