Hertzian dipole formula
Hertzian dipole formula. A diagram of the short dipole antenna of length L appears in the next Dec 17, 2019 · The Hertzian Dipole. … Sep 12, 2022 · Figure 9. Wang) There are multiple approaches to solve this problem. g. ) Nov 1, 2000 · Abstract. As being used generally in numerous numerical calculations and models, Hertzian dipole's calculation show its importance. It is used as a standard to which all other antenna characteristics can be compared. 1: Current distribution of the electrically-short dipole (ESD). 479]. 7m² d) 1. 10. Regardless of the idea of reciprocity, it is often a bad idea to swap the transmitting and the receiving antennas of a system because a transmitter may be designed to handle much more power than a receiver [15, p. The electric and magnetic fields can be calculated as follows: (in Jun 17, 2017 · New asymptotic formulas for the electromagnetic field of a vertical Hertzian dipole above an imperfectly conducting half-space have been derived using two variants of the modified saddle point method, referred to as subtractive and multiplicative, with the latter formulation being slightly simpler. 212 i. 25* ), the impedance is half of that of a half-wave dipole, so Zin = 36. The vector potential A due to a Hertzian dipole, after substituting (34. Published: March 7 2011. Step 5: Calculate the radiation field in the absence of the corner reflector To find the radiation field in the absence of the corner reflector, we can use the formula for 1 day ago · Linear antennas, Hertzian dipole, standing-wave antennas, half-wave dipole, monopole antennas, traveling wave antennas, vee and rhombic antennas, loop antennas, circular and square loops, dipole and quadrupole radiation. You may draw a (2 marks) diagram to illustrate your answer (b) Calculate the radiation resistance of a Herzian dipole with a (3 marks) length of 0. 25) and Figure 10. 1) Without loss of generality, we have assumed the Hertzian dipole to be at the origin. Abstract —Owing to the principle of An N-Element Hertzian Dipole Phase Array x z y N-element Hertzian dipole array a ejα ej2α ej(N−1)α • Current magnitude is the same for all the dipoles • Current phase difference between successive dipoles is α 1 j α m m e I I +1 = () () ()() 1 2 0 2 sin cos 1 0 ˆ . 1: Effective aperture of a half-wave dipole. Obviously this sightly displaced dipole will generate both dipole and quadrupole Jun 6, 2022 · Here, let us focus on the canonical example of a nonstationary Hertzian dipole. This higher value is not for a physically-realizable ESD, but rather for the Hertzian dipole (sometimes also referred to as an “ideal dipole”). The Hertzian dipole or elementary doublet refers to a theoretical construction, rather than a physical antenna design: It is an idealized tiny segment of conductor carrying a RF current with constant amplitude and direction along its entire (short) length; a real antenna can be modeled as the combination of many Hertzian dipoles laid end-to-end. l = L / 2. Far-fields of a localized source, contributing to its electric dipole radiation. 17m² c) 1. Complete Step By Step Answer: The Hertzian dipole is a dipole antenna. 11. 38), and the vector potential A (r, t), equation (20. Field analysis Vector potential Vector potential approach 3. The dielectric spherical nanoparticle of radius a = 70 nm and relative permittivity 15 is located at distance G from the dipole. where: L — the length of the total dipole antenna in feet (ft); l — the length of the arm of the dipole antenna in feet (ft); and. This can be understood since only half the voltage is required to drive a monopole antenna to Step 1/6 For a Hertzian dipole: Step 2/6 1. 5. 2m² Mar 27, 2023 · Power radiated by hertz dipoleHello students, Welcome to our YouTube Channel RTU Wallah. The full-wavelength dipole antenna is more directional than the shorter quarter-wavelength dipole antenna. ) The Hertzian dipole presented earlier has a directivity of 1. 2. Hence, the half-wavelength dipole antenna's length is 0. 3. (a) Explain briefly what is a Hertzian dipole. Sep 12, 2022 · Figure 9. May 2, 2023 · The maximum radiated power density at a distance of 1 km from a center-fed Hertzian dipole can be determined using the formula: Pdmax = (30 * Pi^2 * i0^2 * L^2) / λ^2 * R^2. 18: Radiation from Apertures Radiation from apertures and diffraction. Example Calculation. 1) In the above, we can let the time-harmonic current I= dq=dt= j!q, then Il= j!ql= j!p (34. The current amplitude in the dipole is fixed and denoted by I (we consider the time-harmonic regime, assuming ej!t time dependence). r is the distance from the antenna (in meters) domain, through which a compact asymptotic formula describes the behavior of the EM field, which emanates from a vertical Hertzian radiating dipole, located above flat, lossy ground. 