Weak Coupling Casimir Energies for Finite Plate Configurations
  We derive and use an extremely simplified formula for the interaction Casimir energy for two separate bodies in the weak coupling regime for massless scalar fields. We derive closed form solutions for a general arrangement of two $\delta$-function plates finite in one direction and infinite in another. We examine the situation of two parallel plates finite in both transverse directions. 
Jef Wagner, Kimball A. Milton, Prachi Parashar