21. Gauss’s Law 高斯定律 Electric Field Lines 電場線 Electric Flux & Field 電通量和場 Gauss’s Law 高斯定律 Using Gauss’s Law 使用高斯定律 Fields of Arbitrary Charge Distributions 隨便一個電荷分佈的場 6. Gauss’s Law & Conductors 高斯定律和導體

操作人的籠子受到強烈電擊，但他卻安然無恙。為甚麼？ Huge sparks jump to the operator’s cage, but the operator is unharmed. Why? 操作人的籠子受到強烈電擊，但他卻安然無恙。為甚麼？ Ans: Gauss’ law  E = 0 inside cage 答案： 高斯定律  籠內 E = 0

21.1. Electric Field Lines 電場線 Field line gives direction of E 電場線表示 E 在該點的方向 Vector gives E at point 向量表示該點的 E Spacing gives magnitude of E 間隔表示 E 的大小 Electric field lines = Continuous lines whose tangent is everywhere // E. 電場線 = 連續的線，其切線皆 // E。 They begin at + charges & end at  charges or . 它們都始於 + 電荷，終於  電荷或。 Their density is  field strength or charge magnitude. 它們的密度  場的強度或電荷大小。

Field Lines of Electric Dipole 雙極的場線 Field is strong where lines are dense. 場在線密處較強。 Direction of net field tangent to field line 淨場的方向與場線相切 場線

Field Lines 8 條線始於+q … 而 8 條線終於 q。 故16 條線始於+2q …

21.2. Electric Flux & Field 電通量和場 8 lines out of surfaces 1, 2, & 3. But 22 = 0 out of 4 (2 out, 2 in). 面 1，2，和 3各有8條線外出。面4則有22 = 0條線外出 (2出，2入)。 16 lines out of surfaces 1, 2, & 3. But 0 out of 4. 面 1，2，和 3各有16條線外出。面4則有0條線外出。 8 lines out of (8 into) surfaces 1, 2, & 3. But 0 out of 4. 面 1，2，和 3 有8條線外出(8條進入)。面4則有0條線外出。

Number of field lines out of a closed surface  net charge enclosed. 8 lines out of surfaces 1 & 2. 面 1 和 2 各有8線外出。 16 lines out of surface 3. 面 3 有16 線外出。 0 out of 4. 面 4 則有0 線外出。  8 lines out of surface 1. 面 1 有 8 線外出。 8 lines out of surface 2. 面 2 有8 線外出。 44 = 0 lines out of surface 3. 面 3 則有44 =0 線外出。 Count these. 數這些。 1: 4 2: 8 3: 4 4: 0 Number of field lines out of a closed surface  net charge enclosed. 從密閉面出來的場線數  內藏淨電荷。

Electric Flux 電通量 // 面的法線 [  ]  N m2 / C. A flat surface is represented by a vector where A = area of surface and 平面可由向量 代表，其中 A = 面積 // 面的法線 Electric flux through flat surface A : 穿過平面 A 的電通量 [  ]  N m2 / C. E,  Open surface: can get from 1 side to the other w/o crossing surface. 開放面：不必穿過面本身就可以從面的一邊到另一邊。 Direction of A ambiguous. A 的方向不明確。 A,  Closed surface: can’t get from 1 side to the other w/o crossing surface. 密閉面：必需穿過面本身才可以從面的一邊到另一邊。 A defined to point outward. A 明定為朝外。

GOT IT? 21.1. The figure shows a cube of side s in a uniform electric field E. 圖示一均勻電場 E 中，邊為 s 的方塊。 What is the flux through each of the cube faces A, B, and C with the cube oriented as in (a) ? 通過 (a)圖中方塊面A，B，和 C 的電通量為何？ Repeat for the orientation in (b), with the cube rotated 45 °. 對(b)圖中轉了45 ° 的方塊重覆所作。

21.3. Gauss’s Law 高斯定律 高斯定律：通過任何密閉面的電通量與面內藏的淨電荷成正比。  Gauss’s law: Gauss’s law: The electric flux through any closed surface is proportional to the net charges enclosed. 高斯定律：通過任何密閉面的電通量與面內藏的淨電荷成正比。 depends on units. 視單位而定 For point charge enclosed by a sphere centered on it: 點電荷藏於一圓球的中心 SI units 公制  = vacuum permittivity 真空電容率 Field of point charge: 點電荷的場 Gauss’s law: 高斯定律

