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Related Concept Videos

Gravity between Spherical Bodies01:27

Gravity between Spherical Bodies

Newton's law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, the Moon, and other planets, the law holds with an assumption that masses of spherical objects are concentrated at their respective centers.
This assumption can be proved easily by showing that the expression for gravitational potential energy between a hollow sphere of mass (M) and a point mass (m) is the same as it would be for a pair of extended...
Gravitation Between Spherically Symmetric Masses01:14

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, assuming that the center of mass is concentrated at the respective centers of the bodies.
Electric Field of a Non Uniformly Charged Sphere01:22

Electric 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...
Molecular Shapes01:18

Molecular Shapes

Molecules have characteristic shapes that are crucial for their function. The arrangement of various electron groups around the central atom dictates their molecular geometry. Electron pairs in the valence shell of a central atom will adopt an arrangement that minimizes repulsions between the electron pairs by maximizing the distance between them. The valence electrons form either bonding pairs, located primarily between bonded atoms, or lone pairs.Two regions of electron density in a diatomic...
Gauss's Law: Spherical Symmetry01:26

Gauss'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...
Crystal Field Theory - Octahedral Complexes02:58

Crystal Field Theory - Octahedral Complexes

Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...

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Related Experiment Video

Updated: Jun 29, 2026

Impacts of Free-falling Spheres on a Deep Liquid Pool with Altered Fluid and Impactor Surface Conditions
08:49

Impacts of Free-falling Spheres on a Deep Liquid Pool with Altered Fluid and Impactor Surface Conditions

Published on: February 17, 2019

Exciting hard spheres.

T Antal1, P L Krapivsky, S Redner

  • 1Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2008
PubMed
Summary
This summary is machine-generated.

A single particle entering a hard-sphere gas creates a collision cascade. The number of particles and collisions grows over time, matching hydrodynamic shock wave theories and verified by simulations.

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Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
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Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

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Related Experiment Videos

Last Updated: Jun 29, 2026

Impacts of Free-falling Spheres on a Deep Liquid Pool with Altered Fluid and Impactor Surface Conditions
08:49

Impacts of Free-falling Spheres on a Deep Liquid Pool with Altered Fluid and Impactor Surface Conditions

Published on: February 17, 2019

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
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Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Area of Science:

  • Physics
  • Statistical Mechanics
  • Computational Physics

Background:

  • Collision cascades are fundamental in understanding particle interactions in gases.
  • Previous theories often relied on simplified models or macroscopic approximations.

Purpose of the Study:

  • To analyze the dynamics of a collision cascade initiated by a single particle in a hard-sphere gas.
  • To derive scaling laws for particle and collision number growth over time.
  • To compare these laws with existing hydrodynamic theories.

Main Methods:

  • Theoretical analysis of particle dynamics.
  • Derivation of growth laws based on spatial dimension (d).
  • Molecular dynamics simulations in 1 and 2 spatial dimensions.

Main Results:

  • The number of moving particles grows as t^ξ, and collisions grow as t^η.
  • Derived scaling exponents: ξ=2d/(d+2) and η=2(d+1)/(d+2).
  • Simulations confirmed these theoretical predictions for d=1 and d=2.

Conclusions:

  • The derived growth laws accurately describe collision cascade evolution.
  • The behavior mirrors shock wave phenomena from hydrodynamic theories.
  • A particle impacting a half-space gas results in significant backsplatter, retaining most initial energy.