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Related Concept Videos

Magnetic Field Of A Current Loop01:16

Magnetic Field Of A Current Loop

Consider a circular loop with a radius a, that carries a current I. The magnetic field due to the current at an arbitrary point P along the axis of the loop can be calculated using the Biot-Savart law.
Divergence and Curl of Magnetic Field01:26

Divergence and Curl of Magnetic Field

The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows:
Magnetostatic Boundary Conditions01:28

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...
Ampere's Law: Problem-Solving01:31

Ampere's Law: Problem-Solving

Ampere's law states that for any closed looped path, the line integral of the magnetic field along the path equals the vacuum permeability times the current enclosed in the loop. If the fingers of the right hand curl along the direction of the integration path, the current in the direction of the thumb is considered positive. The current opposite to the thumb direction is considered negative.
Specific steps need to be considered while calculating the symmetric magnetic field distribution using...
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

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,...
Magnetic Field of a Solenoid01:18

Magnetic Field of a Solenoid

A solenoid is a conducting wire coated with an insulating material, wound tightly in the form of a helical coil. The magnetic field due to a solenoid is the vector sum of the magnetic fields due to its individual turns. Therefore, for an ideal solenoid, the magnetic field within the solenoid is directly proportional to the number of turns per unit length and the current. Conversely, the magnetic field outside the solenoid is zero.
Consider a solenoid with 100 turns wrapped around a cylinder of...

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The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

The Diffusion of Passive Tracers in Laminar Shear Flow

Published on: May 1, 2018

Analytical solution for restricted diffusion in circular and spherical layers under inhomogeneous magnetic fields.

Denis S Grebenkov1

  • 1Laboratoire de Physique de la Matière Condensée, CNRS-Ecole Polytechnique, F-91128 Palaiseau, France. denis.grebenkov@polytechnique.edu

The Journal of Chemical Physics
|April 10, 2008
PubMed
Summary

We developed an analytical solution for restricted diffusion in magnetic fields. This reveals a new diffusion regime in thin layers, analogous to porous media, useful for detecting length scales.

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Area of Science:

  • Physics
  • Physical Chemistry
  • Materials Science

Background:

  • Restricted diffusion phenomena are crucial in various scientific fields.
  • Understanding particle diffusion in inhomogeneous magnetic fields requires advanced analytical models.

Purpose of the Study:

  • To propose an analytical solution for restricted diffusion of spin-bearing particles.
  • To investigate diffusion in circular and spherical layers within inhomogeneous magnetic fields.
  • To derive explicit formulas for magnetic field matrices and magnetization evolution.

Main Methods:

  • Derivation of exact and explicit formulas for the magnetic field matrix in the Laplacian eigenbasis.
  • Application of perturbative calculations for thin layers with distinct geometrical length scales.
  • Analysis of magnetization evolution governed by derived formulas.

Main Results:

  • An analytical solution for restricted diffusion in layered geometries was established.
  • A novel intermediate diffusion regime with a constant apparent diffusion coefficient (ADC) was identified in two-scale geometries.
  • This regime's emergence was linked to Laplace operator eigenvalues, analogous to tortuosity in porous media.

Conclusions:

  • The study provides a theoretical framework for understanding restricted diffusion in complex geometries.
  • The observed constant ADC regime offers a potential experimental method for characterizing multiscale geometries.
  • Findings can advance the study of diffusion in systems with varying length scales, such as porous materials and biological tissues.