remain essentially unchanged for the case of time-varying fields, the remaining two

Unit 2 Maxwell s Equations Time-Varying Form While the Gauss law forms for the static electric and steady magnetic field equations remain essentially unchanged for the case of time-varying fields, the remaining two equations (see Summary at the end of Section 1 5) must be revised. Once the proper forms are attained, we then have a general set of field equations which may be applied, in their broadest sense, to any classical electromagnetics problem. 2.1 Faraday s Law We have earlier intimated that the form E dl = 0 is NOT valid for the timevarying electric field. Note that we shall use the symbolism Ẽ to indicate such a field, C remembering, of course, that the field will, in general, be a function of position r and time, t i.e. Ẽ E( r, t). In 1831, Faraday showed that whenever there is relative motion between a closed conducting path and a magnetic field i.e. one or the other or both may be moving an electromotive force, emf is establised in the closed path. The statement of this fact, known as Faraday s law, is customarily written where dφ dt emf = dφ dt (2.1) is the rate of change of magnetic flux. The minus sign indicates that the 1

induced emf is in such a direction as to produce a current whose flux, if added to the original flux, would reduce the magnitude of the emf. This statement is known as Lenz s law. We next define emf, in volts, in terms of a time-varying electric field intensity as emf = C E d L (2.2) where, in general, different paths give different emfs (quite unlike the static case). In electrostatics, we use the term potential difference to indicate the result of the integral, while with time-varying fields, the result is an emf or voltage. Recall that Φ = S B d S (2.3) which is magnetic Gauss law where the B-field now varies with time. Equations (2.1), (2.2) and (2.3) taken together yield (2.4) If we assume a stationary path and a changing flux, then the magnetic flux density is the only time-varying quantity on the RHS of (2.4). Thus, we may use partial differentiation w.r.t. time and write C E d L = B ds (2.5) S t This is one of Maxwell s equations in integral form for time-varying varying fields. Applying Stokes law ( recall, A dl ( ) = A ds ) leads to C S These integrals are perfectly general and the surfaces may be taken to be identical. Therefore, Ẽ = B t (2.6) 2

which is the corresponding point form of equation (2.5). 2.2 Ampère s Law Revisited For the steady magnetic field we had Ampère s law in point form as (2.7) Taking the divergence on both sides yields (2.8) It is straightforward to verify for any vector, A, A = 0. Therefore, (2.9) However, from the conservation of charge, (2.10) Thus, J = 0 can be true only if ρ v = 0. That is, the form of Ampère s law in t equation (2.7) is only valid if things are NOT time-varying. For the time-varying case, the law must be FIXED! We now propose such an adjustment. We have contended that Gauss law (electric) holds for time-varying fields. That is, D = ρ v may be written as D = ρ v (2.11) where D D ( r, t) and ρ v ρ v( r, t). Furthermore, the continuity equation (equation (2.10)) becomes J = t ( D ) (2.12) Now, if we write H = J + D t (2.13) 3

Ampère s law agrees with the conservation of charge since where we have allowed the interchange of the and t on the displacement flux density. Thus, equation (2.13) appears to be a valid adjustment to the non-time-varying case of Ampère s law. Equations (2.11) and (2.13) together imply that charge is conserved. To date, equation (2.13) has not been refuted!! Displacement Current The additional term, D t, in equation (2.13) is referred to as displacement current density. It was Maxwell s big contribution to unifying the existing theory governing electromagnetics. To help with the concept, consider the following filamentary loop through which passes a time-varying B-field. The loop ends are terminated by the plates of a parallelplate capacitor. It is not difficult to deduce that the current between the plates of the capacitor has the same value as the conduction current in the filament. The former is referred to as the displacement current, Ĩ d. where S is the surface area of the capacitor. We have assumed that the D-field is distributed uniformly between the plates in this simple discussion. Otherwise, we may generally write 4

