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[1] We prove the conjecture that α_{e,zz}^{−1} ≥ α_{m,xx}^{−1} + α_{m,yy}^{−1} for a plane aperture of arbitrary shape, where α_{e} and α_{m} are the electric and magnetic polarizabilities.

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[2] Aperture polarizabilities which describe low frequency coupling are useful in many EMI/EMC problems. In 1987 [Lee, 1987], it was conjectured that for an aperture of arbitrary shape in a perfectly conducting plane surface the inverse of the electric polarizability is greater than or equal to the sum of the inverses of magnetic polarizabilities along two principal axes. Since the magnetic polarizability tensor is a two by two real symmetric matrix, two orthogonal principal axes always exist. Let x and y be the principal axes. The conjecture can be stated as follows:

where

with p and m being the electric and magnetic dipole moments, and E_{0} and H_{0} the external fields.

[3] It is known that the equality sign of (1) holds for circular and elliptical apertures [Lee, 1987]. In 1990 [Gluckstern et al., 1990] it was shown that (1) is true for apertures symmetric with respect to the x and y axes, for example, a rectangle, a diamond, and a cross shaped aperture, for which numerical results exist in support of (1) [Gluckstern et al., 1990; Latham, 1972].

[4] In the following we will prove (1) for a plane aperture of any shape. The first step of the proof follows the idea suggested in the work of Gluckstern et al. [1990], i.e., to construct a stationary expression for each of the quantities in (1). We will then prove that these stationary expressions give minimum values when the trial functions are solutions of the associate boundary value problems. We will next manipulate the stationary expression for α_{e,zz}^{−1} to a form similar to those for α_{m,xx}^{−1} and α_{m,yy}^{−1}, which can be done for a plane aperture of any shape. On the other hand, the approach used in the work of Gluckstern et al. [1990] of manipulating the stationary expressions for α_{m,xx}^{−1} or α_{m,yy}^{−1} to a form similar to α_{e,zz}^{−1} needed certain symmetry conditions imposed on the aperture.

2. Proof

[5] The two principal magnetic polarizabilities are found from the solutions of the following magnetostatic boundary value problems formulated in terms of two integral equations [Latham, 1972; Gluckstern et al., 1990]:

where the constants c_{x} and c_{y} are determined by the constraints ∫_{A}gdS = 0 and ∫_{A}hdS = 0, is a two-dimensional position vector, g and h are normalized vertical components of the magnetic fields in the aperture A, i.e., g = H_{z}/H_{0x}, h = H_{z}/H_{0y}. The magnetic polarizabilities are given by [Latham, 1972; Gluckstern et al., 1990; Lee, 1986]:

Multiplying (3) by g and h, respectively, integrating over A and noting that the terms corresponding to c_{x} and c_{y} drop out because of the constraints, we obtain the following stationary representations

and ɛ_{n} is the Neumann number with ɛ_{n} = 1 for n = 0 and ɛ_{n} = 2 for n ≥ 1. Because K has the special form given by (7) we will be able to show that the stationary expressions (5) are minimums when g and h are solutions of the integral equations (3).

[6] Let X_{0} denote the true value of either of equations (5) when the solution g_{0} (or h_{0}) of (3) is used as the trial function in (5), and X denote the value for any other trial function satisfying the constraint ∫_{A}gdS = 0 (or ∫_{A}hdS = 0). We will now prove X ≥ X_{0} following the same procedure given in the work of [Schwinger and Saxon, 1968; Borgnis and Papas, 1955]. Since

we have, after expanding the squares,

Multiplying (3) by the trial function g and integrating over A gives

recalling that g_{0} is the solution of (3). The sum in (8) is then simply X_{0}, since

i.e., any trial function other than g_{0} (or h_{0}) in (5) will give a value greater than or equal to the true value.

[7] The electrostatic boundary value problem can be formulated in terms of the differential-integral equation [Latham, 1972; Gluckstern et al., 1990]

or

where K is given by (6) and f is the electric potential in the aperture divided by the external field E_{0z}. To ensure convergence one may replace K(, ′) by K(, ′, z) = [(x − x′)^{2} + (y − y′)^{2} + z^{2}]^{−1/2}/π, and let z → 0 after integration by parts. The electric polarizability is given by [Latham, 1972; Gluckstern et al., 1990; Lee, 1986]

The stationary representation for α_{e,zz} can be found from (11) and (12) as

Integrating by parts and noticing that f vanishes at the boundary of the aperture, we have

which has been cast in the form of (5) if we use for g and for h and realize that

Notice that ∫_{A}dS = 0, ∫_{A}dS = 0 satisfy the constraints ∫_{A}gdS = 0, ∫_{A}hdS = 0. With (5), (10), (14), and (15) the proof for the conjecture (1) is completed.

3. Conclusion

[8] Using the tangential aperture electric field (E_{x}, E_{y}) as the trial function for the normal component of the aperture magnetic field in the stationary minimum representation of the inverse of the magnetic polarizabilities (i.e., for g and for h) we have thus proved the conjecture expressed in (1) with the help of (10). When these electric trial functions are proportional to the actual solutions of the magnetic integral equations (3) the equality sign of (1) holds. For the simple case of a circular aperture g (or h) is indeed proportional to (or ).

[9] As a corollary for a plane perfectly conducting disk of any shape we have

where the z axis is perpendicular to the plane of the disk; M_{zz} and P_{xx}, P_{yy} are the magnetic and electric polarizabilities (M_{zz} is negative for the perfectly conducting disk).

Acknowledgments

[10] Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.