Do equipotential lines fall on the equiprobability contours?

Click For Summary
SUMMARY

The discussion centers on the relationship between equipotential lines and equiprobability contours in a 2D charge distribution defined by ρ(x,y)=Ne PDF(x,y). It is established that for a 2D Gaussian distribution with equal standard deviations (σx = σy), the equipotential contours coincide with the equiprobability contours. However, as the distance from the core of the distribution increases, the equipotential surfaces become circular, diverging from the equiprobability contours. The conversation also touches on various distribution functions, emphasizing the need for clarity in defining terms like "NDF" and its relation to Gaussian distributions.

PREREQUISITES
  • Understanding of 2D charge distributions and their mathematical representations.
  • Familiarity with probability density functions (PDF) and their properties.
  • Knowledge of Gaussian distributions, including their standard deviations and contour characteristics.
  • Basic concepts of electrostatics, particularly equipotential lines and surfaces.
NEXT STEPS
  • Research the properties of 2D Gaussian distributions and their equipotential contours.
  • Explore the differences between various types of probability density functions, such as bi-Gaussian and super-Gaussian.
  • Learn about the mathematical derivation of equipotential surfaces in electrostatics.
  • Investigate the implications of varying standard deviations (σx and σy) on contour shapes in charge distributions.
USEFUL FOR

Physicists, electrical engineers, and students studying electrostatics or statistical mechanics will benefit from this discussion, particularly those interested in the mathematical relationships between charge distributions and probability contours.

Mikheal
Messages
5
Reaction score
0
TL;DR
Are equipotential lines fall on the equiprobability contours of charge distribution?
For 2D charge distribution ρ(x,y)=Ne PDF(x,y), where PDF is the normalized probability density function with its peak on (0,0) and has standard deviations σ x. and σ y. Are the contours with the equal probability "PDF(x,y)=const" the same as the equipotiential contours?, I tend to think that near the core of the distribution, they will be similar, and as the distance from the core increases, the equipotential surfaces will be circles for σx=σy.

Edit 1: I am speaking in general, not about certain particle distribution functions, such as 2D Gaussian with different σ x and σ y, 2D bi-Gaussian, 2D super-Gaussian, Flat-top, ....

Edit 2: I know that for 2D Gaussian with σ x = σ y, they fall on each other.
 
Last edited:
Physics news on Phys.org
Is NDF a Gaussian? Your question needs to be a little bit more definitive.
 

Similar threads

Show that the equipotential lines are circles
  • · Replies 16 ·
Replies
16
Views
3K
How Does Charge Distribution Affect Electric Field in a Ring?
  • · Replies 1 ·
Replies
1
Views
2K