Formal Methods of Electrostatics


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Firstly, before looking at the formal methods of electrostatics, we would like to look at the basis of electrostatic.What is electrostatic?Electrostatics is a branch of physics in science which deals with the phenomena and the properties of stationary electric charges.What are the laws governing electrostatics?There are three laws of electrostatic;Laws of Electrostatics:First law of electrostatics states that “Like charges of electricity repel each other, while unlike charges attract each other.”Second law of electrostatics state that “The force exerted between two point charges is directly proportional to the product of their strengths and inversely proportional to the square of the distance between them along with the absolute permittivity of the surrounding medium.”Third law of electrostatics state that “The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and is inversely proportional to the square of the distance between them.”Some problems are such that they could be simplified by invoking symmetry or by plausibility arguments. This does not always work, that’s when we consider the formal methods of electrostatics.In solving any electrostatics problem, there are mainly three types of techniques;

  1. Experimental methods
  2. Analytical methods
  3. Numerical methods

Experimental methods are expensive, time consuming, and usually do not allow much flexibility in parameter variation.In this presentation we will consider the last two solving methods. Analytical methods are the most rigorous ones, providing exact solutions, but they become hard to use for complex problems. Numerical methods have become popular with the development of the computing capabilities, and although they give approximate solutions, have sufficient accuracy for engineering purposes. We consider a problem to determine the distribution of electrostatic potential inside a conducting rectangular box having one of the armatures at V0=1V potential and the other three at 0V potential. Three methods were considered in solving the problem:

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  1. Analytical method (the separation of variables)
  2. Numerical method (deterministic finite difference)
  3. Numerical method (probabilistic Monte Carlo)

The numerical algorithms were implemented in matlab program being compared to the results of the analytical method and the number of iterations was raised until a 10-10V iteration order difference was reached between last iterations values. The grid step size was also optimized to minimize the error of the numerical finite difference solutions. This way the same accuracy was obtained with a much smaller number of iterations. All the results obtained with the numerical methods were compared to the exact analytical solution.Analytical methods This is the process of breaking down a problem into the elements necessary to solve it.The two main classifications of analytical methods:

  1. The method of separation of variables
  2. Method of complex analysis for the case of plane fields

The separation method

This allows a simple formulation of the boundary conditions. This calls for a coordinate transformation. With a few exceptions, we have thus far only used Cartesian coordinates. Also, the vector operators (grad ( ∇ ), div (∇. A), curl (∇ × A ), Laplacian ( ∇2 ) have only been expressed in their Cartesian coordinates.Usually the boundary value problems are solved using the method of separation of variables, which is illustrated, for different types of coordinate systems.

Field energy method is the recent method used for calculating the whole electrostatic free energy of a macromolecule. It is applicable to particles of random shape and size and also membranes or macromolecular assemblies along with the substrate molecules and ions. This method is derived by integrating the energy density of an electrostatic field. It is constructed on the dielectric model, by which the solute and the nearby water are regarded as changed continuous dielectrics. This method yields both the communication energy among all charge pairs and the self energy of solitary charges, efficiently accounting for the contact with water. First, the dielectric boundary and mirror charges are. for all charges of the solute. The energy is then given as a simple function of the interatomic distances, and the standard atomic partial charges and volumes. The interaction and self energy are shown to result from three-body and pairwise interactions. Both energy terms explicitly involve a polar atom, revealing that a polar group are also subject to electrostatic forces. We applied the field energy method to a spherical model protein.

Comparison with the Kirkwood solution shows that errors are within a small percentage. As a further test, the field energy method was used to calculate the electrostatic potential of the protein superoxide dismutase. We attained good contract with the result from a program that gears the numerical finite difference algorithm. The field energy method delivers a basis for energy minimization and dynamics programs that account for the solvent and screening effect of water at little computational expense.

Analytical methods in solving electrostatic problemsa. Integral method b. Method of images c. Separation of variables d. Conformal mapping e. Green’s function f. Purely numerical a. The integral method: (Simple but needs special symmetry) The integral method works well for a charge distribution that is specified and there are no conductors or dielectrics analytically or numerically. The shape and distribution of the charge density can be multi-dimensional. The integral procedure works well when the charge distributions are isolated in free space but is not too useful when conductors are present, particularly when they do not have a very simple geometric shape.b. The method of images: (Simple but needs special symmetry) The method of images is a simple technique that is very useful, but only in very specialized geometries with a lot of symmetry. A nice feature is that when the method works the solution is usually in the form of a simple closed analytic form. It can be used to solve Poisson’s equation but they have extremely limited range of applicationc. Separation of variables: (Slightly complicated, more general geometry) Separation of variables is a good way to solve a reasonably large class of problems. In general, the solution is obtained as a summation of individual separated solutions; that is the solution can be expressed as a sum of expansion functions with appropriately determined coefficients (e.g. Fourier series, Bessel series)This possesses a wider range of applicationFor separation of variables to work the geometry must have some symmetry although the charge density can have an arbitrary shape and distribution.Let us examine a specific example.

The Green’s function procedure is a very powerful technique that works in a wide variety of cases. It is a great procedure, worth understanding. The technique is relatively complicated mathematically. We have to understand Green’s theorem. We have to learn how to solve an integral equation. f. Pure numerical solution (General, but hidden difficulties) The pure numerical solution works very well if you know what you are doing. Most of my experience has led me to the conclusion that relying on the numerical procedure too early, without really thinking about the problem, will result in unsatisfactory results.Once you have a sufficient understanding of the problem and a reasonable expectation of the qualitative behavior of the solution, then the numerical approach can be very powerful and useful.

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