Question

3 Casimir effect We will derive the Casinir effect in three dimen- sions, making use of the Euler-Maclaurin formula where θη _ 1 for n > 0,4,-1/2, and θη 0 for n <0. (You don't need to prove this formula.) Let us consider walls of length L. Let EL be the vacuum energy inside the box. We now insert a conducting plate at a distance RL parallel to one of the walls, dividing the volume into two. Denote ER and a square box with conducting R the vacuum energy in the two volumes. e in vacuum energy due to insertion The change i of the wall is evidently a. Show that (The function n is needed because there are two polarization states for The wavenumbers ky,k can take any values, a good approximation if L is large. Since R is small, however, only wavenumbers k, /R are allowed in -direction. b. Since L- R and L are both large, we can find simplified gies in the larger volumes by assuming that now aTy values are allowed even for k replacing the surmmation by an integration. Show that expressions for the vacuum ener- Me di,dk.dk.y4呺呺 L-(L

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