3]. $ Power Gain\left ( dB \right)=10\times \log \left ( power gain \right) $. kuehn@aurinovo. 3, we obtain (10. 1. The dipole length is l. derivation of V from A using Lorentz Gauge 3. Hertzian dipole 1. Let us consider a Hertzian dipole antenna located in an infinite homogeneous space, as illustrated in Fig. direction of the current and also the orientation of the dipole in space. 6. 3 THE HERTZIAN DIPOLE The Hertzian or infinitesimal dipole is a very short length of thin wire which carries a current 1= 10 exp(jwt) that is uniform along its length. ii. Consider the following thought experiment: we represent a very slightly displaced dipole as the original dipole at the origin plus a quadrupole consisting of the displaced dipole and the dipole at the origin subtracted. 4. e. Oct 17, 2016 · The Hertzian dipole is fundamental to the understanding of dipole radiation. antenna array, we assume a simple linear dipole array as shown in Figure 27. The principal stresses occur on the z-axis and can be determined from the following equations: Equation 3: Undefined control sequence \taninv. The length of the total wire, which is being used as a dipole, equals half of the wavelength (i. (1078) Suppose that the wire is aligned along the -axis, and extends from to . First, the expressions for the scalar potential Ø (r, t), equation (20. 76 dB), owing to the lack of energy transmitted in the z-direction. Calculate the power density for a target at a range of 3 km and elevation of 15o above the x􀀀y plane of the antenna. Aug 10, 2015 · The quantum source is modeled as a Hertzian dipole of length 10 nm. This means they would lag by a phase Contact of bodies occurs in all interfaces that transmit force or motion, or both. Equation 4: σz = −Pmax 1 + z2 a2. We model a Hertzian dipole as a short wire antenna of length l, assuming that the current is Determine the radiation efficiency of a center-fed dipole of length 1. Thus, in order to define a finite physically meaningful quantity Antenna Theory - Half-Wave Dipole. dipole moment on the small particle. Directly substituting values, we have. College of Engineering - Purdue University Mar 5, 2022 · The effective length d eff of a short dipole is approximately half its physical length [see (10. Wang) This is the distribution expected on a thin straight wire having length L = λ/2 L = λ / 2, where λ λ is wavelength. 131λ2 (half-wave dipole, max. Figure 3. Such an antenna is called as half-wave dipole antenna. Equations for the electric and magnetic fields of such a dipole can be derived mathematically. The energy density is. Ste ffen K. Equation 2: Pmax = 3F 2πa2. 1), is A(r) = 4ˇ V dr0 J(r0) jr r0j ej jrr 0j Need genetric algorithm to randomly vary vector so its metric can be computed. In the far eld of the Hertzian dipole, we can start with E = j! Ar = j! A 1 j! " rrA (34. Determine the radiated power when the direction is (2 marks) along the axis of the dipole, ie, when θ 0°. 8) The magnetic field due to the Hertzian dipole is then given by or (10. Normalized radiation patterns for dipole antennas of specified length. This object can then be used to plot the electric field and radiation pattern of the dipole. The latter characteristic is Dec 1, 2015 · New asymptotic formulas for the electromagnetic fields of a vertical Hertzian dipole over an imperfectly conducting half-space are derived using two variants of the modified saddle point method Apr 6, 2023 · Interpretation of the solution of Maxwell’ s. April 6, 2023. Although its radiation characteristics are some what less than desirable it can be realised in practice, it can play a useful role as a viable antenna in situations where space is limited, and (conveniently) its fields can be expressed in closed form. For half-wave dipole, set L = λ/2 or kl = π. Figure 9. Generation and reception Animated diagram of a half-wave dipole antenna receiving a radio wave. E and H fields 1. Elliptical contact is also in the category Jul 30, 2023 · Interpretation of the solution of Maxwell’ s. Fig. The modern term "radio wave" replaced the original name "Hertzian wave" around 1912. 