The relative permittivity of tetragonal arrays of perfectly conducting thin discs

The methods available for the calculation of the interaction field of an array of dipoles are discussed with reference to an artificial dielectric formed from a tetragonal array of conducting discs. A modified method is described and used to derive a formula for the relative permittivity, valid when the discs are not too closely spaced. For closely packed discs, the fields due to higher-order multipoles become important and a second formula for the relative permittivity is obtained for this case. Comparison with experimental values already obtained shows that the two formulae together enable the calculation of the relative permittivity over the whole range of spacings likely to be used in practice to be made.