= Solution
For weak potentials, Boltzmann <Electron>[electrons] give $\delta n_e=n_ee\phi/(k_BT)$ against fixed <Ion>[ions]. <Poisson equation>[Poisson's equation] becomes $\nabla^2\phi=\phi/\lambda_D^2$ outside a test <Electric charge>[charge], giving a <Yukawa potential> with $\boxed{\lambda_D=\sqrt{\epsilon_0k_BT/(n_ee^2)}}$. For a small <electron> displacement $x$ relative to fixed <Ion>[ions], <Electric charge>[charge] separation produces restoring field $E=n_eex/\epsilon_0$, so $m\ddot x=-n_ee^2x/\epsilon_0$. Hence $\boxed{\omega_p=\sqrt{n_ee^2/(\epsilon_0m_e)}}$. The two formulas respectively encode thermal screening and <electron> <inertia>.
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