NationMaster - Encyclopedia: Scalar potential

A scalar potential is, mathematically, a scalar field whose negative gradient is a given vector field. If the scalar potential is denoted by the Greek letter φ and the vector field it generates as v, then Wikipedia does not have an article with this exact name. ... It has been suggested that this article or section be merged with Scalar potential. ... In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

$mathbf{v} = - nabla phi qquad qquad (1)$ .

The vector field can be a velocity field or a force field. Equation (1) therefore means a movement or acceleration towards the direction in which there will be a decrease of potential.

Physically, the scalar potential is similar or identical to potential energy. Any conservative force field can be represented as the negative gradient of some scalar potential. Potential energy is stored energy. ... A conservative force is a force which is path-independent. ...

Any lamellar field can be represented as having a scalar potential, but a solenoidal field generally does not have a scalar potential (except the degenerate case when it is Laplacian). In vector analysis and in fluid dynamics, a lamellar vector field is a vector field with no rotational component. ... This article is in need of attention. ... In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. ...

1 Altitude as gravitational potential
2 Pressure as buoyant potential
3 Calculating the scalar potential
4 See also

Altitude as gravitational potential

An example is the (nearly) uniform gravitational field near the Earth's surface. It has a potential The gravitational field is a field that causes bodies with mass to attract each other. ...

U = m g h

where U is the gravitational potential and h is the height above the surface. This means that gravitational potential on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface which cancels out the component of gravity which is perpendicular to the hill's surface. The component of gravity which remains to move the ball is parallel to the surface: Example of a topographic map with contour lines Topographic maps, also called contour maps, topo maps or topo quads (for quadrangles), are maps that show topography, or land contours, by means of contour lines. ...

$F_S = - m g sin theta$

where θ is the angle of inclination, and the component of F_S perpendicular to gravity is

$F_P = - m g sin theta cos theta = - {1 over 2} m g sin 2 theta$

This force F_P, parallel to the ground, will be greatest when θ is 45 degrees.

Let Δh be the uniform interval of altitude between contours on the contour map, and let Δx be the distance between two contours. Then

$theta = tan^{-1}{Delta h over Delta x}$

so that

$F_P = - m g { Delta x Delta h over Delta x^2 + Delta h^2 }$ .

However, on a contour map, the gradient will be inversely proportional to Δx, which is not similar to force F_P: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.

Pressure as buoyant potential

In fluid mechanics, a fluid in equilibrium but in the presence of a uniform gravitational field will be permeated by a uniform buoyant force which will cancel out the gravitational force: that is how the fluid maintains its equilibrium. This buoyant force is the negative gradient of pressure: Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ... In physics, buoyancy is an upward force on an object immersed in a fluid (i. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ...

$mathbf{f_B} = - nabla p$ .

The buoyant force points upwards, in the direction opposite to gravity. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes which are normal to the gravitational vector.

If the liquid has a vertical vortex (whose axis of rotation is perpendicular to the ground), then the vortex will cause a depression in the pressure field. The surfaces of constant pressure will be normal to the gravitational vector far away from the vortex, but near and inside the vortex the surfaces of constant pressure will be pulled downwards. This will also happen to the surface of zero pressure: therefore, inside the vortex, the top surface of the liquid dips into a depression, or even into a tube (a solenoid). Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex is a spinning turbulent flow (or any spiral whirling motion) with closed streamlines. ...

Calculating the scalar potential

Given a vector field E, its scalar potential can be calculated to be

$phi(mathbf{R_0}) = {1 over 4 pi} int_tau {nabla cdot mathbf{E}(tau) over | mathbf{R}(tau) - mathbf{R_0} |} , dtau$

where τ is volume. Then, if E is irrotational, In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ...

$mathbf{E} = -nabla phi = - {1 over 4 pi} nabla int_tau {nabla cdot mathbf{E}(tau) over | mathbf{R}(tau) - mathbf{R_0} |} , dtau$ .

This formula is known to be correct if E is continuous and vanishes asymptotically to zero towards infinity, decaying faster than 1/r and if the divergence of E likewise vanishes towards infinity, decaying faster than 1/r². In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

Categories: Articles to be merged | Classical mechanics | Electrostatics | Fluid mechanics | Multivariate calculus Electric potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... The fundamental theorem of vector analysis states that any vector field meeting certain conditions (of decaying towards infinity) can be resolved into irrotational and solenoidal component vector fields. ... A Yukawa potential (also called a screened Coulomb potential) is a potential of the form Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of the pion whose mass is . ...

Results from FactBites:

PowerPedia:Scalar field theory - PESWiki (4720 words)
	Scalar waves are hypothetical waves, which differ from the conventional electromagnetic transverse waves by one oscillation level parallel to the direction of propagation; they thus have characteristics of longitudinal waves.
	Scalar field theory (SWT) is a set of theories in a abstract model which posits that there is a basic mechanism that produces the electric field and the magnetic field.
	Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space.

Healing America Scalar Energy (1729 words)
	The term scalar was used by Nikola Tesla at the end of the last century as a part of powerful non-Hertzian energy (without frequencies) which he referred to as standing energy or cosmic waves.
	A survey of the literature suggests that 'scalar' energy is the most appropriate description of healing energy up-to-date, as it allows for the fact that healing energy has been shown to be not part of the electromagnetic spectrum as we understand it.
	According to Robert Jacobs, scalar waves are capable of acting on living organisms at a sub-atomic level, and certain frequencies of scalar have been shown to destroy viruses and bacteria.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Altitude as gravitational potential GA_googleFillSlot("encyclopedia_square");

Pressure as buoyant potential

Calculating the scalar potential

See also

Altitude as gravitational potential