07m² b) 0. The power density at a distance r from a Hertzian dipole is given by the formula: Pd = (Pr * G) / (4 * pi * r^2) where Pr is the radiated power, G is the antenna gain (which is 1 for a Hertzian dipole), and pi is the mathematical constant pi. uhn. ste ff en. The algorithm hill-climbs efficiently toward optimum design, especially if the antenna is not too reactive. Antenna Engineering CHO, Yong Heui Analysis of Hertzian Dipole 2. (d) The power density of a Hertzian dipole is given by the following formula. Hence, we expect the half-wave dipole to exhibit slightly more directivity than its Hertzian counterpart. , l = λ/2). We first demonstrate an efficient technique for the accurate numerical calculation of the well-known Sommerfeld integrals. Using a standard Cartesian coordinate system, a Hertzian dipole at the origin is depicted as: This type of antenna is called a Hertzian dipole, after the German physicist Heinrich Hertz. . The plot depicts the time evolution of the lines of force of the electric field around a Hertzian dipole It straightforwardly visualizes the function see Details for further informationThus it recomputes Hertzs famous figures that he had created for several instants of time in the The magnetic vector potential ˜A is a vector field, defined by Equation 9. 15); it results primarily from the energy stored in the near fields. Electric Current on a half-wave dipole antenna. Where the phasor The current density at the origin seen by the The radiation resistance of an “electrically-short dipole” is sometimes said to be 80π2(L/λ)2 80 π 2 ( L / λ) 2; i. 25 meters and an operating frequency of 600 MHz, the radiation resistances for different types of dipoles are calculated and presented. E = magnitude of field strength (μs/m) Sep 30, 2015 · The Sommerfeld halfspace problem is revisited allowing for multilayer and plasmonic media, with a focus on the surface field computation. where, Cn is a constant chosen to make maximum of g (θ) equal to unity and θ is the polar angle in the spherical coordinate system. From the formula above, since , we are just left with. The vector potential A due to a Hertzian The general radiation formula for a Hertzian dipole immersed in an isotropic dissipative medium of infinite extent has been derived. 1: Illustration of power radiating from an isotropic antenna. Wang) The two characteristics that define the ESD are (1) the current is aligned along a straight line, and (2) the length L L of the line is much less than one-half of a wavelength; i. The Hertzian Dipole Antenna. Electrical Engineering questions and answers. Here’s an example: Example 10. Thus, in order to define a finite physically meaningful quantity The far-fields from a dipole antenna of length L are given by: The normalized radiation patterns for dipole antennas of various lengths are shown in Figure 3. Jun 2, 2023 · The simplest source of electromagnetic waves in three dimensions is the Hertzian dipole. 5 (m) operating at 100 (M H z) 100(\mathrm{MHz}) 100 (MHz). 25 Ohms. Sep 21, 2020 · #infinitesimaldipole#HertzianDipole#Hertzian#antenna The Electric Field for Hertzian Dipole formula states that the electric field strength is directly proportional to the magnetic field strength, with the intrinsic impedance of the medium acting as the proportionality constant and is represented as E Φ = η*H Φ or Electric Field Component = Intrinsic Impedance*Magnetic Field Component. RTU Wallah channel covers subjects specifically designed for Rajastha Apr 27, 2024 · Where \(L_{eff}\) is the effective length of the dipole, and \(\lambda\) is the wavelength of the operating frequency. The electromagnetic fields of a Hertzian dipole can be analyzed using the electrical Hertz vector: where is the dipole moment (a vector in the direction in this analysis), and is the distance to the observation point. The Hertzian dipole can be thought of as being composed of two point charges with charge quantities +q and −q, both moving in proximity to the coordinate origin. So the absolute size of the above dipole antenna does not matter, only the May 17, 2024 · The formula to convert the power gain to decibel is. 32Tr2 (2 marks) i. 3. Metrics and vector descriptions favoring simplicity bias the design accordingly. The unit of frequency, cycle per second, was named the Nov 24, 2018 · The Elementary Doublet (Hertzian Dipole): The doublet is a theoretical antenna shorter than a wavelength (Figure 9-3a). The general radiation formula for a Hertzian dipole immersed in an isotropic dissipative medium of infinite extent has been derived. 0012, where is the wavelength in free space Use the given formula Hint: In free space, the intrinsic impedance, η 120π 2T (1 (c) Suggest which type of antenna (omnidirectional, sectorial or directive A Hertzian dipole can be approximated by a small current source so that J(r) = ^zIl (r) (34. It provides basic insights into the genesis of electromagnetic waves and lays the groundwork for an understanding of half-wave antennae and other types. 