Gauss & Coulomb 高斯和庫倫 For a point charge 點電荷 : E  r 2 A  r 2 Outer sphere has 4 times area 外球有4倍面積 But E is 4 times weaker. 但 E 是4倍弱 So  is the same 所以  是一樣 For a point charge 點電荷 : E  r 2 A  r 2   indep of r. 與 r 無關。 Principle of superposition  argument holds for all charge distributions 叠加原理  推論適用於所有電荷分佈 Gauss’ & Coulomb’s laws are both expression of the inverse square law. 高斯和庫倫定律都是表達反平方定律。 For a given set of field lines going out of / into a point charge, inverse square law  density of field lines  E in 3-D. 對一組離開／進入一點電荷的場線來說，反平方定律  在3-D中，場線密度  E。

GOT IT? 21.2. A spherical surface surrounds an isolated positive charge, as shown. 如圖所示，一個球面包著一顆孤立的正電荷。 If a 2nd charge is placed outside the surface, which of the following will be true of the total flux though the surface? 如果球面外放了第二顆電荷，通過球面的總通量適合下列那一項？ it doesn’t change 不變 ; it increases 增加; it decreases 減少; it increases or decreases depending on the sign of the second charge. 增減由第二顆電荷的符號决定 Repeat for the electric field on the surface at the point between the charges. 對球面上兩電荷之間那點的電場重覆上述問題。 Opposite charges 異號電荷 Same charges 同號電荷

21.4. Using Gauss’s Law 使用高斯定律 Useful only for symmetric charge distributions. 在電荷分佈對稱時才有用。 Spherical symmetry: 球形對稱 ( point of symmetry at origin ) 對稱點在座標原點 

Example 21.1. Uniformily Charged Sphere 均勻帶電的球 A charge Q is spreaded uniformily throughout a sphere of radius R. 電荷 Q 均勻分佈在一半徑為 R 的球體中。 Find the electric field at all points, first inside and then outside the sphere. 求所有位置的電場，先算球內的，再算球外的。   True for arbitrary spherical (r). 任何球形 (r) 皆然。

Example 21.2. Hollow Spherical Shell 中空球殼 A thin, hollow spherical shell of radius R contains a total charge of Q distributed uniformly over its surface. 一個半徑為 R 的中空薄球殼的表面上均勻地分佈了總電荷 Q 。 Find the electric field both inside and outside the sphere. 求球內外的電場。 Reflection symmetry  E is radial. 鏡面對稱  E 為徑向  Contributions from A & B cancel. A 與 B 的貢獻互相抵銷。

GOT IT? 21.3. A spherical shell carries charge Q uniformly distributed over its surface. 一個球殼表面上均勻地分佈了Q 電荷。 If the charge on the shell doubles, what happens to the electric field 如果殼上的電荷加倍， inside and 殼內和 outside 殼外 the shell? 的電場會怎樣？ Stays 還是 0. Doubles 加倍.

Example 21.3. Point Charge Within a Shell 殼內點電荷 A positive point charge +q is at the center of a spherical shell of radius R carrying charge 2q, distributed uniformly over its surface. 一半徑為 R 的球殼，其表面上均勻地分佈了總電荷 2q ；其球心則有一正點電荷 +q 。 Find the field strength both inside and outside the shell. 求殼內外的電場。 

Tip: Symmetry Matters 祕訣：對稱性是重點 Spherical charge distribution inside a spherical shell is zero  E = 0 inside shell 球殼內球型分佈的總電荷為零  殼內 E = 0 E  0 if either shell or distribution is not spherical. 若殼或分佈不是球形則 E  0。 Q = qq = 0 But E  0 on or inside surface 但球面上和球面內的 E  0

Line Symmetry 線對稱 Line symmetry: 線對稱: r = perpendicular distance to the symm. axis. 與對稱軸的垂直距離 Distribution is independent of r//  it must extend to infinity along symm. axis. 分佈與 r// 無關  它在對稱軸方向必定延伸至無窮遠。 

