| Schl(?)milch’s formula is generalized to several forms and applied into thermal Casimir effect.The Casimir energy and Casimir force of scalar field,electromagnetic field and fermionic field are derived.The fermionic field is analysed detailedly.At low temperature limit,for forces between three pairs of faces,the two-dimensional edge sizes space of a rectangular box is split into four regions.In one region,all three forces are attractive.In another two regions,the force along the longest edge becomes repulsive.In the last region,only the force along the shortest edge is attractive.The results of a cube box and parallel plates agree with other papers.For a waveguide,the force along the longer side of the cross section transforms from attractive to repulsive with the increase of this edge.At high temperature limit,both the Casimir energy and force approach zero.For finite temperature case,the parameter space is separated into four parts:(1)In the case similar to parallel plates with relatively high temperature,the Casimir force is always attractive and decreases with the increase of the shortest edge.(2)In the case similar to a waveguide with relatively high temperature,along the longer side of the cross section the Casimir force transforms from attractive to repulsive when this side exceed a critical value which decreases with respect to temperature increase.(3)For the change from parallel plates to a waveguide at low temperature,the attractive Casimir force becomes repulsive when the length exceeds certain critical values which change with respect to temperature in a nontrivial way.(4)In the case similar to a waveguide at low temperature,the Casimir energy is proportional to the the waveguide length,and Casimir force along the longitudinal direction is a positive constant.Lastly,there exists an L-shaped surface in the parameter space of edge sizes separating the repulsive and attractive regions for a fixed temperature. |
