= Solution
The <Debye–Hückel approximation> linearizes the hyperbolic sine, giving $(\nabla^2-\kappa^2)\psi=0$ with
$$
\boxed{\kappa^2=\frac{2n_0q^2}{\varepsilon k_BT},\qquad \lambda_D=\kappa^{-1}.}
$$
The <planar Debye-Hückel double layer> has $\psi(s)=\psi_0e^{-\kappa s}$. For an isolated sphere of radius $a$, the <Debye-Hückel potential of a charged sphere> is $\psi(r)=\psi_0(a/r)e^{-\kappa(r-a)}$ outside the sphere; the factor $a/r$ accounts for spherical spreading as well as screening.
The <Debye–Hückel screening length> is the characteristic distance over which the ion cloud suppresses a weak surface potential. Adding salt increases <ionic strength> and therefore increases $\kappa$, shortening the interaction range. Linearization requires \b[$|q\psi|\ll k_BT$], particularly near the surface where the potential is largest. It means the ionic electrostatic energy is small compared with thermal energy; for monovalent ions near room temperature, $k_BT/e\simeq26\,\mathrm{mV}$. This condition alone does not remove other limitations of point-ion mean-field theory at high concentration or strong correlations.
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