We study, via computer simulations, the fluctuations in the net electric charge, in a two dimensional one component plasma (OCP) with uniform background charge density $-e rho$, in a region $Lambda$ inside a much larger overall neutral system. Setting $e=1$ this is the same as the fluctuations in $N_Lambda$, the number of mobile particles of charge $e$. As expected the distribution of $ N_Lambda$ has, for large $Lambda$, a Gaussian form with a variance which grows only as $hat kappa |partial Lambda|$, where $|partial Lambda|$ is the length of the perimeter of $Lambda$. The properties of this system depend only on the coupling parameter $Gamma = kT$ which is the same as the reciprocal temperature in our units. Our simulations show that when the coupling parameter $Gamma$ increases, $hat kappa(Gamma)$ decreases to an asymptotic value $hat kappa(infty) sim hat kappa(2)/2$ which is equal (or very close) to that obtained for the corresponding variance of particles on a rigid triangular lattice. Thus, for large $Gamma$, the characteristic length $xi_L = 2hat kappa/rho$ associated with charge fluctuations behaves very differently from that of the Debye length, $xi_D sim 1/sqrt Gamma$, which it approaches as $Gamma to 0$. The pair correlation function of the OCP is also studied.
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