5 meters. 1: A Hertzian dipole located at the origin, represented as a current moment. 1 4. The electrically-thin half-wave dipole exhibits radiation resistance ≅ 73 Ω and effective length λ / π. 1. Heinrich Rudolf Hertz ( / hɜːrts / HURTS; German: [ˈhaɪnʁɪç ˈhɛʁts]; [1] [2] 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell 's equations of electromagnetism. do ds Is jR S = e− β z Ja i(t) =Ito cos(ω+α) j IIeos = α The stored charge at the ends resembles an electric dipole, and the short line of oscillating current is then referred to as a Hertzian Dipole. The theoretical May 30, 2015 · Expand. , 4 times the right side of Equation 10. In fact, the HPBW of a Hertzian dipole is 90 , while that of a half-wave dipole is only 78 . The dipole is made of brass and has a radius of 1 (mm). 1: A potential is induced at the terminals of a thin straight dipole in response to an incident plane wave. A Hertzian dipole is a dipole which is much smaller than the wavelength under consideration so that we can approximate it by a point current distribution, mathematically given by [31,38] J(r) = zIl ^ (r) (25. A direct attack is to invoke the principle that potential is equal to the integral of the electric field intensity over a path. The understanding of this basic scatterer can be extended to small time-varying inclusions which have electric response. Yields rats-nest configurations that perform well. Two Hertzian Dipoles on X-Axis – Some Physical Reasoning J1 r J2 r y x lcos()φ φ 1 =I I =I Aejα 2 When traveling in the direction φ: • The waves from dipole 2 have a phase lead of α compared to those from dipole 1 • But the waves from dipole 2 travel a distance ℓcos(φ) more than from dipole 1. The reactance X of a short dipole antenna can be found using (10. The dipole antenna is cut and bent for effective radiation. 2 days ago · The antenna length formulas used in our dipole length calculator are as follows: L = 468 / f. Assume the current on the dipole is 1 A. • Gain and radiation pattern of an antenna. • A Hertzian dipole is represented by an arrow whose direction indicates the positive. , a simple antenna, is mathematically Graßmann formula, which is perfectly appropriate for electro-and magnetostatics, retains its validity. Where Pdmax is the maximum radiated power density, i0 is the current through the dipole, L is the length of the dipole, λ is the wavelength , and R is the distance from Radio communication began to be used commercially around 1900. The current can then be described mathematically as follows: J(r0) = ^zIl[A 0 (x0) + A 1 (x0 d 1) + A Abstract: An explicit formula is derived for the change of input impedance of a vertical electric dipole as a function of the distance of the dipole to a horizontal boundary with a prescribed surface impedance n. 85 inches if you don't like the metric system). 13) Once again looking back at Figure \(\PageIndex{1}\), note that each Hertzian dipole representing current on one conductor has an associated Hertzian dipole representing the current on the other conductor, and is only a tiny fraction of a wavelength distant. f — the desired frequency. A potential source of confusion is that the Hertzian dipole is also a “dipole” which is “electrically-short. Dec 7, 2014 · 1. It is simply an open-circuited wire, fed at its center as shown in Figure 1. 12 for ˜A and then use Equation 9. 7. The analysis of the resulting formula breaks new scientific ground and indicates that Maxwell's equations predict that electromagnetic waves in vacuum propagate at the speed of light, notably for each receiver, even when these receivers have relative velocities with Figure 4. This distribution is described mathematically as follows: the Hertzian dipole. Recently, such a calculation was first performed for the Hertzian dipole. 1 Hertzian Dipole 679 Expressing A in (10. 7. e. 0; C. First, without loss of generality and for simplicity to elucidate the physics, we assume that this is a linear array of point Hertzian dipoles aligned on the xaxis. 10)-b2 sin The simplest polarized antenna is the dipole antenna which consist of a split length of wire coupled at the middle to a coaxial cable. Alternative field representations are discussed and an improved Sommerfeld-Norton groundwave formula is developed. Figure 1. Other practical wire antennas, such as the half-wave dipole, can be analysed as a number of short elements, each of Hertzian dipole type and with an appropriate excitation, the Aug 6, 2021 · The more difficult part is the translation. The half-wave dipole antenna is as you may expect, a simple half-wavelength wire fed at the center as shown in Figure 1: Figure 1. The paper is divided into two parts. Ch. The midpoint q and –q is called the centre of the dipole. 