高斯面（、縮寫：），又稱高斯曲面，是三維空間一閉合曲面，用於運用高斯定理計算向量場的通量，例如重力場、電場和磁場。Essential Principles of Physics PM Whelan MJ Hodgeson 2nd Edition 1978 John Murray 是任意形狀的封閉曲面（3 維 流形邊界），透過對其進行曲面積分運算，可以求出曲面中包含的場源總量，例如重力場中包含的物質總量和靜電場場源中包含的總電荷量等等，也可以反過來從場源推算它產生的場。例如這裡所舉的最常見的情況，運用高斯曲面和高斯定理計算電場的時候，運用對稱性選擇恰當的高斯面，可以簡化所研究的問題，使曲面積分更簡單。如果高斯曲面上的每一點都能使該點垂直曲面的電場分量為常數，進行曲面積分的時候就能大大簡化運算，因為常數可以從積分式中被提取出來。 常見的高斯曲面 大多數使用高斯曲面計算都是從研究高斯定律電開始：Introduction to electrodynamics 4th Edition D J Griffiths 2012 從而是被高斯曲面包圍的電荷。 這是結合了高斯散度定理和庫侖定律的高斯定律。 高斯球面 當找到由以下任何一種產生的電場或通量時，可使用高斯球面：Physics for Scientists and Engineers with Modern Physics 6th Edition P A Tipler G Mosca Freeman 2008 a point charge a uniformly distributed spherical shell of charge any other charge distribution with 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 of negligible thickness with a uniformly distributed charge and radius We can use Gausss law to find the magnitude of the resultant electric field at a distance from the center of the charged shell It is immediately apparent that for a spherical Gaussian surface of radius 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 in Gausss law where is the charge enclosed by the Gaussian surface With the same example using a larger Gaussian surface outside the shell where Gausss law will produce a nonzero electric field This is determined as follows 球面通量為： The surface area of the sphere of radius is iint_S dA 4 pi r2 which implies Phi_E E 4pi r2 By Gausss law the flux is also Phi_E frac Q_Avarepsilon_0 finally equating the expression for gives the magnitude of the field at position E 4pi r2 frac Q_Avarepsilon_0 quad Rightarrow quad Efrac Q_A4pivarepsilon_0r2 This nontrivial 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 Coulombs law And as mentioned any exterior charges do not count 高斯圓柱面 當找到由以下任何一種產生的電場或通量時，可使用高斯圓柱面： 一條無限長的均勻電荷線 一個無限均勻電荷平面 一個無限長的均勻電荷圓柱體 例如 無限線電荷附近的場 如下所示； Consider a point P at a distance from an infinite line charge having charge density charge per unit length λ Imagine a closed surface in the form of cylinder whose axis of rotation is the line charge If is the length of the cylinder then the charge enclosed in the cylinder is q lambda h where is the charge enclosed in the Gaussian surface There are three surfaces a b and c as shown in the figure The differential vector area is on each surface a b and c The flux passing consists of the three contributions For surfaces a and b and will be perpendicular For surface c and will be parallel as shown in the figure begin align Phi_E iint_a E dAcos 90circ iint_b E d A cos 90circ iint_c E d Acos 0circ E iint_c dA end align The surface area of the cylinder is iint_c dA 2 pi r h which implies Phi_E E 2 pi r h By Gausss law Phi_E frac qvarepsilon_0 equating for yields E 2 pi rh frac lambda hvarepsilon_0 quad Rightarrow quad E frac lambda2 pivarepsilon_0 r 高斯盒 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 at one end of the cylinder with area 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 Gausss 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 另見 表面積 微積分 積分 法拉第籠 經典場論 場線 參考 進一步閱讀 Electromagnetism 2nd Edition IS Grant WR Phillips Manchester Physics John Wiley Sons 2008 外部連結 Fields 線上教科書 Category 曲面 Category 靜電學 Category 卡爾弗里德里希高斯