Quantum electrodynamics which connects quantum theory with the relativity brought revolution into our understanding of the vacuum. Vacuum isn't emptiness and nothingness anymore, it's a state with the lowest energy. The lowest possible energy state cannot be zero since we have to take into account the uncertainty principle – which leads us to the fluctuations of the vacuum. The strongest mechanical demonstration of quantum fluctuations is the attraction force between two mirrors (two parallel metallic plates without any charge) separated by a narrow gap. Around the system, there are waves with all frequencies but inside the gap, there are only such waves which correspond to the resonance frequency of the gap. This results in a small, yet measurable force which pushes the mirrors closer to each other. It's magnitude depends on the area of the mirrors S (it's natural to assume linear dependence here) and the width of the gap d. There are two fundamental constants in the relation, c and h:
F =KcαhβdγS.
Using dimensional analysis, determine the coefficients α, β and γ and the force for the values S = 1cm2 and d = 1μm. K is a dimensionless constant, its value is K = π/480 as can be derived with precise calculations.