2) where the dipole moment p= ql. When the conductivity of the medium is finite, the total radiating power is found to be infinite. The magnetic vector potential generated by a current distribution is given by the well-known formula (see Sect. 4 0. Furthermore, these pairs of Hertzian dipoles are identical in magnitude but opposite in Mar 7, 2011 · Wolfram Demonstrations Project. Now, we want to consider an application of the Liénard-Wiechert potentials, namely, the oscillating electric dipole with the dipole moment p = p (t), first investigated by Heinrich Hertz. equations for a moving Hertzian dipole. 6 0. This condition is rarely and only approximately seen in practice, whereas the triangular magnitude distribution is a relatively good Oct 12, 2016 · An antenna that radiated equally well in all directions would have a directivity of 1 (0 dB. Also: To determine the electromagnetic fields radiated by a current distribution ˜J, one may solve Equation 9. 1: Current distribution of the half-wave dipole (HWD). 6, that is able to represent both the electric and magnetic fields simultaneously. derivation of E and B using A and V The following diagram presents an electric dipole antenna: [𝑅 𝑎 𝑉] [𝑅 𝑎 𝐴] Sep 12, 2022 · The radiation resistance of an “electrically-short dipole” is sometimes said to be 80π2(L/λ)2 80 π 2 ( L / λ) 2; i. The field strength of this antenna can be calculated as follows: Where. Wang) and is given by: Δ˜J(r) = ˆl ˜I Δl δ(r) where the product ˜IΔl (SI base units of A ⋅ m) is the current moment, ˆl is the direction of current flow, and δ(r) is the volumetric sampling function Oct 28, 2013 · The maximum power density radiated by a Hertzian dipole is given by. It is defined as a line (electric) current in a wire of infinitesimal length and diameter. 12 ) (1077) where. 13. The short dipole antenna is the simplest of all antennas. ( CC BY-SA 4. 25 meters (or 9. , L ≪ λ/2 L ≪ λ / 2. Dec 20, 2016 · Discussion of the basic Hertzian dipole solution to the radiation equations, introducing basic concepts of radiation impedance, directivity, gain, and other Mar 17, 2013 · In Matlab, a Hertzian Dipole can be created by defining its parameters such as length, position, and orientation, and then using the "dipole" function to generate a dipole object. 39), are noted again: A Hertzian dipole can be approximated by a small current source so that J(r) = ^zIl (r) (34. 5 + j21. For the Hertzian dipole, the dependence indicates that the radiation is maximum in the broadside direction (i. Hertzian Dipole Let us consider a short line of current placed along the z-axis. ), corresponding to the azimuth plane. The most important feature of this result is that the time-dependent field decreases very slowly (only as \ 1 / r) with the distance from the source, so that the radial component of the corresponding Poynting vector (7. 2). 8 1 30 210 60 240 90 270 120 300 150 330 180 0 Let’s evaluate the directivity and input impedance of the half Question: (d) The power density of a Hertzian dipole can be expressed as the formula below:- 1212 -nB sin? a. Short dipole antenna of length L. 5) in terms of its components in spherical coordinates, as shown in Fig. Antennas can be used in different applications based on their directivity: 10. As a boundary condition of the source, it is assumed that the dipole moment is a given quantity. In this case, ˆI = ˆz. By default, the direction of electric dipole in space is always from negative charge -q to positive charge q. 1) The dipole may look like the following schematically. de. The ideal/Hertzian/elemental dipole is the simplest radiative element studied as the basis for more complex antenna constructions. Ideal (Hertzian) Dipole Page 5 Strength factor { determined solely by material parameters, magnitude of excitation current, and dipole length Distance factor { purely the amplitude decay and phase shift incurred with distance Shape factor { determined the radiation pattern of the antenna, or the part that is a function of ;˚. ,, 0 ⇒ = ∑ = =∑ − = + − = − = N m j k a m N moving Hertzian dipole, i. 9b), 8. The simplest example of an electric dipole is a pair of electric charges of Hertzian dipole; Near and far field of the Hertzian dipole; Near field; Field lines of the Hertzian dipole; electrodynamics; Antenna parameters; Energy flow of the Hertzian dipole; First and second Maxwell´s equation; Ampère´s