Image charge

The most readily solved electrostatics problems have some intrinsic geometric symmetry—spheres and point charges, cylinders and line charges, or planes and parallel-plate capacitors, for example.. Some situations involve two or more different types of symmetry, as when a spherical ion approaches a planar interface such as an oil/water interface. Electrostatic fields and potentials can be found in such cases by the method of image charges. 

Ions are attracted to metal or other conductors that have no charge on them. The attraction arises from a process of induction. An ion in the neighborhood of a conducting surface induces a field. That field attracts the ion to the surface. Induction is readily modelled by recognizing an extraordinary mathematical property, the uniqueness of differential equations such as Poisson s equation. According to the uniqueness theorem, if you can find any solution to Poisson s equation that satisfies the boundary conditions for the problem of interest, you have found the only solution, even if you found it by wild guesses or clever tricks. 

Let s use this approach to find the electrostatic potential that attracts a point charge Q at a distance a. from a conducting planar surface. By definition, charges move freely in a conducting surface. As long as no electric current is flowing, the charges in a conductor experience no electrostatic force within the 

Now let s apply the same reasoning to a charge near a dielectric boundary. A Charge Near a Dielectric Interface 

We begin by recognizing that there is a fundamental symmetry in this problem. If we violate that symmetry by our invention of image charge positions, we will be unable to satisfy the boundarv conditions. The oil/water interface is not a conductor, so the field lines need not be perpendicular to it, as they were in Example 21.8. (Indeed, the field lines look much like the light rays at optical boundaries Snell s law of optical refraction has its roots in similar physics 2. ) 

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Schematic diagram showing how placing a thin layer of highly dispersed carbon onto the surface of a metal filament leads to an induced dipolar field having positive and negative image charges. The positive side is always on the metal, which is much less electronegative than carbon. This positive charge makes it much more difficult to remove electrons from the metal surface. The higher the value of a work function, the more difficult it is to remove an electron. Effectively, the layer of carbon increases the work function of the filament metal. Very finely divided silicon dioxide can be used in place of carbon.

To a first approximation, the ions in both Helmholtz layers can be considered point charges. They induce an equal and opposite image charge inside the conductive electrode. When the electrode is negative to the point of zero charge, cations populate the inner Helmholtz layer. 

Fig. 5. ElecUostatic image charges indueed in a grounded, condueting substrate due to the presence of 1 (a) and 2 (b) eharged spherical particles.

The electric field that generates the multipoles does not have to be externally applied. Rather, the image charge generated by the particle causes a non-uniform electric field that, not only polarizes the particle, but can result in attractive forces. 

Now let us consider a spherical particle of radius R that has a charge q uniformly distributed either on its surface or within its volume. The field produced by its image charge at some distance z above the surface of the substrate is given simply by... 

The first simulation studies of full double layers with molecular models of ions and solvent were performed by Philpott and coworkers [51,54,158] for the NaCl solution, using the fast multipole method for the calculation of Coulomb interactions. The authors studied the screening of a negative surface charge by free ions in several highly concentrated NaCl solutions. A combination of (9-3) LJ potential and image charges was used to describe the metal surface. 

The experiment is performed at various voltages and the barrier height values are corrected for the image charge potential by extrapolating to zero voltage. 

As discussed already in Chapter 2 the work function, , of a solid surface is one of the most important parameters dictating its chemisorptive and catalytic properties. The work function, (eV/atom) of a surface is the minimum energy which an electron must have to escape from the surface when the surface is electrically neutral. More precisely is defined as the energy to bring an electron from the Fermi level, EF, of the solid at a distance of a few pm outside of the surface under consideration so that image charge interactions are negligible. 

Consider a hydrogen atom with its nucleus at the origin located above the surface of a conducting metal at the position d = (0,0,d) and an electron at r = (x,y,z) (Fig. 6.1).The nucleus and the electron both induce image charges in the metal equal to... 

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