Related Experiment Video
Updated: Jun 25, 2026
08:23
Finite Element Modelling of a Cellular Electric Microenvironment
Published on: May 18, 2021
Packings of a charged line on a sphere
1School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA. alben@math.gatech.edu
Summary
Lines of charge on a sphere form stable spiral shapes as the sphere shrinks. These configurations minimize electrostatic energy, revealing fundamental principles of charge behavior on curved surfaces.
Area of Science:
- Computational Physics
- Electrostatics
- Condensed Matter Physics
Background:
- Understanding charge distribution on curved surfaces is crucial for various physical phenomena.
- Previous studies have explored simplified models, but complex equilibrium configurations remain an active research area.
Purpose of the Study:
- To determine equilibrium configurations of open and closed lines of charge on a sphere.
- To analyze how these configurations evolve with changes in sphere radius.
- To investigate the impact of different repulsive potentials on charge line behavior.
Main Methods:
- Numerical simulations were employed to find equilibrium configurations of charged lines.
- The study tracked these configurations as the sphere radius was systematically varied.
- Analysis included examining bifurcations and stability of different charge arrangements.
Main Results:
- Closed lines of charge transition from circular to spiral shapes via 'baseball seam' and 'twist' bifurcations, minimizing Coulomb energy.
- The spiral configuration is the unique stable equilibrium for closed lines; unstable equilibria arise from tip-splitting.
- Open lines smoothly transition to spiral shapes, and under various repulsive potentials, converge to a uniform spiral spacing during shrinkage.
Conclusions:
- The sphere's geometry dictates stable charge line configurations, favoring spiral patterns.
- Bifurcation events play a critical role in the transition between equilibrium states.
- The study reveals universal behavior in charge distribution under different repulsive forces at later stages of sphere shrinkage.
Related Concept Videos
01:22Electric Field of a Non Uniformly Charged Sphere
Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...
01:17Continuous Charge Distributions
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.
The electric charge can also be subjected to an analogical...
The electric charge can also be subjected to an analogical...
01:26Gauss's Law: Spherical Symmetry
A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...
01:23Electric Field of a Charged Disk
The simplest case of a surface charge distribution is the uniformly charged disk. Calculating its electric field also helps us calculate the electric field of a large plane of charge.
The system's symmetry is in the cylindrical directions across the plane of the charge. As a result, the electric fields created by various surface charge elements nullify each other in the direction parallel to the surface. Thereby, the resulting electric field is perpendicular to the plane. Since the disk is...
The system's symmetry is in the cylindrical directions across the plane of the charge. As a result, the electric fields created by various surface charge elements nullify each other in the direction parallel to the surface. Thereby, the resulting electric field is perpendicular to the plane. Since the disk is...
01:25Electric Field Lines
The three-dimensional representation of the electric field of a positive point charge requires tracing the electric field vectors, whose lengths decrease as the square of their distance from the charge and which point away from the charge at each point. This vector field is no doubt challenging to visualize. The visualization of electric fields becomes quickly intractable as the number of charges increases.
The solution to this problem is to use electric field lines, which are not vectors but...
The solution to this problem is to use electric field lines, which are not vectors but...
01:19Electric Field of a Continuous Line Charge
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.
In calculations of electric fields, symmetry is of great use. For example, while calculating electric fields of continuous charge distributions.
Consider a line element with a...
In calculations of electric fields, symmetry is of great use. For example, while calculating electric fields of continuous charge distributions.
Consider a line element with a...