Summary of Maxwell s Equations The results which have been arrived at from Section 1.3 to Section 2.2 may now be written concisely as Maxwell s equations in time-varying form for a stationary surface S. Integral Form Faraday s Law: E C d L = B ds S t Ampère s Law (Modified): dl C H ( = J S + D ) ds t Gauss s Law (Electric): ds S D = ρ v dv = Q v Gauss s Law (Magnetic): B S d S = 0 Point Form Faraday s Law: Ẽ = B t Ampère s Law (Modified): H = J + D t Gauss s Law (Electric): D = ρ v Gauss s Law (Magnetic): B = 0 Along with the constitutive relationships, the Lorentz force equation and Ohm s law, these equations form the basis for all classical electrodynamics. In their most general form they are, at best, difficult to solve. Fortunately, however, there are many important cases where approximations yield valid results. We note that the electrostatic and magnetostatic forms of the equations are simply arrived at by removing the and setting the time derivatives to zero. Finally, we note that while the existence of electrostatic fields is independent of the magnetostatic fields, the same is not true for time-varying fields. This is obvious as the Ẽ and or and are seen to be coupled together in the above equations. B H D For example, in radio communications a transmitting antenna generates waves that 5

are electromagnetic in nature. The remainder of this course involves the application and interpretation of the time-varying Maxwell equations in a variety of contexts. 2.3 Maxwell s Equations for Time-Harmonic Fields Time-harmonic fields (see Section 1.1.2 of these notes) represent a very important class of fields for which Maxwell s equations may be considered. Recall that we shall use Ẽ, H etc. to represent the time-harmonic (complex) form of the fields. For any time-harmonic vector field Ṽ we had Ṽ = Re { Ṽ e jωt} (2.14) where the phasor notation for the (complex) vector is defined on page 9 of Unit 1 these notes as Ṽ = ( V x (x, y, z)e jφx ) ˆx + ( Vy (x, y, z)e jφy ) ŷ + ( Vz (x, y, z)e jφz ) ẑ Clearly, each component of the complex vector Ṽ has real and imaginary parts. Recall that t is replaced in the complex domain by jω. Therefore, if the fields are timeharmonic, the point form of the Maxwell equations in the previous summary may be written as Ẽ = jω B (2.15) H = J + jω D (2.16) D = ρ v (2.17) B = 0 (2.18) Whenever analysis is carried out in the complex vector domain, returning to the time-domain is readily accomplished via (2.14). 6

2.4 Poynting s Theorem Conservation of Power The following analysis derives a power relationship from the equations and Take the dot product of (2.19) with H to get Ẽ = B t (2.19) H = J + D t (2.20) (2.21) and take the dot product of (2.20) with Ẽ to get (2.22) A useful vector identity here is Therefore, subtracting (2.22) from (2.21) gives (2.23) Recalling that B = µ H and D = ɛ Ẽ and noting that equation (2.23) becomes ( ) E = H t µh 2 2 + t ɛẽ 2 + Ẽ Poynting s Theorem (2.24) 2 J This important statement is referred to as Poynting s theorem. We shall now attemt to answer the question What does it mean?. First we notice the units of the terms: 7

The other terms have the same units. Next let s integrate (2.24) over a volume, V, enclosed by a surface, S. which from the divergence theorem becomes We note the following: ( ) E S ds H = V t ɛẽ 2 µh 2 + 2 2 dv + V J Ẽ dv. (2.25) 1. J V Ẽ dv is (if there are no sources in V ) the power being dissipated by ohmic losses within V or (if there are sources) it is the power being delivered by the sources. ɛẽ 2 2 2. µh V t 2 + dv is the power being stored in the electromagnetic field 2 within V. ( ) 3. E S ds H is (recalling that ds is along the outward normal) power flow into V across S from external sources. Therefore, equation (2.25) represents a power balance equation for the volume, V. The Poynting Vector The cross product, Ẽ, whose units are watts/metre H 2, is referred to as the Poynting vector, S. That is, S = Ẽ H (2.26) S may be thought of as a vector whose magnitude is a power density and which is in the direction of positive power flow. The power,, crossing a surface S (not P necessarily closed) due to an electromagnetic (e-m) source is then = P S ( ) S d S = E S ds H (2.27) 8

For time-harmonic fields, the time-averaged Poynting vector, S av, is given from Section 1.1.2 as S av = S = 1 2 Re { Ẽ H }. (2.28) Example: The electric field intensity in a certain region is specified as Ẽ = ˆx E 0 e jkz V/m. where E 0 is a real constant. Determine Ẽ, H, S and S av. (Also, as an exercise, determine the stored electric and magnetic energy per unit volume.) 9

