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The Debye length is the distance over which a small, slowly varying electrostatic potential is screened by mobile thermal charges. For classical electrons in a weak potential $\varphi$, their Boltzmann distribution givesTake fixed singly charged ions of density $n_0$. Then $\rho =e(n_0-n_e)\simeq -n_0{e}^2\varphi /(k_{\mathrm{B}}T)$, and Poisson's equation outside a test charge becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, withThe screened point-charge potential is proportional to $e^{-r/{\lambda }_D}/r$. The derivation assumes nondegenerate statistics and $|e\varphi |\ll k_{\mathrm{B}}T$. Mobile ions would add their own contribution to ${\lambda }_D^{-2}$; the expression here is the electron screening length.

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.

Poisson's equation is $\Delta u=f$, where $\Delta$ is the Laplace operator and $f$ a prescribed source. It generalizes the Laplace equation, recovered when $f=0$. Boundary conditions are needed to determine a unique solution; Neumann boundary conditions can leave an additive constant.

For electrostatics in vacuum, the convention is $\Delta V=-\rho /{ϵ}_0$. For Newtonian gravity, $\Delta \Phi =4\pi G\rho$. The opposite signs reflect the specified force and potential conventions. A simple mathematical example is $u(x)=x^2/2$ on $[0,1]$, which solves $u^{\prime \prime }=1$ with endpoint values zero and $1/2$. On a periodic domain the source must have zero mean, by integrating the equation.

Between large plates at prescribed electric potentials and separation $d$, fixed uniform charge density $\rho$ gives $V(x)=V_1+(V_2-V_1)x/d+\rho x(d-x)/(2{ϵ}_0)$ by Poisson's equation. The signed electric field is linear. Its magnitude can have an interior V-shaped minimum, so a sketch of signed field alone need not answer a request for field magnitude.

Uniqueness for Poisson's equation depends on the specified boundary conditions. If $\Delta u=f$ and $\Delta v=f$ on a bounded connected domain, their difference $w=u-v$ is harmonic. With identical Dirichlet boundary conditions and continuity up to the boundary, the maximum principle gives $w=0$.

For identical Neumann boundary conditions, the difference can instead be any constant. Under sufficient smoothness, Green's first identity gives ${\int }_{\Omega }|\nabla w{|}^2={\int }_{\partial \Omega }w{\partial }_{n}w-{\int }_{\Omega }w\Delta w=0$, so those constants are the only freedom. A prescribed mean fixes this remaining ambiguity. Stating the partial differential equation without boundary data gives neither of these uniqueness conclusions.