A simple macroscopic derivation is given of the non-retarded Van der Waals interaction between two semi-infinite dielectric media.Neutral macroscopic bodies, when placed at a distance of the order of 1000 A, attract each other as a result of the Van der Waals forces between their atoms. Lifshitz [1] has developed a macroscopic theory for this phenomenon by introducing fluctuating terms in the Maxwell equations for a dielectric medium. For the interaction energy per unit area at T = 0 between two semi-infinite media a distance d apart IAfshitz finds in the case that d << k (;~ is the principal absorption wave length of the material); ; 1 Starting from the Drude-Lorentz model of an atom (harmonic oscillator model) Renne and Nijboer [2] gave an atomistic derivation of the non-retarded interaction between one atom and a semi-infinite medium by summing two-particle, three-particle, etc., interactions. Their derivation can be extended to the case of two semi-infinite media and for arbitrary atoms [3]. In this note we will give a simple alternative macroscopic derivation of eq. (1), which can easily be generalized to more complicated configurations.Consider two semi-infinite dielectric media a distance d apart. We look for those solutions of the equations of electrostatics: div D = 0, curl/~ = 0, which oscillate harmonically in time. It will be obvious that these equations can be satisfied by: 1°: curl E = O, ~(¢o) = 0 (longitudinal bulk modes); 20: div D = 0, e(c0) = co (transverse bulk modes); 30: div~ = 0, curl E = 0 (surface modes). At T = 0 the interaction between the media may be found from the sum of the zero point energies ~k ½~wk of all modes evaluated at a distance d minus the corresponding sum for infinite d. This procedure, due to Casimir [4], who applied it to the case of ideal conductors in vacuo, can be justified in our case by considering a macroscopic Hamiltonian of the system which is equivalent to a sum of harmonic oscillator Hamiltonians with the frequencies ¢o k as eigenfrequencies. A similar Hamiltonian was used by Hopfield [5]. Now the frequencies of the bulk modes 1 ° and 2 ° are evidently independent of the distance d. Hence for the evaluation of the Van der Waals interaction only the surface modes 3 ° need be considered.Introducing the electrostatic potential ~ (x,y,z,t) by E = -grad ~ we have to solve the equation V2~ = 0, subject to the boundary conditions q~ and ~(w)a~/Sz continuous at z = 0 and z = d (the z-axis is chosen perpendicular to the plane-parallel boundaries of the media).Assuming ¢ to be of the form

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SynopsisAn expression is derived for the Van der Waals force between two semi-infinite bodies with small surface irregularities. Calculations are given both for the plane-plane and the plane-sphere configuration. The value of the correction from the surface irregularities upon the Van der Waals force is shown to amount easily to lo-50%.