(9420) Casimir Effect

Consider two conducting parallel plates of area \( A \) located at a distance \( r \) in vacuum. Calculate the force between the plates due to vacuum zero-point-fluctuations (ZPF).

(1) Perform simplified calculation assuming that the the problem can be treated as one-dimenisonal.

(2) Perform proper claculation that takes into accout the actual 3D geometry.

Tip: Write an expression for the ZPF energy \( E(r) \) of the confined modes. For plates that have very large distance \( r=L \) the result is proportional to \( L \), and the prefactor can be written as a diverging \( d\omega \) integral over the modes. For small \( r \), the energy of the system is a sum of innner contribution \( E(r) \) and outer contribution \( E(L-d) \). A well defined finite result can be obtianed for the difference \( U(r) = [E(r)+E(L-r)]-E(L) \) in the \( L \rightarrow \infty \) limit. The force is \( F(r) = -(d/dr)U(r) \).

Solution of the first part here.
Solution of the second part -- see [C12.2].