2.5 Plane Waves in Free Space In order to investigate the phenomena which Maxwell s equations predict, let us initially seek a solution to their time-harmonic form in free space. For such a region, ρ v = 0 and J = 0 (i.e., no sources in the region of interest) and Maxwell s equations in point form given in (2.15 2.18) become Ẽ = jω B = jωµ 0 H (2.29) H = jω D = jωɛ 0 Ẽ (2.30) D = 0 (2.31) B = 0 (2.32) We have used B = µ 0 H and D = ɛ0 Ẽ here. Taking the curl of (2.29) we get which, on substitution into (2.30) gives Ẽ = ω2 µ 0 ɛ 0 Ẽ (2.33) Next we apply the vector identity A = ( A) 2 A for any vector, A, to equation (2.33) to get From equation (2.31), the first term on the LHS of the last expression is zero so that 2 Ẽ + ω 2 µ 0 ɛ 0 Ẽ = 0 (2.34) This is the famous (homogeneous) Helmoltz wave equation in complex vector form. A similar equation could have been derived for H. Equation (2.34) is a second-order 10

partial differential equation. Let s seek a particularly simple solution to (2.34), in which the E-field has only a ˆx component which depends only on z. It is easily verified (DO THIS) that (2.35) is the solution where E x0 + and Ex0 are constants determined from boundary conditions and k = ω µ 0 ɛ 0. (2.36) Now, the two pieces of the general solution in (2.35) are linearly independent and each is a solution by itself. Therefore, let s consider the solution Ẽ = ˆx E + x0e jkz (2.37) Returning to the time domain, or E = ˆx E+ x0 cos(ωt ω µ 0 ɛ 0 z) V/m (2.38) Of course, E + x0 may be complex (with magnitude E + x0 and phase φ + ) in which case E = ˆx E+ x0 cos(ωt ω µ 0 ɛ 0 z + φ + ) V/m (2.39) What about the H-field? Well, from (2.6) and using this with (2.38) Ẽ = B t = µ 0 H t Integrating with respect to time, H = ŷ ɛ0 µ 0 E + x0 cos(ωt ω µ 0 ɛ 0 z) A/m 11

Finally, defining the intrinsic impedance, η 0, of the free space to be η 0 = µ0 ɛ 0 (2.40) H = ŷ E+ x0 η 0 cos(ωt ω µ 0 ɛ 0 z) A/m (2.41) It is easily seen that η 0 has units of ohms and its value is approximately η 0 120π Ω. Of course, the H-field corresponding to (2.39) is H = ŷ E+ x0 η 0 cos(ωt ω µ 0 ɛ 0 z + φ + ) A/m (2.42) We note that equations (2.38) or (2.39) and (2.41) or (2.42) indicate that if there is a time-varying E-field then there is a time-varying H-field. Now, let s seek an interpretation for (2.38) and (2.41). At a given instant (i.e. fixed t), all points, P, in the x y plane have a constant phase. Similarly all points, P 1, in the z = z 1 plane have a (different) constant phase. That is, all observers in a particular plane parallel to the x y plane see the same fields and these time-varying E and H fields are phase delayed as one progresses in the 12

z-direction. Thus, since all observers in a plane see the same fields, equations (2.38) and (2.41) combined are referred to as plane waves. Furthermore a plane wave with uniform amplitudes over its constant phase planes is called a uniform plane wave. Temporal Period, T : The time period, T, of the sinusoidal fields is defined as that time such that ωt = 2π or 2πfT = 2π f = 1 T where f is the frequency in hertz. Wavelength, λ, and wavenumber, k: We may regard the ω µ 0 ɛ 0 coefficient on z as a spatial frequency, k, commonly referred to as the wavenumber and define the spatial period (i.e. wavelength), λ, as or ω µ 0 ɛ 0 λ = 2π or λ = 2π ω µ 0 ɛ 0 = 2π 2πf µ 0 ɛ 0 λ = 2π k (2.43) All observers in planes separated by integer multiples of λ see the same fields (i.e. no phase difference). We note, too, that a wave vector k may be defined as k = kˆk where ˆk is a unit vector in the direction of travel of the plane wave (see below). In the illustration we are using, ˆk = ẑ. Phase velocity Suppose, next, that the observer travels along with the e-m field so that a constant field is observed. This clearly requires that ωt ω µ 0 ɛ 0 z = constant 13