Example 21.4. Infinite Line of Charge 無限長線電荷 Use Gauss’ law to find the electric field of an infinite line charge carrying charge density  in C/m. 用高斯定律找出一條電荷密度為  C/m 的無限長線電荷的電場。 (radial field 徑向場) No flux thru ends 末端無通量  c.f. Eg. 20.7  True outside arbitrary radial (r). 任何徑式 (r) 外皆然。 高斯面

Example 21.5. A Hollow Pipe 空管 A thin-walled pipe 3.0 m long & 2.0 cm in radius carries a net charge q = 5.7 C distributed uniformly over its surface. 一條薄管長 3.0 m，半徑 2.0 cm ，且表面均勻帶電 q = 5.7 C 。 Fine the electric field both 1.0 cm & 3.0 cm from the pipe axis, far from either end. 求離兩端很遠，距管軸 1.0 cm & 3.0 cm 處的電場。  at r = 1.0 cm at r = 3.0 cm

Plane Symmetry 面對稱 Plane symmetry: r = perpendicular distance to the symm. plane. 到對稱面的垂直距離 Distribution is independent of r//  it must extend to infinity in symm. plane. 分佈與 r// 無關  它在對稱面上必定延伸至無窮遠。 

Example 21.6. A Sheet of Charge 電荷片 An infinite sheet of charge carries uniform surface charge density  in C/m2. 一無限寬的電荷片上有均勻的面電荷密度  C/m2 。 Find the resulting electric field. 求其電場。   E > 0 if it points away from sheet. 如果它的方向是離開片子。

21.5. Fields of Arbitrary Charge Distributions 隨便一個電荷分佈的場 Dipole : E  r 3 雙極 Point charge : E  r 2 點電荷 Line charge : E  r 1 線電荷 Surface charge : E  const 面電荷

Conceptual Example 21.1. Charged Disk 帶電圓盤 Sketch some electric field lines for a uniformly charged disk, 把一個均勻帶電圓盤的電場線劃幾條出來， starting at the disk and extending out to several disk diameters. 從圓盤開始，再伸延到幾個圓盤直徑的距離。 infinite-plane-charge-like 像無限寬的面電荷 point-charge-like 像點電荷

Making the Connection 連起來 Suppose the disk is 1.0 cm in diameter & carries charge 20 nC spread uniformly over its surface. 假設盤子直徑為 1.0 cm，而且帶着 20 nC 均勻分佈於其表面的電荷。 Find the electric field strength 求下列位置的電場 1.0 mm from the disk surface and 離盤面 1.0 mm 處 1.0 m from the disk. 離盤面 1.0 m 處 Close to disk : 靠近盤面處 : Far from disk : 遠離盤面處 :

GOT IT? 21.4. A uniformly charged sheet measures 1 m on each side, and you are told the total charge is Q. 一帶電均勻的薄片每邊 1 m 長；你被告知總電荷是 Q 。 What expression would you use to get approximate values for the field magnitude 你會用甚麼式子來估計下列位置的电場大小？ 1 cm from the sheet (but not near an edge) and 離薄片 1 cm 處 ( 但不靠近邊緣 ) 1 km from the sheet? 離薄片 1 km 處 (a) (b)

21.6. Gauss’s Law & Conductors 高斯定律和導體 Electrostatic Equilibrium 靜電平衡 Conductor = material with free charges 導體 = 有自由電荷的物質 E.g., free electrons in metals. 例：金屬的自由電子。 Neutral conductor 中性導體 Uniform field 均勻場 Induced polarization cancels field inside 誘發偏極抵消了內部的場 External E  Polarization 外加 E  偏極  Internal E  內 E Total E = 0 : Electrostatic equilibrium 總 E = 0 ：靜電平衡 ( All charges stationary ) (所有電荷都不動) Microscopic view: replace above with averaged values. 微觀：上述各量以平均值代替。 Net field淨場

Charged Conductors 帶電導體 Excess charges in conductor tend to keep away from each other 導體內的多餘電荷喜歡彼此保持距離。  they stay at the surface. 它們都留在導體表面。  = 0 thru this surface 通過此面的  = 0 More rigorously 嚴格來說 : Gauss’ law with E = 0 inside conductor 高斯定律加起導體內 E = 0  qenclosed = 0 For a conductor in electrostatic equilibrium, all charges are on the surface. 靜電平衡下的導體中，全部的電荷都在表面。

