free energy per unit area of the plates is given by [14, 21–24] F ( a, T ) = k$_{B}$T 2 π ∞ ∑ l =0 ′ ∫ ∞ 0 k ⊥ dk ⊥ { ln [ 1 − r (1) $_{TM}$( iξ$_{l}$, k ⊥ ) r (2) $_{TM}$( iξ$_{l}$, k ⊥ ) e − 2 aq l ] + ln [ 1 − r (1) $_{TE}$( iξ$_{l}$, k ⊥ ) r (2) $_{TE}$( iξ$_{l}$, k ⊥ ) e − 2 aq l ]} . (1) q 2 l ≡ q 2 $_{l}$( iξ$_{l}$, k ⊥ ) = k 2 ⊥ + ξ 2 l c $_{2}$. (2) r ( n ) $_{TM}$( iξ$_{l}$, k ⊥ ) = ε ( n ) l q$_{l}$ − k ( n ) l ε ( n ) l q$_{l}$ + k ( n ) l , r ( n ) $_{TE}$( iξ$_{l}$, k ⊥ ) = µ ( n ) l q$_{l}$ − k ( n ) l µ ( n ) l q$_{l}$ + k ( n ) l , (3) k ( n ) l 2 ≡ k ( n ) $^{2}$( iξ$_{l}$, k ⊥ ) = k 2 ⊥ + ε ( n ) l µ ( n ) l ξ 2 l c $_{2}$. (4) P ( a, T ) = − k$_{B}$T π ∞ ∑ l =0 ′ ∫ ∞ 0 q$_{l}$k ⊥ dk ⊥    [ e 2 aq l r (1) $_{TM}$( iξ$_{l}$, k ⊥ ) r (2) $_{TM}$( iξ$_{l}$, k ⊥ ) − 1 ] − 1 + [ e 2 aq l r (1) $_{TE}$( iξ$_{l}$, k ⊥ ) r (2) $_{TE}$( iξ$_{l}$, k ⊥ ) − 1 ] − 1   $^{.}$ (5) 6 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 The reflection coefficients for the two independent polarizations of the electromagnetic field (transverse magnetic, TM, and transverse electric, TE) are given by 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 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 ,