Temperature dependence of the Casimir effect

The temperature dependence of the Casimir force between a real metallic plate and a metallic sphere is analyzed on the basis of optical data concerning the dispersion relation of metals such as gold and copper. Realistic permittivities imply, together with basic thermodynamic considerations, that the transverse electric zero mode does not contribute. This results in observable differences from the conventional prediction, which does not take this physical requirement into account. The results are shown to be consistent with the third law of thermodynamics, as well as being not inconsistent with current experiments. However, the predicted temperature dependence should be detectable in future experiments. The inadequacies of approaches based on ad hoc assumptions, such as the plasma dispersion relation and the use of surface impedance without transverse momentum dependence, are discussed.

Figure 1

Force $\mathcal{F}$ between a gold sphere and a gold plate versus gap $a$, when $T=300\mathrm{K}$ and $R=296\mu \mathrm{m}$.

Figure 2

Force difference $\Delta \mathcal{F}=\mid \mathcal{F}\left(1\mathrm{K}\right)\mid -\mid \mathcal{F}\left(300\mathrm{K}\right)\mid$ between a gold sphere and a gold plate, versus gap $a$ for $R=296\mu \mathrm{m}$.

Figure 3

Force $\mathcal{F}$ between a gold sphere and a copper plate versus $a$, when $T=300\mathrm{K}$ and $R=296\mu \mathrm{m}$. Bottom curve is the $m=0$ contribution.