circuital law; Equations of Potential; Electromagnetic Field Theory Sep 4, 2012 · Snapshot 3: a DC electric field. As a boundary condition of the source, it is assumed Figure 10. ” The distinction is that the current comprising a Hertzian dipole is constant over its length. Sep 14, 2021 · Defining k = 2π/λ as the wave number and h = L/2 as the half-length of the antenna, the current distribution and the normalized power gain g (θ) are given by. Elliptical contact is the general case of the contact of surfaces of arbitrary shapes, and the circular contact of spherical surfaces and line contact of cylindrical surfaces are two special cases of elliptical contact. Gain for an isotropic antenna is $ 1 $ . Sketch the approximate shape for the radiated power (3 marks) pattern in the x-z plane. For a dipole antenna with a physical length of 0. 2 0. An electric dipole is defined as a couple of opposite charges q and –q separated by a distance d. From the equation above, what is the value for radiated power when the direction is along the axis of the dipole, ie, when A=0°. Lecture 29. What is the effective aperture of Hertzian dipole antenna operating at frequency 100 MHz? a) 1. 8. For a quarter-wave monopole (L=0. May 22, 2022 · 1. sin cos 1 0. Abstract —Owing to the principle of Nov 8, 2004 · In some recent publications, King (1990, 1992) and King and Sandler (1994) have provided formulas for the electromagnetic field radiated by an infinitesimal vertical Hertzian dipole above a lossy homogeneous half space (earth) and above a lossy half space coated by a thin dielectric layer. For simplicity, we assume that the medium is non-magnetic ( = 0), which does not May 21, 2024 · 1 Consider a radar system employing a Hertzian dipole of length 0:1 m operating at a frequency of 100 MHz. 1 9. The simplest dipole, from a mathematical perspective, is the Hertzian dipole, in which the length of wire is much shorter than a wavelength. The words "short" or "small" in antenna engineering always imply "relative to a wavelength". We also used the simple wavelength formula: Wavelength = c / f. The maximum shear stress in any of the two spheres can be determined from: Equation 5: τmax = σx–σz 2 = σy–σz 2. Ground plane. 5 yields: Ae ≈ 0. Now, having the tools to examine the Hertzian dipole, we will follow this sequence: 1. Hertzian Dipole Antenna - Review. The antenna consists of two metal rods connected to a receiver R. May 6, 2023 · For a center-fed Hertzian dipole, the maximum radiated power can be calculated using the formula: Pradmax = (I^2 * l^2 * pi * eta) / (8 * lambda^2) where I is the current, l is the length of the dipole, lambda is the wavelength, and eta is the impedance of free space. Assuming the dipole is lossless and in free space, Equation 10. Then Il= j!ql= j!p (34. 4. In the above, we let the time-harmonic current I= dq=dt= j!q. May 12, 2023 · To determine the maximum radiated power density of a center-fed Hertzian dipole at a distance of 1 km, we can use the following formula: Pd = (Pr * G) / (4πr²) Where: Pd is the power density (in watts per square meter) Pr is the radiated power (in watts) G is the gain of the antenna. 0. derivation of A 2. If the incident eld is time-harmonic, the the small dipole will be time-oscillating and it will radiate like a time-varying Hertzian dipole whose far eld is quite di erent from its near eld (see Section 25. In this lecture you will learn: • Hertzian dipole antenna. Several authors have questioned the accuracy of the King-Sandler formulas, particularly for field points One wavelength at 600 MHz is = c / f = 0. It is shown that, if n \ll 120\pi, the result is consistent with the formula obtained by the compensation-theorem approach. Jan 24, 2024 · When a Hertzian dipole is placed near a corner reflector, the reflected waves interfere with the waves radiated by the dipole, resulting in changes to the radiation field. This is the most widely used antenna because of its advantages. This paper begin with the very basic level to deduct the equations of a Hertzian dipole fed by a current in time-varying fields in spherical co-ordinates, and then a simulation in Matlab helps understanding how The only change that needs to be noted is that the impedance of a monopole antenna is one half of that of a full dipole antenna. 4 10. 5 (1. 9) Using Maxwell’s curl equation for H with J set equal to zero in view of perfect dielectric medium, we then have or (10. The role of the Zenneck wave and surface plasmon polariton in the surface field and their relationship is elucidated. 6 to determine ˜H and subsequently ˜E. iq dx mo rk as ox xe wb hi wn