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(krt) (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 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 , , 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 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(zh)  e(zh)F(), z  h The Hertz potential satisfies the equation:  0   (14)  0    2  z   prim  sec        2 2   (zh)  (zh)   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:   zh 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 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. Thus we have: 0 J  eˆ x p (x) (y) (z  h) (30)   2  k 2 2 F  1 , F  (22) where p = Il is proportional to the of   E 2      E  kE   E the antenna. The boundary conditions in terms of the Hertz potential for the vertical and horizontal dipoles are the same as those of Sommerfeld. e jkR e jkR  d     2 J (r)e (zh) Banos points out that the formalism can be expressed in x  0 R R    E 0 (23) terms of a triple Fourier transform. Apparent differences 2  reflect the choice of coordinate systems in the spatial domain, k  zh d   2 J (r)e E Ex k 2  0    the transform domain or both. As we have seen, Sommerfeld’s E 0 E model is based on cylindrical coordinate systems in both the where R is the distance to the image of the dipole in the spatial domain and the transform domain. ground. His treatment focuses on the Green’s function, G. for a unit For the z-component of the Hertz potential, we set the axis electric dipole moment, this is given by: of a cylindrical coordinate system in the z-direction. The wave jkR e equation is a Helmholtz equation: G  , (31) 2 2 R (  k ) z  0 (24) which is a solution of the inhomogeneous Helmholtz equation: This has solutions in cylindrical coordinates of the form (see (2  k 2 )G  4(x) (y) (z) (32) Mathews and Walker [3], or any standard text on differential In the case of the horizontal dipole, Banos indicates the equations, including [1]): z jm following relations:  z  J m (r)e e (25) Π  eˆ r  x cos  eˆ   x sin  eˆ z  z

 2 From the boundary conditions, Sommerfeld shows that the Er  (  Π)  k  x cos azimuthal factor is cosϕ; therefore the eigenfunctions now r 1  2 involve J1 rather than J0. The primary field does not affect z E  (  Π)  k  x sin because it arises only from currents parallel to the x-axis. On r  the other hand, the currents in the ground are created by the  E  (  Π)  k 2 (33) changing , which induces currents in the z z z x-direction. The radial electric field due to the wire also 2 jk   x 1  z  produces currents. These currents have a component in the z- H r   sin     z r   direction. 0   jk 2     Sommerfeld’s results are: H   cos x  z    2    0  z r     cos J (r)e (zh) E 2 d, z  0 z k 2  1 k 2 / k 2   2 0 E E (26) jk   x   H z  sin  2  zh     r    cos J (r)e E E 2 d, z  0 0   z k 2  1 k 2 / k 2   E 0 E E Results for the electric and magnetic fields in the air and ground are found in terms of various integrals related to Green’s functions for Hertzian dipoles located at different III. BANOS THEORY heights in each of the two media. There is also a discussion on Banos [2] also treats the Hertzian dipole and uses a time- choices of branch cuts and the handling of poles in the frequency factor equal to e-jt. He notes that, as well as complex plane; this is useful when explicit results are needed. Maxwell’s equations, the problem requires the continuity Most of Banos’s book is devoted to saddle point methods equation: and their utility in deriving results in particular cases. Saddle  J  j (27) point methods are instructive in that they offer physical insight into how fields propagate. For example the surface waves of where  is the . He introduces the intrinsic Sommerfeld emerge naturally from this type of treatment. impedance, , and admittance, , of the medium; these are However, it is equally obvious that practical problems must defined by: often be sub-divided into various categories, such as the field k  , k   j (28) near the interface and the field near the vertical axis. This The inhomogeneous Helmholtz equation for the Hertz renders the treatment of long antennas rather difficult and potential is shown to be: suggests that numerical methods are much simpler. Therefore (2  k 2 )Π   jJ0 / k (29) saddle point methods will not be pursued here. where J0 is the impressed . For a horizontal dipole on the x-axis we have: LRDC 2012-01-18-001 4

IV. WAIT THEORY involving the detection of command wires: Contract number Wait provides a useful summary of the basic W7702-125254/001/EDM. electromagnetic theory in the appendix of his book [4]. This begins with the usual theory of radiation from an elementary REFERENCES impressed source of current. The Hertz vector is introduced [1] A. Sommerfeld, Partial Differential Equations in Physics, Volume VI, and it is asserted that, in free space, only one component of the Academic Press, London, 1964. [2] A. Banos Jr., Dipole Radiation in the Presence of a Conducting Half- Hertz potential is required and this is parallel to the impressed Space, Pergamon Press, Oxford, 1966. current on the z-axis, say. This is reasonably obvious because [3] J. Mathews and R.L. Walker, Mathematical Methods of Physics, 2nd we know that the magnetic field lines are all perpendicular to Ed., W.A. Benjamin Inc., New York, 1970. the radial and axial vectors and circle the z-axis. Only the curl [4] J.R. Wait, Electromagnetic Radiation from Cylindrical Structures, Pergamon Press, New York, 1959. of the z-component of the Hertz vector is purely in the azimuthal direction, so that this is all that is required. It should be noted that, in contrast to the other two authors, Wait employs a time dependence of the form ejt. The concept of magnetic current, M, is discussed by exploiting the symmetry between electric and magnetic vector equations, viz:  E  ( j H  M) 0 (34)  H  (  j0 )E  J Here J and M are impressed currents, though the latter is fictitious. This leads directly to the introduction of the magnetic Hertz vector, *, which is given by: 1 e jkR Π*   Mdv (35) 4j0 V R

Wait then considers the radiation from a small current loop using the previous type of approach with the usual Hertz vector (by integrating around the current loop). He shows that the result can equally be derived using the magnetic Hertz potential:  jkR * Kl e  z  (36) 4j0 R where

Kl  j0IS (37) Here, K is the magnetic current of an element of length l and IS is the of the loop. He then gives an example of the radiation from a solenoid, which can be represented by a magnetic current of finite length. Wait also provides insight into the fields of line sources in a homogeneous medium with no interface nearby. However, this is similar to the treatments in other standard texts. He shows that the fields are proportional to modified Bessel functions of the second kind (which are proportional in turn to Hankel functions). In later papers, Wait shows that the magnetic Hertz vector is a useful tool for representing circulating currents when an infinite line source is parallel to the interface with a conducting medium.

ACKNOWLEDGMENT This report forms a part of the support for a Defence Research and Development Canada (DRDC Suffield) contract