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[[File:Gaussian2.jpg|thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire.]]
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A '''Gaussian surface''' is a [[closed surface]] in three dimensional space through which the [[flux]] of a [[vector field]] is calculated; usually the [[gravitational field]], the [[electric]] field, or [[magnetic]] field.<ref>Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1</ref> It is an arbitrary [[closed surface]] ''S'' = ∂V''V'' used in conjunction with Gauss's law for the corresponding field ([[Gauss's law]], [[Gauss's law for magnetism]], or [[Gauss's law for gravity]]) by performing a [[surface integral]], in order to calculate the total amount of the source quantity enclosed, i.e. amount of [[gravitational mass]] as the source of the gravitational field or amount of [[electric charge]] as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.
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For concreteness, the electric field is considered in this article, as this is the most frequent type of field the surface concept is used for.
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Gaussian surfaces are usually carefully chosen to exploit [[symmetry|symmetries]] of a situation to simplify the calculation of the [[surface integral]]. If the Gaussian surface is chosen such that for every point on the surface the component of the [[electric field]] along the [[normal vector]] is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral.
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== Common Gaussian surfaces ==
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{{see also|charge density}}
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[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Examples of valid (left) and invalid (right) Gaussian surfaces. Left: Some valid Gaussian surfaces include the surface of a sphere, surface of a torus, and surface of a cube. They are [[closed surface]]s that fully enclose a 3D volume. Right: Some surfaces '''cannot''' be used as Gaussian surfaces, such as the [[disk (mathematics)|disk surface]], square surface, or hemisphere surface. They do not fully enclose a 3D volume, and have boundaries (red).]]
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Most calculations using Gaussian surfaces begin by implementing [[Gauss's law]] (for electricity):<ref>Introduction to electrodynamics  By: Griffiths D.J</ref>
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| preintegral = <math>\Phi_E = \,\!</math>
 
| intsubscpt = <math>\scriptstyle \partial V\,\!</math>
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| integrand = <math>\mathbf E\;\cdot\mathrm{d}\mathbf A = \frac{Q(V)}{\varepsilon_0}.\,\!</math>}}
 
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Thereby ''Q''(''V'') is the electrical charge contained in the interior, ''V'', of the closed surface.
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This is Gauss's law, combining both the [[divergence theorem]] and [[Coulomb's law]].
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=== Spherical surface ===
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A [[sphere|spherical]] Gaussian surface is used when finding the electric field or the flux produced by any of the following:<ref>Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7</ref>
* a [[point charge]]
* a uniformly distributed [[spherical shell]] of charge
* any other charge distribution with [[Circular symmetry|spherical symmetry]]
The spherical Gaussian surface is chosen so that it is concentric with the charge distribution.


As an example, consider a charged spherical shell ''S'' of negligible thickness, with a uniformly distributed charge ''Q'' and radius ''R''. We can use Gauss's law to find the magnitude of the resultant electric field ''E'' at a distance ''r'' from the center of the charged shell. It is immediately apparent that for a spherical Gaussian surface of radius ''r'' < ''R'' the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting ''Q''<sub>''A''</sub> = 0 in Gauss's law, where ''Q''<sub>''A''</sub> is the charge enclosed by the Gaussian surface).
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With the same example, using a larger Gaussian surface outside the shell where ''r'' > ''R'', Gauss's law will produce a non-zero electric field.  This is determined as follows.
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| preintegral = <math>\Phi_E = \,\!</math>
| intsubscpt = <math>\scriptstyle \partial S\,\!</math>
| integrand = <math> \mathbf{E}\cdot d \mathbf{A} = \int\!\!\!\!\int_c E dA\cos 0^\circ = E \int\!\!\!\!\int_S dA \,\!</math>}}


The [[Surface area#Common formulas|surface area of the sphere]] of radius ''r'' is
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:<math>\Phi_E =\frac{Q_A}{\varepsilon_0}</math>
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:<math>E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}.</math>
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This non-trivial result shows that any spherical distribution of charge ''acts as a point charge'' when observed from the outside of the charge distribution; this is in fact a verification of [[Coulomb's law]]. And, as mentioned, any exterior charges do not count.
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* an infinitely long [[line (geometry)|line]] of uniform charge
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* an infinite [[Plane (geometry)|plane]] of uniform charge
 
