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Related Concept Videos

Electric Field of a Continuous Line Charge01:19

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In physics, symmetry in a system means that something in the considered system remains unchanged due to a specific operation to which it is subjected. For example, consider a horizontal square. The square looks the same if its right and left sides are interchanged. Hence, it is symmetric under a right-left interchange.
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An electric dipole is a system of two equal but opposite charges, separated by a fixed distance. This system is used to model many real-world systems, including atomic and molecular interactions. One of these systems is the water molecule, but only under certain circumstances. These circumstances are met inside a microwave oven, where electric fields with alternating directions make the water molecules change orientation. This vibration is equivalent to heat at the molecular level.
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Consider a ring of radius R with a uniform charge density λ. What will the electric potential be at point M, which is located on the axis of the ring at a distance x from the center of the ring?
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Imagine a bucket of water. It contains many molecules, of the order of 1026 molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.
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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The equipotential surface is always perpendicular to the electric field lines, and while it is three-dimensional, it can be treated as an equipotential line in a two-dimensional case. These equipotential lines are also always perpendicular to electric field lines. The term equipotential is often used as a noun, referring to an equipotential line or...
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A FAST SIMPLE ALGORITHM FOR COMPUTING THE POTENTIAL OF CHARGES ON A LINE.

Zydrunas Gimbutas1, Nicholas F Marshall2, Vladimir Rokhlin3

  • 1National Institute of Standards and Technology, Boulder, CO 80305, USA.

Applied and Computational Harmonic Analysis
|October 22, 2020
PubMed
Summary

This paper introduces a fast computational method for calculating electrostatic potentials from charges distributed along a line. The technique offers significant speed improvements for specific physics simulations.

Keywords:
31C20 (primary) and 41A5541A50 (secondary)Chebyshev systemFast multipole methodgeneralized Gaussian quadrature

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Area of Science:

  • Computational Physics
  • Numerical Analysis
  • Electromagnetism

Background:

  • Evaluating sums of inverse differences is common in physics simulations.
  • Existing methods for electrostatic potential calculation can be computationally intensive.
  • A need exists for efficient algorithms for line charge distributions.

Purpose of the Study:

  • To develop a fast and simple method for evaluating expressions of the form u_j = sum(alpha_i / (x_i - x_j)).
  • To apply this method to calculating electrostatic potentials of charges on a line in R^3.
  • To demonstrate the method's efficiency and provide numerical examples.

Main Methods:

  • The study presents a novel algorithm for evaluating the specified summation.
  • The method is designed for speed and simplicity.
  • Numerical results are reported to validate the approach.

Main Results:

  • A computationally efficient method for evaluating the target expression was developed.
  • The method significantly speeds up calculations for line charge distributions.
  • Numerical examples confirm the accuracy and speed of the proposed technique.

Conclusions:

  • The presented method provides a fast and effective way to compute electrostatic potentials for linear charge distributions.
  • This algorithm is particularly useful in computational physics scenarios requiring high efficiency.
  • The findings offer a valuable tool for researchers working with charge distributions on a line.