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Velocity Potential

Velocity Potential

In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:

Divergence and Curl of Electric Field

Divergence and Curl of Electric Field

The divergence of a vector is a measure of how much the vector spreads out (diverges) from a point. For example, an electric field vector diverges from the positive charge and converges at the negative charge. The divergence of an electric field is derived using Gauss's law and is equal to the charge density divided by the permittivity of space. Mathematically, it is expressed as

Magnetic Vector Potential

Magnetic Vector Potential

In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic field ensures that the magnetic field can be expressed as the curl of a vector potential. This potential is known as the magnetic vector potential.
Consider an ideal solenoid with n turns per unit length and radius R. If I is the current through the solenoid, the magnetic field inside the solenoid is expressed as the product of vacuum...

Plane Potential Flows

Plane Potential Flows

Plane potential flows simplify fluid motion by assuming the fluid to be irrotational and incompressible. These characteristics allow these flows to be described by a velocity potential function, ϕ, representing the flow speed in a given direction, and a stream function, ψ, that visualizes the flow path, both governed by Laplace's equation. These parameters help in estimating flow patterns, velocity distributions, and pressure fields around various hydraulic structures.
Uniform...

Electric Field of a Non Uniformly Charged Sphere

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...

Magnetostatic Boundary Conditions

Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...

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Related Experiment Video

Updated: May 7, 2026

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Vortex distribution in a confining potential.

Matheus Girotto¹, Alexandre P Dos Santos, Yan Levin

¹Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 16, 2013

Summary

This summary is machine-generated.

We studied interacting vortices in type II superconductors. Our findings show Boltzmann-Gibbs statistical mechanics remains valid even when mean-field theory fails in the strong coupling limit.

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Published on: July 28, 2023

Area of Science:

Condensed Matter Physics
Superconductivity Theory

Background:

Type II superconductors host interacting vortices.
Mean-field theory is applicable in the weak coupling limit.
Vortex correlations become significant in the strong coupling limit.

Purpose of the Study:

To investigate the behavior of interacting vortices in type II superconductors.
To determine the validity of mean-field theory and statistical mechanics in different coupling limits.
To analyze vortex density distribution within a confining potential.

Main Methods:

Construction of a mean-field theory for the weak coupling limit.
Application of molecular dynamics simulations.
Utilization of Monte Carlo simulations for comparison.

Main Results:

Accurate calculation of vortex density distribution in the weak coupling limit.
Demonstration that mean-field theory fails in the strong coupling limit due to particle correlations.
Validation of Boltzmann-Gibbs statistical mechanics through simulation results.

Conclusions:

Interacting vortices in type II superconductors exhibit complex behavior dependent on coupling strength.
Boltzmann-Gibbs statistical mechanics is robust and applicable beyond mean-field approximations.
Simulations confirm the continued validity of statistical mechanics where mean-field theory breaks down.