The Hamiltonian of one- and two-component plasmas is calculated in the negligible radiation Darwin approximation. Since the Hamiltonian is the phase space energy of the system its form indicates, according to statistical mechanics, the nature of the thermal equilibrium that plasmas strive to attain. The main issue is the length scale of the magnetic interaction energy. In the past a screening length λ=1/√(ren), with n number density and re classical electron radius, has been derived. We address the question whether the corresponding longer screening range obtained from the classical proton radius is physically relevant and the answer is affirmative. Starting from the Darwin Lagrangian it is nontrivial to find the Darwin Hamiltonian of a macroscopic system. For a homogeneous system we resolve the difficulty by temporarily approximating the particle number density by a smooth constant density. This leads to Yukawa-type screened vector potential. The nontrivial problem of finding the corresponding, divergence free, Coulomb gauge version is solved.