Solution of the Linearized P-B Equation

The rather messy-looking linearized P-B equation (3.21) can be tidied up by a mathematical trick. Introducing a new variable /U defined by 

To solve this differential equation, it is recalled that the differentiation of an exponential function results in the multiplication of that function by the constant in the exponent. For example. 

if fi is an exponential function of r, one will obtain a differential equation of the form of Eq. (3.25). In other words, the primitive or origin of the differential equation must have had an exponential in Kr. 

Two possible exponential functions, however, would lead to the same final differential equation one of them would have a positive exponent and the oto a negative one [Eq. (3.26)]. The general solution of the linearized P-B equation can therefore be written as 

The constants is evaluated by using the boundary condition that far enough from a central ion situated at r = 0, the thermal forces completely dominate the Coulombic forces, which decrease as and there is electroneutrality (i.e., the elechostatic potential vanishes at distances sufficiently far from such an ion, v r 0 as r °°). This condition would he satisfied only if S = 0. Thus, if B had a finite value, Eq. (3.28) shows that the electrostatic potential would shoot up to infinity (i.e., physically unreasonable proposition. Hence, 

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By assuming that ions can be regarded as point charges, the solution of the linearized P-B equation turns out to be (Fig. 3.37)... 

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