 Solution

The electron Debye length is the scale over which a weak electrostatic disturbance is screened. Take the ions as an immobile neutralizing background and the electrons as an isothermal Boltzmann distribution. For a small potential $\varphi$,Poisson's equation then becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$ away from the perturbing source, withA test charge has a screened potential proportional to $e^{-r/{\lambda }_D}/r$. For fully static screening with mobile ions, inverse squared Debye lengths from all species add; the displayed expression is the electron scale used below.

The electron plasma frequency is the natural collective oscillation frequency when electrons are displaced from a nearly fixed ion background. Displace a uniform electron fluid by $\xi$ along a slab normal. The charge separation produces the restoring electric force $-n_e{e}^2\xi /{ϵ}_0$ on an electron. Hence $m_e\ddot{\xi }=-n_e{e}^2\xi /{ϵ}_0$ andIt sets a response time $1/{\omega }_p$ and the cold unmagnetized electromagnetic cutoff frequency. These collective effects arise from long-range electric fields, even when individual binary collisions are infrequent.