which on taking the time derivative yields ω ω µ 0 ɛ 0 z t = 0. Defining the derivative to be the phase velocity, u p, u p = ω k = 1 µ0 ɛ 0 = c 3 10 8 m/s (2.44) where c is the speed of e-m waves (i.e. light ) in free space (vacuum). Intrinsic Impedance, η 0, of Free Space: It has been noted already that the quantity µ0 quantity is referred to as the intrinsic impedance of free space: ɛ 0 has ohms as its unit, and this η 0 = µ0 ɛ 0 120π Ω Power Density, S: We note that the power density as specified by the Poynting vector, S, is given by S = Ẽ = ẑ H E+2 x0 cos 2 (ωt kz) (2.45) η 0 from which the time average power density in W/m 2 is clearly S av =< >= ẑ S E+2 x0 = 1 2η 0 2 Re { Ẽ H } (2.46) (Note the analogy between (2.46) and the average power, V 0 2, dissipated in a resistor, 2R R, where V 0 is the peak voltage). It is evident from equation (2.45) that the plane wave energy travels in a direction perpendicular to the plane containing Ẽ and H ; i.e., Ẽ, H, and S are mutually perpendicular. These uniform plane waves are referred to as transverse electromagnetic (TEM) waves. In free space, we see from the above discussion that this uniform plane wave travels travels with a speed of c, the speed of light in a vacuum. 14

2.6 Polarization The tip of the time-harmonic electric field vector, Ẽ, will trace a certain locus as time progresses. This locus defines the polarization of the uniform plane wave. We shall consider three such cases: (1) linear polarization, (2) circular polarization, and (3) elliptical polarization. To introduce the idea, consider our solution to the wave equation given by (2.38) E = ˆx E+ x0 cos(ωt kz). For a given plane, z = constant, it is clear that the tip of the Ẽ-vector traces a line segment of length 2E x0 + along the x-axis as time progresses. For this case, we say that the plane wave is linearly polarized. To generalize this idea, consider an E-field which has two components and which may be represented in complex vector form as Ẽ = ˆx a e j(kz φa) + ŷ b e j(kz φ b) (2.47) where the amplitudes, a and b, and the phases, φ a and φ b, are constants. It may be verified that this is a solution to the wave equation (2.34) with the same dispersion relationship i.e. k 2 = ω 2 µ 0 ɛ 0. You may quickly verify that H has ˆx and ŷ components also and that the uniform plane wave described by (2.47) propagates in the + z direction. The real E vector may be specified by its components using Re { Ẽe jωt} as Ẽ x = a cos(ωt kz + φ a ) Ẽ y = b cos(ωt kz + φ b ) (2.48) As we shall see, specification of a, b, φ a, and φ b will determine a particular polarization. 15

Linear Polarization Let φ = φ b φ a and A = b a. If φ = 0 (i.e. φ a = φ b ), then equation (2.48) clearly yields Ẽ y AND if φ = π (i.e. φ b = φ a + π), Ẽ x = A Ẽ y = b a Ẽ x Ẽ y Ẽ x = A Ẽ y = b a Ẽ x Therefore, the two cases may be summarized as Ẽ y = ± b a Ẽ x (2.49) Circular Polarization Suppose A = 1 (i.e. b = a in equation (2.48)). A special case arises when φ = ± π 2. If φ = π, equation (2.48) becomes 2 Ẽ x = a cos(ωt kz + φ a ) Ẽ y = a cos(ωt kz + φ a + π 2 ) = a sin(ωt kz + φ a ) Squaring and adding yields Ẽ x 2 + Ẽ y 2 = a 2 (2.50) The tip of the Ẽ progresses clockwise around a circle of radius a in the x-y plane as shown. This is referred to as left-hand circular polarization because curling the fingers of the left hand in the direction in which the tip of Ẽ progresses results in the thumb pointing in the direction of propagation (z direction here). 16