Example 21.7. A Hollow Conductor 空心導體 An irregularly shaped conductor has a hollow cavity. 一個不規則形狀的導體內有一空洞。 The conductor itself carries a net charge of 1 C, and there’s a 2 C point charge inside the cavity. 導體本身帶有 1 C 淨電荷，而空洞內則有一 2 C點電荷。 Find the net charge on the cavity wall & on the outer surface of the conductor, assuming electrostatic equilibrium. 求在定靜電平衡下，洞壁及導體外部表面上的淨電荷。 E = 0 inside conductor 導體內E = 0  = 0 through dotted surface 穿過虛線面的 = 0 qenclosed = 0 Net charge on the cavity wall 洞壁上的淨電荷 qin = 2 C qout qin +2C Net charge in conductor 導體的淨電荷 = 1 C = qout + qin charge on outer surface of the conductor 導體外部表面上的淨電荷為 qout = +3 C

GOT IT? 21.5. A conductor carries a net charge +Q 一導體帶有淨電荷 +Q。 There is a hollow cavity inside the conductor that contains a point charge Q. 導體內有一空洞，內藏點電荷 Q。 In electrostatic equilibrium, is the charge on the outer surface of the conductor 在定靜電平衡下，導體外部表面上的電荷是 2Q Q Q 2Q

Experimental Tests of Gauss’ Law 高斯定律的實驗証明 帶電 不帶電 Measuring charge on ball is equivalent to testing the inverse square law. 量球上的電荷相當於測試反平方定律。 The exponent 2 was found to be accurate to 1016 . 結果是指數 2 準確到 1016 . 電荷跑到外面… … 球變成不帶電

Application: Shielding & Lightning Safety 應用：電磁場蔽體和避雷設備 Coaxial cable 同軸電纜 High frequency EM field easily blocked by metal (skin effect). 高頻電磁場容易被金屬阻擋 (表皮效應) 。

Field at a Conductor Surface 導體表面的場 At static equilibrium 在定靜電平衡下， 導體內 E = 0 inside conductor, 導體表面 E = E at surface of conductor. E 垂直於導體表面 Gauss’ law applied to pillbox surface: 高斯定律用在药盒型面上  表面上一塊，小到是平的 The local character of E ~  is incidental. E ~  的局部性是偶然的。 E always dependent on ALL the charges present. E 永遠由所有在場的電荷决定。 高斯面

Dilemma 兩難? Resolution 解答: 故導體外 E outside charged sheet of charge density was found to be 在電荷密度為  的帶電片之外的 E 是 E just outside conductor of surface charge density  is 在電荷表面密度為  的導體之外的 E 是 What gives 幹啥 ? 對稱性要求兩面的電荷密度相同 … …所以實際上有兩 塊帶電片 裏面的場相抵銷… …但外面的場相加乘 Resolution 解答: There’re 2 surfaces on the conductor plate. 導體板有兩個面。 The surface charge density on either surface is . 每個面的電荷密度都是  。 Each surface is a charge sheet giving E = /20. 每個面都是帶電片，其 E = /20。 Fields inside the conductor cancel, while those outside reinforce. 在導體裏面的場相抵銷，但在外面的則相加乘 Hence, outside the conductor 故導體外

Application: Shielding & Lightning Safety 應用：電磁場蔽體和避雷設備 Coaxial cable 同軸電纜 Car hit by lightning, 車被雷打， driver inside unharmed. 其內駕駛卻無損。 Strictly speaking, Gauss law applies only to static E. 嚴格來說，高斯定律祗適用於靜 E。 However, e in metal can respond so quickly that high frequency EM field ( radio, TV, MW ) can also be blocked (skin effect). 可是，金屬內的電子反應之快，連高頻電磁場 ( 電台，電視，微波 ) 也可以阻擋 (表皮效應)。

Plate Capacitor 板電容器 inside 裏面 outside 外面 Charge on inner surfaces only 裏面的表面才有電荷