The Interaction of Two Charged Particles

According to the preceding section the potential energy of a particle with charge q and velocity ui in an electromagnetic field is 

We now consider the situation where the field is due to another particle with charge q2 and velocity U2. According to section 3.3, the potential set up by this particle is 

The obvious problem with this expression is that the variables relating to particle 2 are retarded quantities, and we really need to know something about the motion of particle 2 before we can use it. Another less obvious problem appears if we carry out the same exercise starting with particle 2 instead of particle 1. Then we get the interaction 

For the case at hand the relative motion of the two particles may be arbitrary, and we need the general form of the Lorentz transformation for the position vector r given in the previous chapter  

We may cast this in a more convenient form by expanding the triple cross product. For that purpose we use the general expression 

---

From these conditions, the allowed energies of the electron can be calculated. Coulomb s law (Equation 0.5) gives the potential energy, V, due to the interaction of two charged particles with charges qi and q2, separated by a distance r... 

The rate constants, measured at 300 K are high since they refer to the interaction of two charged particles approaching under Coulomb attraction. The possibility of such a recombination mechanism was first directly demonstrated for process e + HeJ He + He [399] which is considered to be responsible for the large cross section of electron-ion recombination under discharge in helium. 

Figure 21 represents the interaction of two charged particles. The appropriate equations for the NLDH potential and pressure were derived previously and are given by Eqs. [196], [197], [201], [202], and [209]. The development here is similar to that used previously for two interacting cylinders. To implement the general expressions to treat two interacting spheres, we use the Debye-Hiickel solution [307] to put... 

Fig. 1 Illustration of the DLVO theory interaction of two charged particles as a function of the interparticle distance (attractive energy curve, VA, repulsive energy curve, VR and net or total potential energy curve, Vj).

Ionic compounds are often soluble in polar solvents with high permittivities (dielectric constants). The energy of interaction of two charged particles is given by... 

The difference between the asymptotic forms (3.2) and (3.4) can be traced back the difference in the associated forms of the orbit-orbit interactions mentioned above. Thus we see that in the case of two charged particles the leading asymptotic behavior of V2 depends on the precise definition of Vj,. This observation resolves a longstanding puzzle concerning conflicting results for the value of C2 Further, as was noted some time ago by L. Spruch, 03" is classical in character, i.e. if h and c are restored, Cj turns out to be independent of h. One should therefore try to understand the source of this term from classical electrodynamics. It turns out that this is indeed possible by a reexamination of the work of Darwin [1], but I will not enter into the details here [10]. 

The classical DLVO theory of interparticle forces considers the interaction between two charged particles in terms of the overlap of their electric double layers leading to a repulsive force which is combined with the attractive London-van der Waals term to give the total potential energy as a function of distance for the system. To calculate the potential energy of attraction Va between solid spherical particles we may use the Hamaker expression ... 

Coulomb interaction The interaction between two charged particles. According to Coulomb s law of electrostatics, the energy of such interaction is q q2/R, where g, is the charge of the ith particle and R is the particles separation. 

Dielectric constant A dimensionless constant that expresses the screening effect of an intervening medium on the interaction between two charged particles. Every medium (such as a water solution or an intervening portion of an organic molecule) has a characteristic dielectric constant. 

The only remaining possible cause for particle asymmetry lies in the definition of r as ri — F2, but from the expression above it is clear that this is of no consequence. In addition to the Coulomb term, the expression for the interaction between two charged particles contains two relativistic terms. These will be discussed in greater detail when we later introduce similar expressions in the relativistic Hamiltonian. 

Two of the three SI base units have in the meantime acquired redefinitions in atomic terms (e.g., the second is now defined as 9 192 631 770 hyperfine oscillations of a cesium atom). However, the definitions (C.2a)-(C.2c) conceal another unfortunate aspect of SI units that cannot be overcome merely by atomic redefinitions. In the theory of classical or quantal electrical interactions, the most fundamental equation is Coulomb s law, which expresses the potential energy V of two charged particles of charge q and 2 at separation R as... 

Calculation of Coagulation Rate. Here we discuss an interaction potential of two charged particles in a liquid within a framework of DLVO theory. Following this theory, the overall interaction potential U, of charged spherical particles of the same radius R and surface distance d is a sum of a coulombic repulsive force of charged particles and a van der Waals attractive force given by the equation (28) ... 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

For the case of two spherical particles of radii a and a2, Stern potentials, iftdi and i//d2, and a shortest distance, H, between their Stern layers, Healy and co-workers195 have derived the following expressions for constant-potential, V, and constant-charge, Fr, double-layer interactions. The low-potential form of the Poisson-Boltzmann distribution (equation 7.12) is assumed to hold and Kax and xa2 are assumed to be large compared with unity ... 

Fig. 1. Formation of the Aab bound state of two charged particles a+ and b due to the Coulomb interaction in the final state...

At separations large compared with particle size, the interaction between two charged colloids is due to a correlation between fluctuations in net charge around each of them. (At shorter distances there are multipole terms, fluctuations in potential over the space of the colloid, which lead to additional forces.) This is a monopole-monopole correlation energy [see Table S.9.c and Eq. (L2. 206)] ... 

The simplest examples of such systems in quantum physics are the interaction of the charged quantum particle with the Lagrangian L = mr2/2 + ejcAr and the solenoidal magnetic field A — e x r/r 2 (the Aharonov-Bohm effect) or the interaction of two anions in 2 + 1-field theory [14]. In both cases, the configurational space is the plane with one point removed. 

---