content_image/349562.jpg content_image/349563.jpg content_image/349564.jpg content_image/349565.jpg content_image/349566.jpg
free energy per unit area of the plates is given by [14, 21–24] 

<img src='content_image/349562.jpg'>

Here, k$_{B}$ is the Boltzmann constant, ξ$_{l}$ = 2 πk$_{B}$T l/ ℏ with l = 0 , 1 , 2 , . . . are the Matsubara frequencies, the prime near the summation sign multiplies the term with l = 0 by a factor of 1/2, k ⊥ is the modulus of the wave vector projection on the plate (i.e., perpendicular to the z -axis) and 2 

<img src='content_image/349563.jpg'>

The reflection coefficients for the two independent polarizations of the electromagnetic field (transverse magnetic, TM, and transverse electric, TE) are given by 

<img src='content_image/349564.jpg'>

where ε ( n ) l ≡ ε ( n $^{)}$( iξ$_{l}$ ), µ ( n ) l ≡ µ ( n $^{)}$( iξ$_{l}$ ) with n = 1 , 2 are the dielectric permittivity and magnetic permeability of the first and second plates, respectively, calculated at the imaginary Matsubara frequencies, and 

<img src='content_image/349565.jpg'>

Recently the Lifshitz formula (1) for magnetodielectric media was rigorously rederived [41] under the same assumptions, as formulated above, in the framework of quantum field- theoretical scattering approach. 

The Casimir force per unit area of the plates (i.e., the Casimir pressure) is obtained from Eq. (1) by the negative differentiation with respect to a , 

<img src='content_image/349566.jpg'>

