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A Summary of EM Theory for Dipole Fields Near a Conducting Half-Space

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A Summary of EM Theory for Dipole Fields Near a Conducting Half-Space LRDC 2012-01-18-001 1 A Summary of EM Theory for Dipole Fields near a Conducting Half-Space James K.E. Tunaley The basic theory is actually fairly straightforward but is Abstract—Estimates of the electric and magnetic fields arising algebraically quite tedious especially when matching the from an impressed current source above or within the ground are boundary conditions at an interface. important for estimating the detectability of wires. Important contributions to the theory have been made by Sommerfeld, II. SOMMERFELD THEORY Banos, Wait and King. However, some of the basic concepts are described in books that are over 50 years old and are becoming Sommerfeld deals with radio propagation over the ground in difficult to access. Therefore the concepts, as described by the chapter 6 of his book on partial differential equations [1]. His first three authors, are summarized. treatment applies to the Hertzian dipole (comprising two very short elements of constant current) and is based on the Hertz Index Terms—Electomagnetic Propagation, Conducting Half- Space, Buried Wire Detection. potential, . In the present context we consider an oscillatory excitation so that, apart from a constant of proportionality, this I. INTRODUCTION potential can be identified as the magnetic vector potential, A. The solutions of the electrodynamics equations for an HE application of electromagnetic theory to estimate the oscillating charge in free space are assumed to be of the form: T fields from conducting wires close to or beneath the 1 ground is important for the detection of Improvised Explosive e j(krt) (1) Devices (IEDs), which are often triggered by command wires. r The command wires can be excited by external radio and the fields can be obtained by differentiation. The actual frequency signals and detected by small changes in the relation between the two potentials is: j scattered fields close to the wire. Π A (2) The theoretical basis is usually that adopted by Sommerfeld k 2 in his pioneering work on the propagation of radio broadcasts In the presence of a conducting ground, the general concept over the conducting earth. The finite conductivity of the earth is to treat the Hertz potential as the sum of two parts. The first is important because the propagating waves induce currents in part represents the field created by the elementary current in the earth. These create a component of the electric field the dipole in free space without the presence of the ground. parallel to the ground and permit the transmission of signals The second can be regarded as the contribution due to the over much greater distances than if the earth were infinitely ground and includes the effects of induced currents. The form conducting. of the second part is derived by matching the tangential Mainly two groups of researchers have been involved with electric and magnetic fields at all points across the plane the theory of fields from horizontal wires near a conducting boundary. These are initially unknown and the key to the interface. These groups were headed by James Wait and solution is to express the two Hertz potentials as Fourier- Ronold King. There were also several important independent Bessel transforms both of which are in the form of an integral. workers, such as Carson and Banos: here I shall focus It then proves to be possible to apply the boundary conditions primarily on the theoretical basis using the results of in a straightforward manner. Sommerfeld, Banos and Wait. Their papers often refer back to Sommerfeld shows that: early work and the details of basic concepts are not repeated in E k 2Π Π recent literature. k 2 (3) Unfortunately much of the early theory is located in books, H Π which are becoming difficult to obtain by the independent 0 j researcher, and that is the reason for this summary. I also add These relations are similar to those involving the magnetic explanatory comments where I feel it is appropriate. vector potential in which the Lorentz gauge condition is invoked to specify A in terms of an electric potential, , i.e. J.K.E. Tunaley is President of London Research and Development j A (4) Corporation, Ottawa, ON, and Prof. Emeritus, Physics Dept., University of c 2 Western Ontario, London. Email: [email protected] As with A, both the electric and magnetic fields can be derived Published January 20th, 2012. LRDC 2012-01-18-001 2 from just the Hertz potential and the electric potential is not When a vertical dipole is located above a conducting earth, required. we augment the permittivity in the ground by a complex In general, the angular wave number, k, is given by: conductivity. Thus we have two media and in the conducting k 2 2 j (5) medium, currents are induced that give rise to secondary fields. In air, the primary potential is given by (13) and there is where ε and are the actual permittivity and permeability of a secondary potential from the ground: the medium and σ is the conductivity. J (r)d e(zh) e(zh)F(), z h The Hertz potential satisfies the equation: 0 (14) 0 2 z prim sec 2 2 (zh) (zh) Π Π k Π (6) J (r)d e e F(), h z 0 2 0 t t 0 When this wave equation is written out in cylindrical In the earth, there is no primary potential and we have: zh coordinates and /ϕ = 0 (such as for the vertical dipole), we J (r)d e E F (), 2 k 2 , 0 z (15) z E 0 E E E have for the z-component of the Hertz potential: 0 -h 2 2 Notice that the terms F and FE have been multiplied by e . 1 2 k Π z 0 (7) This is simply for convenience ( is a function of so that the r 2 r r z 2 factor can be absorbed into F’s). The eigenfunctions, u, can be determined by a separation of To determine F and FE, the tangential components of E and variables and so we have: H must be continuous across the air-ground interface. For the 2 2 2 u J 0 (r)cos(z), k (8) vertical dipole, Sommerfeld shows that the continuity where is no longer the permeability but is given by: conditions at z = 0 are: m k 2 (9) E , E (16) h z z k 2 E and h is the length of the cylinder on which the normal These lead immediately to: derivative is zero; m is an integer. The length of the cylinder is 2 kE now allowed to become infinite so that we have a continuous F E FE , F 2 FE (17) k distribution of eigenfunctions and: Solving the two equations yields: z 2 2 z F()J 0 (r)e d, k (10) 2 E 2 F 1 , FE (18) 0 2 2 2 2 kE / k E kE / k E where the cosine is replaced by exponentials. The function F Sommerfeld demonstrates that, when the ground becomes has to be determined. highly conducting, the Hertz potential reduces to that predicted For a source located at z = 0, the Hertz potential is given by: by image theory. jkR e 2 2 2 For the horizontal dipole, the calculation is more z , R r z (11) R complicated because, according to Sommerfeld, two By performing an inverse transformation, Sommerfeld then components of the Hertz potential must be considered. The shows that, for this case, the function, F, is given by: dipole lies in the x-direction and the two components are x F() (12) and z. This choice can be justified by the fact that it is possible to satisfy the boundary conditions and because of the and therefore: uniqueness of the solutions to the wave equation. The boundary conditions for the electric field become: |z| d 1 (1) |z| d z J 0 (r)e H 0 (r)e (13) 2 2 Π ΠE , k x kE Ex (19) 0 2 The boundary conditions for the magnetic field become: The parameter, , which plays the role of a propagation 2 2 constant in the z-direction, is ambiguous because of the square k E x k E Ex z Ez , (20) root; the sign with a positive real part must be taken. k 2 z k 2 z When an antenna is located above an infinitely conducting and, combining all these, gives for the two components: earth, the effect of boundary conditions at the earth surface can k 2 k 2 E , x E Ex be represented by images. The image of a vertical dipole is x k 2 Ex z k 2 z another vertical dipole with its polarity in the same direction as (21) k 2 the actual dipole. Therefore, if the actual dipole is located very E z Ez Ex x z 2 Ez , close to the surface, the Hertz potential is doubled. In contrast, k z z x x the polarity of the image of a horizontal dipole is opposite to Since the dipole is aligned along the x-direction, we can that of the actual dipole. When a horizontal dipole is placed consider that a cylindrical coordinate axis is also in this close to the surface, cancellation occurs; the system and its direction and the wave equation is similar to (7) with x and z fields can be represented by a quadrupole if needed. interchanged. The final result for x can be determined in an LRDC 2012-01-18-001 3 analogous way to the vertical dipole.
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