  When described in a grand canonical ensemble, a finite Coulomb system
exhibits charge fluctuations. These fluctuations are studied in the case of a
classical (i.e. non-quantum) system with no macroscopic average charge.
Assuming the validity of macroscopic electrostatics gives, on a
three-dimensional finite large conductor of volume $V$, a mean square charge
$<Q^2>$ which goes as $V^{1/3}$. More generally, in a short-circuited capacitor
of capacitance $C$, made of two conductors, the mean square charge on one
conductor is $<Q^2>=TC$, where $T$ is the temperature and $C$ the capacitance
of the capacitor. The case of only one conductor in the grand canonical
ensemble is obtained by removing the other conductor to infinity. The general
formula is checked in the weak-coupling (Debye-H\"uckel) limit for a spherical
capacitor. For two-dimensional Coulomb systems (with logarithmic interactions),
there are exactly solvable models which reveal that, in some cases, macroscopic
electrostatics is not applicable even for large conductors. This is when the
charge fluctuations involve only a small number of particles. The mean square
charge on one two-dimensional system alone, in the grand canonical ensemble, is
expected to be, at most, one squared elementary charge.
