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Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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Related Experiment Video

Updated: Apr 16, 2026

Fabrication and Visualization of Capillary Bridges in Slit Pore Geometry
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Capillary-bridge-derived particles with negative Gaussian curvature.

Liming Wang1, Thomas J McCarthy2

  • 1Polymer Science and Engineering Department, University of Massachusetts, Amherst, MA 01003.

Proceedings of the National Academy of Sciences of the United States of America
|March 3, 2015
PubMed
Summary

Researchers created millimeter-scale, catenoid-shaped particles with negative Gaussian curvature by polymerizing liquid monomer capillary bridges. These particles show tunable optical properties and self-assemble into unique structures, with potential for microscale applications and anisotropic adhesion.

Keywords:
capillary bridgecatenoidnegative Gaussian curvatureparticlewetting

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Area of Science:

  • Materials Science
  • Soft Matter Physics
  • Nanotechnology

Background:

  • Fabrication of complex micro/nanostructures is crucial for advanced materials.
  • Controlling particle shape and properties at small scales remains a challenge.
  • Capillary forces offer a route to self-assembly and structure formation.

Purpose of the Study:

  • To develop a method for preparing millimeter-scale particles with negative Gaussian curvature.
  • To investigate the tunability of particle shape and optical properties.
  • To explore the self-assembly behavior and potential applications of these particles.

Main Methods:

  • Thermal polymerization of liquid monomer capillary bridges between patterned surfaces.
  • Utilizing lithographic pinning features, surface separation, and lateral shear to control shape.
  • Observation of self-assembly in the presence of condensing water.

Main Results:

  • Successfully prepared millimeter-scale catenoid-shaped particles exhibiting negative Gaussian curvature.
  • Demonstrated fine control over particle shape and optical properties via tunable parameters.
  • Observed self-assembly into reversible neck-to-neck pairs and aggregates.

Conclusions:

  • The thermal polymerization of capillary bridges is an effective method for creating complex particle shapes.
  • Tunable optical properties and self-assembly behavior suggest potential for advanced material design.
  • The approach is scalable to micrometer dimensions, offering possibilities for anisotropic adhesion studies.