Volume 43A, number 3

12 March 1973

PHYSICS LETTERS

ON THE MACROSCOPIC THEORY OF RETARDED VAN DER WAALS FORCES K. SCHRAM Instituut voor theoretische fysica, Rijksuniversiteit Utrecht, The Netherlands Received 26 January 1973 The Lifshitz formula for the retarded dispersion force between two dielectric media at zero temperature is derived by a simple macroscopic method.

Some time ago Van Kampen, Nijboer and Schram [ 1] gave a simple macroscopic derivation of the Lifshitz formula [2] for the non-retarded dispersion force between two non absorbing dielectric halfspaces. The method was based on a calculation of the zero-point energy of the electrostatic surface modes. In some recent attempts [3,4] to generalize the surface mode method to the retarded case the derivations are mathematically incorrect because contour integrations are performed in the complex o-plane without taking into account branch points of non-analytic functions which occur in the integrals. In fact it has not been proved that the surface modes in semiinfinite dielectrics give rise to the correct retarded dispersion force and from the following it appears that indeed the surface modes alone do not yield the right expression. In this letter it is shown how one may obtain Lifshitz’ formula for the retarded case by a macroscopic method analogous to Casimir’s calculation [ 51 for metals. Consider two dielectric slabs of finite thickness d parallel to the x, y-plane and separated by a vacuum gap of width a. The system is bounded by perfectly conducting plates at z = 0 and z = a+2d. A third perfectly conducting plate is situated at z = L + 2d with L S a+2d. The total electromagnetic zero-point energy of this configuration may be written as ,!?(a, d) + &5-u, 0). This energy should be compared with the energy E(L, d) of the configuration where the upper slab (a+d