# Electrostatic interaction inside and outside the source

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• kelly0303
In summary, the conversation discusses the calculation of electrostatic interaction between an electron and a nucleus, taking into account the possibility of the electron being inside the nucleus. The resulting integral includes both the electron and nuclear coordinates, as well as the nuclear radius. To simplify the function, it can be expanded using Legendre polynomials, with the smaller coordinate in the numerator.
kelly0303
Hello! I want to get the electrostatic interaction (between and electron and a nucleus), while accounting for the fact that the electron can also be inside the nucleus (e.g. in an S##_{1/2}## state). I ended up with this double integral:

$$\int_{r_e=0}^{r_e=\infty}\int_{r_n=0}^{r_n=R}\frac{\rho(r_n)}{|r_e-r_n|}d^3r_ed^3r_n$$

where ##r_e## and ##r_n## are the electron and nuclear coordinates and ##R## is the nuclear radius. Please note that we are not necessarily assuming that the nucleus is a perfect sphere (although it is usually very close to it). How can I expand the ##\frac{1}{|r_e-r_n|}## and get this into a simpler form that I can also truncate as needed? Thank you!

That function is expanded in Legendre polynomials as
$$\frac{1}{|{\bf r}-{\bf r'}|}=\sum_l\frac{r'^l}{r^{l+1}}P_l(\cos\theta)$$,
with the smaller r in the numerator.

vanhees71

## What is electrostatic interaction?

Electrostatic interaction refers to the force between charged particles. These forces can be attractive or repulsive depending on the nature of the charges involved. Like charges repel each other, while opposite charges attract.

## How does electrostatic interaction occur inside a source?

Inside a source, electrostatic interactions occur between charged particles such as electrons and protons within atoms or molecules. These interactions are governed by Coulomb's law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

## What factors affect electrostatic interaction outside a source?

Outside a source, electrostatic interactions are influenced by factors such as the distance between the charges, the medium through which the interaction occurs (e.g., air, vacuum, or water), and the presence of other charged or polarizable objects that can affect the electric field.

## How can electrostatic interaction be measured?

Electrostatic interaction can be measured using various techniques such as Coulomb's law calculations, electric field mapping, and instruments like electrometers or voltmeters that measure the potential difference. Additionally, techniques like atomic force microscopy (AFM) can be used to measure forces at the nanoscale.

## What are some practical applications of electrostatic interactions?

Electrostatic interactions have numerous practical applications, including in electrostatic precipitators for air pollution control, in photocopiers and laser printers, in the design of capacitors and other electronic components, and in various scientific instruments like mass spectrometers. They also play a crucial role in biological processes, such as the structure and function of proteins and DNA.

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