If φ = π 2, (i.e. φ b = φ a π ), equation (2.50) again results but with 2 Ẽ y = + a sin(ωt kz + φ a ) The result is that the wave is right-hand circularly polarized because curling the fingers of the right hand in the direction in which the tip of E progresses results in the thumb pointing in the direction of propagation (z direction here). Elliptical Polarization For φ and A combinations other than those given above, the wave is generally elliptically polarized. Suppose that φ = π 2 and A = 0.5. Since, in this case, φ b = φ a + π, equation (2.48) becomes 2 Ẽ x = a cos(ωt kz + φ a ) Ẽ y = a 2 cos(ωt kz + φ a + π 2 ) = a 2 sin(ωt kz + φ a) Squaring and adding gives 2 2 Ẽ x Ẽ y a + 2 (a/2) = 1 (2.51) 2 which is the equation of an ellipse centred at the origin. A little thought given to Ẽ x and Ẽ y quickly reveals a left-hand elliptical polarization for this example. [NOTE: A more extensive treatment on polarization is given in Section 7-3 of the text.] The polarization of a particular plane wave will dictate the orientation of the receiving antenna which maximizes the received power. For example, for linear po- 17

larization, the receiving antenna should optimally be parallel to the polarization direction (and perpendicular to the direction of propagation). (What corresponding statement could be made for circular polarization?) 2.7 Plane Waves in Lossy (Dissipative) Media Let us now consider a region in space which has a finite conductivity, σ i.e. the medium is dissipative or lossy. Furthermore, we shall assume that the region of interest is source free. (A source current density, J0 say, could be of the conduction or convection type, but we will consider the observation point, P, to be placed where J 0 = 0). Because there is a finite conductivity throughout the region, we shall still have to include a conduction current density term ( J c = σẽ) in Ampere s law as was understood in the J of equation (2.16). That is, using complex vector forms for the time-harmonic fields, Ampere s law becomes Notice that the conduction and displacement currents are 90 out of phase. The above equation may be written as H [ = jω ɛ j σ ] Ẽ (2.52) ω Defining a complex permittivity, ɛ c, by (2.53) 18

where ɛ = ɛ and ɛ = σ, we have ω H = jωɛ c Ẽ (2.54) With reference to the diagram above, we note that the quantity σ/ωɛ, is referred as the loss tangent of the lossy medium and tan δ = σ ωɛ Maxwell s equations in complex vector form may now be written Ẽ = jωµ H H = jωɛ c Ẽ D = 0 (source free) B = 0 These equations have exactly the same form as the free-space equations, (2.29) (2.32), but ɛ has been replaced by its complex counterpart, ɛ c. We may repeat the analysis of Section 2.5 to obtain the following for planes waves travelling in dissipative media: Helmoltz Wave Equation: or 2 Ẽ γ 2 Ẽ = 0 (2.55) and γ is now a complex quantity given, in general, by γ 2 = ω 2 µɛ c = ω 2 µ(ɛ jɛ ) (2.56) We write γ = α + jβ (2.57) where α is called the attenuation constant (this is a good name, as shall be seen shortly) and β is the phase constant. It is straightforward to show that α = (2.58) 19

and β = (2.59) It is easily observed that for a lossless media, in which ɛ = 0, β is synonomous with the wavenumber k. The β unit is therefore rad/m. A solution to the Helmoltz equation given in (2.55) representing a plane wave having an E-field with only an x-component but which is travelling in the + z-direction analogous to our earlier work for free space may be verified to take the form Ẽ = ˆxE + x0 e γz = (2.60) Clearly, from the e αz factor, it may be observed that the wave decays or attenuates as it travels in the +z-direction, and hence the name assigned to α. The unit on α is the neper/metre (Np/m) where the neper is really a dimensionless descriptor added to the 1/m unit as a reminder that we are referring to the attenuation constant. (The neper is a poorly spelled form of the last name of John Napier, a Scottish mathematician who first proposed the use of logarithms.) The H-field is readily deduced from H = 1 η c (ˆk Ẽ) as H = ŷ E+ x0 η c e αz e jβz (2.61) with η c being the complex intrinsic impedance of the medium given by η c = This may be written in magnitude-phase form as µ ɛ c =. (2.62) η c = η c e jφ (2.63) where φ is the phase angle of this complex intrinsic impedance. 20

Time domain Fields: From equations (2.60), (2.61) and (2.63), conversion to the time domain is effected as follows: which yields E = Re { } { Ẽe jωt = Re E + x0 e αz e jβz e jωt} ˆx (2.64) For the H-field, (2.65) It is entirely possible that there may be a non-zero phase angle associated with the E-field as in (2.39). This would also appear in the H-field. Note that investigation of the arguments of the cosine in equations (2.64) or (2.65) leads to a phase velocity, u p, of (2.66) and in this analysis it is in the + ẑ direction. The wavelength is given by Skin Depth, δ s : λ = u p f = 2π β We noted above that, as the wave given by (2.64) or (2.65) travels into a lossy medium, it decays exponentially. The skin depth, δ s, is defined to be that distance, z, for which the Ẽ-field decays to 1/e of its value at some reference position which is here taken to be z = 0; i.e. which implies e αδs = e 1 δ s = 1 α (2.67) where α is to be determined from equation (2.58). 21

