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Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

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...

Gauss's Law: Spherical Symmetry

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...

Plane Electromagnetic Waves I

Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of Maxwell's equations. As Maxwell's equations hold in free space, the predicted electromagnetic waves do not require a medium for their propagation. An EM wave comprises an electric field, defined as the force per charge on a stationary charge, and a magnetic field, which is the force per charge on a moving charge.
The EM field is assumed to be a...

Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Gauss's Law: Cylindrical Symmetry

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,...

Gauss's Law in Dielectrics

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Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...

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Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data

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Spatial pattern formation induced by Gaussian white noise.

Stefania Scarsoglio¹, Francesco Laio, Paolo D'Odorico

¹Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Turin, Italy. stefania.scarsoglio@polito.it

Mathematical Biosciences

|December 21, 2010

Summary

This summary is machine-generated.

Gaussian noise can create ordered states and spatial patterns in dynamical systems. This study explores stochastic mechanisms involving local dynamics, environmental disturbances, and diffusion to understand pattern formation.

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Area of Science:

Physics
Mathematics
Complex Systems

Background:

Dynamical systems can exhibit complex behaviors.
Stochastic processes, like noise, can influence system dynamics.
Understanding pattern formation is crucial in various scientific fields.

Purpose of the Study:

To provide an overview of stochastic mechanisms for spatial pattern generation in dynamical systems.
To investigate the role of Gaussian noise in inducing ordered states.
To analyze the interplay between deterministic dynamics, noise, and spatial coupling.

Main Methods:

Analytical tools including mean-field theory, linear stability analysis, and structure function analysis.
Numerical simulations to validate analytical findings.
Investigation of additive and multiplicative noise components.

Main Results:

Gaussian noise can indeed induce ordered states and spatial patterns.
The interplay of local dynamics, noise, and spatial coupling is key to pattern formation.
Analytical and numerical methods confirm the proposed stochastic mechanisms.

Conclusions:

Stochastic mechanisms, particularly Gaussian noise, are effective in generating spatial patterns in dynamical systems.
The study highlights the importance of considering noise and spatial coupling in understanding complex system behavior.
Further research can explore diverse noise types and system complexities.