## I. INTRODUCTION 

At present physical phenomena caused by the zero-point oscillations of quantized fields attract much experimental and theoretical attention. One of the most prospective subjects in this area is the Casimir effect [1], i.e., the attractive force acting between two neutral parallel ideal metal plates in vacuum arising due to the existence of zero-point oscillations of the electromagnetic field and thermal photons. The Casimir force is a version of the van der Waals force in the case when the separation distances between the interacting bodies are large enough so that the relativistic effects contribute essentially. Lifshitz [2, 3] developed the general theory of the van der Waals and Casimir forces for the case of two dielectric semispaces separated with a gap of width a . The material of semispaces was described by the frequency-dependent dielectric permittivity ε ( ω ). In recent years several reviews [4–7] and books [8–11] on different aspects of the Casimir effect have been published. A new stage in the measurement of the Casimir force was started by the two experiments [12, 13]. During the last few years significant progress has been made both in the measurement of the Casimir force and in the development of new calculational methods applicable to nontrivial geometries and taking into account real material properties of the interacting bodies. This progress is reflected in the monograph [14]. 

The seminal paper by Casimir [1] treated the configuration of two parallel ideal metal plates which do not posses magnetic properties and found that the force is always attrac- tive. The possibility to obtain the repulsive Casimir force has agitated scientists for several decades. It is of high promise for solving the problems of stiction and friction in micro- and nanoelectromechanical devices [15]. Boyer [16, 17] was the first who considered config- urations of an ideal metal spherical shell and of two parallel plates one of which is made of an ideal metal and another one is infinitely permeable. In both cases the Casimir force was shown to be repulsive. The latter configuration which is better adapted for possible applications in microdevices was often discussed in the literature as an unusual, hybrid or mixed pair of plates [18–20]. 

The investigation of the influence of magnetic properties on the Casimir force in the case of real materials requires the generalization of the Lifshitz theory for magnetodielectric media possessing frequency-dependent dielectric permittivity ε ( ω ) and magnetic permeability µ ( ω ). Such generalization was performed by Richmond and Ninham [21] and later formulated for 