2.7.1 Special Cases of Lossy Media 1. Slightly Conducting Media A slightly conducting medium (a low-loss dielectric) is defined as one for which the loss tangent has the property Using the binomial series for x 1 tan δ = σ ωɛ = ɛ ɛ 1. (1+x) n = equation (2.56) may be written (with n = 1/2) as γ = ( ) ɛ Neglecting terms higher than first order in (such will be negligible by definition ɛ of slightly conducting ) we have [ ] γ jω µɛ 1 j ɛ = σ µ 2ɛ 2 ɛ + jω µɛ from which, to a good approximation, we identify α = σ 2 µ ɛ (2.68) β = ω µɛ (2.69) A similar approximation on the complex impedance leads to η c = (2.70) For a slightly conducting medium, we see from (2.67) and (2.68) that the wave will thus decay to 1/e of its original (z = 0) value at δ s = 1 α = 2 σ ɛ µ (2.71) For low-loss materials this depth is referred to as the penetration depth. 22

Example: Ice has a conductivity of about 10 6 /m and a relative permittivity of 3.2. Would a frequency of 4 MHz be useful for interrogation of the interior of a glacier? Initially, it may seem that, since δ s appears to be independent of frequency as long as the medium is slightly conducting, e-m waves exceeding a certain minimum frequency may be used to give any desired δ s. However, this ISN T so because effects, other than simple conduction, may become important. For example, ice contains entrapped air bubbles and, when the wavelengths of the interrogating signal ( GHz) become comparable to the bubble dimensions, e-m scattering effects become important and the conduction model breaks down. 2. Plane Waves in Highly Conducting Media We define a highly conducting medium (or a good ) conductor as one for which the loss tangent has the property From equation (2.58), tan δ = σ ωɛ = ɛ ɛ 1. α = (2.72) Also, from (2.59) β = (2.73) Therefore, for highly conducting media, to a good aproximation, α = β. 23

Additionally, for this case, it is easy to show from (2.62) that the intrinsic impedance may be approximated as πfµ η c = (1 + j) σ = (1 + j)α σ Finally, we note that the skin depth of a good conductor becomes δ s = 1 α = 1 2 = πfµσ σωµ (2.74) (2.75) Now, for these highly conducting media, δ s is very small and it is for this reason it is usualy referred to as the skin depth. This indicates that, for highly conducting surfaces, most of the conduction current will have to concentrate near the surface with very little flowing in the interior of the conductor. This is referred to as the skin effect. This skin effect has very important consequences for such applications as electromagnetic shielding. Read Section 7-5 of the Text to see the implication for current flow in good conductors. Extrapolating this idea to the perfect conductor where σ, equation (2.75) indicates that δ s 0 i.e. all the current must flow on the surface of a perfect conductor. Recall Ohm s law in point form : J = σ E. While in the perfect conductor, E 0, since σ, a finite current density may result, but this is on the conductor s surface. 3. Lossless Media A lossless or non-dissipative medium is similar to free space in that the conductivity, σ, of each is effectively zero. However, the former will, in general, have constitutive parameters different from the latter. That is, µ and ɛ, will replace µ 0 and ɛ 0, respectively. Furthermore, we see from equation (2.58) that the attenuation constant α = 0 and from (2.59) The phase velocity becomes β = ω µɛ (2.76) u p = ω β = 1 µɛ = c µr ɛ R (2.77) 24

Since σ = 0, the loss tangent becomes tan δ = 0 Finally, for the non-dissipative medium, the intrinsic impedance from equation (2.62) is real and is given by η = µ ɛ = µr ɛ R η 0. (2.78) This concludes our study of plane waves in unbounded media. In the next unit, we consider such waves impinging the boundary between media of differing electric and magnetic properties. Such a study will help provide further fundamental principles required for many electromagnetic applications such as two-wire transmission lines, optic fibres, and antennas. 25