* an infinitely long [[Cylinder (geometry)|cylinder]] of uniform charge
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As example "field near infinite line charge" is given below;
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Consider a point ''P'' at a distance ''r'' from an infinite line charge having [[charge density]] (charge per unit length) λ (lambda). Imagine a closed surface in the form of cylinder whose axis of rotation is the line charge. If ''h'' is the length of the cylinder, then the charge enclosed in the cylinder is
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:<math> q = \lambda h </math>,
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where ''q'' is the charge enclosed in the Gaussian surface. There are three surfaces ''a'', ''b'' and ''c'' as shown in the figure. The [[differential of a function|differential]] [[vector area]] is d'''A''', on each surface ''a'', ''b'' and ''c''.
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[[File:Gaussian surface.jpg|right|frame|Closed surface in the form of a cylinder having line charge in the center and showing differential areas d'''A'''of all three surfaces.]]
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The flux passing consists of the three contributions
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| preintegral = :<math> \Phi_E  = \,\!</math>
 
| intsubscpt = <math>\scriptstyle A\,\!</math>
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| integrand = <math>\mathbf{E} \cdot d\mathbf{A} = \int\!\!\!\!\int_a \mathbf{E} \cdot d\mathbf{A} + \int\!\!\!\!\int_b\mathbf{E} \cdot d\mathbf{A} + \int\!\!\!\!\int_c\mathbf{E} \cdot d\mathbf{A} </math>}}
 
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For surfaces a and b, '''E''' and d'''A''' will be [[perpendicular]].
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For surface c, '''E''' and d'''A''' will be [[Parallel (geometry)|parallel]], as shown in the figure.


:<math> \begin{align}
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\Phi_E & = \int\!\!\!\!\int_a E dA\cos 90^\circ + \int\!\!\!\!\int_b E d A \cos 90^\circ + \int\!\!\!\!\int_c E d A\cos 0^\circ \\
 
& = E \int\!\!\!\!\int_c dA\\
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\end{align}
 
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The [[Surface area#Common formulas|surface area of the cylinder]] is
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:<math> \int\!\!\!\!\int_c dA = 2 \pi r h </math>
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:<math> \Phi_E = E 2 \pi r h </math>
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By Gauss's law
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:<math> \Phi_E = \frac{q}{\varepsilon_0}</math>
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:<math> E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} </math>
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This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. The [[pillbox]] has a cylindrical shape, and can be thought of as consisting of three components: the [[Disk (mathematics)|disk]] at one end of the cylinder with area πR², the disk at the other end with equal area, and the side of the cylinder. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Because the field close to the sheet can be approximated as constant, the pillbox is oriented in a way so that the field lines penetrate the disks at the ends of the field at a perpendicular angle and the side of the cylinder are parallel to the field lines.


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* [[Area]]
  <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li>
* [[Surface area]]
 
* [[Vector calculus]]
  <li>[http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/ http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/]</li>
* [[Integration (calculus)|Integration]]
 
* [[Divergence theorem]]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24593894 http://enseignement-lsf.com/spip.php?article64#forum24593894]</li>
* [[Faraday cage]]
 
* [[Classical field theory|Field theory]]
  <li>[http://verdamilio.net/tonio/spip.php?article227/ http://verdamilio.net/tonio/spip.php?article227/]</li>
* [[Field line]]
 
 
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
== References ==
 
 
</ul>
{{reflist}}
 
*{{cite book |author=Purcell, Edward M.|title=Electricity and Magnetism|publisher=McGraw-Hill|year=1985|isbn=0-07-004908-4}}
*{{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|isbn=0-471-30932-X}}
 
==Further reading==
 
* ''Electromagnetism (2nd Edition)'', I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
 
==External links==
* [http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html Fields] - a chapter from an online textbook
 
{{DEFAULTSORT:Gaussian Surface}}
[[Category:Surfaces]]
[[Category:Electrostatics]]

Revision as of 15:29, 24 February 2014

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Fake Tiffany Jewelry Australia $A1.72 billion

Clark spins off health care industry

Kimberly Clark's stock jumped five per cent in after hours buying and selling following the announcement on Thurs Fake Tiffany Jewelry Australia night.

The health care business, making medical devices and medical products, has about $US1.Six billion ($A1.72 billion) within annual sales.

The spinoff is predicted to be completed at the end of the 3rd quarter of 2014 and is governed by market, regulatory and other ailments. Kimberly Clark is working with Morgan Stanley on the deal.

Robert Abernathy, who is president regarding Kimberly Clark's Europe group, will Mbt Masai Australia become CEO of the new health care organization. Abernathy has worked at Dallas based mostly Kimberly Clark since 1982.

"While K Chemical Health Care has been part of our company since the 1970s, its proper fit and growth main concerns have changed over time Timberland Stockists Brisbane therefore we now think that pursuing your spin off makes sense for the shareholders," said Kimberly Clark Founder and Chairman Thomas Falk.

Medical care business is based in Roswell, Atlanta, and has more than Nike Free Run 3 Online Australia 16,1000 employees around the world.

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