# On the Debye–Hückel effect of electric screening

## Abstract

The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potential vanishes differs from the Debye-Hückel radius by a factor of √(2). The preceding (Secs. II–VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalizedmore »

- Authors:

- Centro de Ciências e Tecnologias Aeronauticas e Espaciais (CCTAE) and Área Científica de Mecânica Aplicada e Aeroespacial (ACMAA), Instituto Superior Técnico IST, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal)

- Publication Date:

- OSTI Identifier:
- 22304077

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Plasmas; Journal Volume: 21; Journal Issue: 7; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; CHARGE DENSITY; CHARGE DISTRIBUTION; ELECTRIC CHARGES; ELECTRIC FIELDS; ELECTRIC POTENTIAL; EQUATIONS; EXACT SOLUTIONS; NONLINEAR PROBLEMS; POTENTIALS; SYMMETRY

### Citation Formats

```
Campos, L. M. B. C., and Lau, F. J. P.
```*On the Debye–Hückel effect of electric screening*. United States: N. p., 2014.
Web. doi:10.1063/1.4886363.

```
Campos, L. M. B. C., & Lau, F. J. P.
```*On the Debye–Hückel effect of electric screening*. United States. doi:10.1063/1.4886363.

```
Campos, L. M. B. C., and Lau, F. J. P. Tue .
"On the Debye–Hückel effect of electric screening". United States.
doi:10.1063/1.4886363.
```

```
@article{osti_22304077,
```

title = {On the Debye–Hückel effect of electric screening},

author = {Campos, L. M. B. C. and Lau, F. J. P.},

abstractNote = {The paper considers non-linear self-consistent electric potential equation (Sec. I), due to a cloud made of a single species of electric charges, satisfying a Boltzmann distribution law (Sec. II). Exact solutions are obtained in a simple logarithmic form, in three cases: (Sec. III) spherical radial symmetry; (Sec. IV) plane parallel symmetry; (Sec. V) a special case of azimuthal-cylindrical symmetry. All these solutions, and their transformations (Sec. VI), involve the Debye-Hückel radius; the latter was originally defined from a solution of the linearized self-consistent potential equation. Using an exact solution of the self-consistent potential equation, the distance at which the potential vanishes differs from the Debye-Hückel radius by a factor of √(2). The preceding (Secs. II–VI) simple logarithmic exact solutions of the self-consistent potential equations involve no arbitrary constants, and thus are special or singular integrals not the general integral. The general solution of the self-consistent potential equation is obtained in the plane parallel case (Sec. VII), and it involves two arbitrary constants that can be reduced to one via a translation (Sec. VIII). The plots of dimensionless potential (Figure 1), electric field (Figure 2), charge density (Figure 3), and total charge between ζ and infinity (Figure 4), versus distance normalized to Debye-Hückel radius ζ ≡ z/a, show that (Sec. IX) there is a continuum of solutions, ranging from a charge distribution concentrated inside the Debye-Hückel radius to one spread-out beyond it. The latter case leads to the limiting case of logarithmic potential, and stronger electric field; the former case, of very concentrated charge distribution, leads to a fratricide effect and weaker electric field.},

doi = {10.1063/1.4886363},

journal = {Physics of Plasmas},

number = 7,

volume = 21,

place = {United States},

year = {Tue Jul 15 00:00:00 EDT 2014},

month = {Tue Jul 15 00:00:00 EDT 2014}

}