Short Version : 20. Electric Charge, Force, & Fields 短版: 20. 電荷，力，和場

20.1. Electric Charge 電荷 2 kinds of charges: + & . 兩種電荷: 正和負 Total charge = algebraic sum of all charges. 總電荷 = 全部電荷的和 Like charges repel. Opposite charges attract. 同性相斥，異性相吸。 = elementary charge 基本電荷 All electrons have charge e. 所有電子都帶 e 電荷. All protons have charge +e. 所有電子都帶 +e 電荷. 1st measured by Millikan on oil drops. 首先由密立根在油滴上量得。 Theory (standard model) : basic unit of charge (carried by quark) = 1/3 e. 理論 (標準模型) ： 基本電荷單位 (由夸克帶著) = 1/3 e. Quark confinement  no free quark can be observed. 夸克禁閉  無法量得自由夸克  Smallest observable charge is e.  量得到的最小電荷是 e。 Conservation of charge: total charge in a closed region is always the same. 電荷守恆：在一個密閉系统中總電荷永遠不變。

Coulomb’s law (force between 2 point charges) : 庫倫定律 (兩點電荷間的力) : [q] = Coulomb = C

Conceptual Example 20.1. Gravity & Electric Force 重力和電力 The electric force is far stronger than the gravitational force, 電力比重力強很多， yet gravity is much more obvious in everyday life. 可是重力在日常生活中卻比較明顯。 Why? 為甚麼? Only 1 kind of gravitational “charge” 祗有一種重力 “荷” forces from different parts of a source tend to reinforce. 一個質源各部份產生的力傾向於相增。 2 kinds of electric charges 有兩種電荷 forces from different parts of a neutral source tend to cancel out. 一個中性電源各部份產生的力傾向於互相抵消。

Making the Connection 連起來 Compare the magnitudes of the electric & gravitational forces between an electron & a proton. 比較一粒電子和一粒質子之間的電和重力的大小。

Point Charges & the Superposition Principle 點電荷和疊加原理 Extension of Coulomb’s law (point charges) to charge distributions. 推廣庫倫定律 (點電荷) 至電荷分佈。 Superposition principle: 疊加(重疊)原理: Fnet F23 F13 Independent of each other 互為獨立 Task: Find net force on q3 . 任務：求 q3 的淨力。

Example 20.2. Raindrops 雨點 y x Charged raindrops are responsible for thunderstorms. 帶電雨點為雷雨的成因。 Two drops with equal charge q are on the x-axis at x = a. 兩滴帶同樣電荷的雨水在 x 軸上 x = a 處。 Find the electric force on a 3rd drop with charge Q at any point on the y-axis. 求第三滴位於 y 軸上任一點，且帶 Q 電荷的雨水所受的電力。 y F1 F2 Q r r y q q   x x1 = a x2 = a

20.3. The Electric Field 電場 Electric field E at r = Electric force on unit point charge at r. 在 r 處的電場 E = 在 r 處單位電荷所受電力。 F = electric force on point charge q. = 點電荷 q 所受電力。 E = F/q [ E ] = N / C = V / m V = Volt g = F/m Implicit assumption: q doesn’t disturb E. 不明文假定：q 沒有干擾 E. Rigorous definition: 精確的定義 Gravitational field 重力場 Electric field 電場

Force approach 力的做法: Charges interact at a distance (difficult to manage when many charges are present). 電荷遠距離作用 ( 電荷很多時難以處理 ) 。 Fails when charge distributions are not known. 電荷分佈不詳時即失敗。 Field approach 場的做法: Charge interacts only with field at its position. 電荷祇與在它位置上的場作用。 No need to know how field is generated. 無需知道場是怎麼產生的。 Given E: 曉得 E:

The Field of a Point Charge 點電荷的場 Field at r from point charge q : 與點電荷 q 相隔 r 處的場： Field vectors for a negative point charge. 負點電荷的場向量。

20.4. Fields of Charge Distributions 電荷分佈的場 (Discrete sources 離散電源) Superposition principle  疊加原理 (Point charges 點電荷)

The Electric Dipole 電雙極 Electric dipole = Two point charges of equal magnitude but opposite charges separated by a small distance. 電雙極 = 兩個大小一樣但符號相反的近距離電荷。 Examples 範例: Polar molecules 極性分子. Heart muscle during contraction 心肌收縮時  Electrocardiograph (EKG) 心電圖儀 Radio & TV antennas 收音機和電視的天線. H2O 水

Example 20.5. Modeling a Molecule 模擬一個分子 A molecule is modeled as a positive charge q at x = a, and a negative charge q at x =  a. 一個分子被模擬成一個在 x = a 的正電荷和一個在 x =  a 的負電荷。 Evaluate the electric field on the y-axis. 求在 y-軸上的電場。 Find an approximate expression valid at large distances (y >> a). 求在極遠處 (y >> a) 的近似值。 y E2 E Q = 1 E1 r r y q q   x x1 = a x2 = a (y >> a)

Dipole ( q with separation d ): 雙極 ( q 相隔 d ) for r >> d = 2a Typical of neutral, non-spherical, charge distributions ( d ~ size ). 典型為一般中性，非球狀的電荷分佈 ( d ~ 大小)。 Dipole moment : p = q d. 雙極矩 d = vector from q to +q 從 q 到 +q 的向量 y On perpendicular bisector: 在垂直中分線上 E2 E Q = 1 On dipole axis: 在雙極軸上 E1 r r y (Prob 習題 51) q q   x x1 = d/2 p x2 = d/2

Continuous Charge Distributions 連續電荷分佈 All charge distributions are ultimately discrete ( mostly protons & electrons ). 所有電荷分佈終究都是離散的 (多是質子和電子) 。 Continuum approximation: Good for macroscopic bodies. 連續體近似法：巨觀物體適用 Volume charge density 體電荷密度  [ C/m3 ] Surface charge density 面電荷密度  [ C/m2 ] Line charge density 線電荷密度  [ C/m ] 電荷分佈

Example 20.6. Charged Ring 帶電環 A ring of radius a carries a uniformly distributed charge Q. 一個半徑為 a 的環上均勻分佈著電荷 Q. Find E at any point on the axis of the ring. 求在環軸上任一點的 E 。 By symmetry, E has only axial (x-) component. 由對稱性可知 E 祇有 x-軸分量。 On axis of uniformly charged ring 在均勻帶電環的軸上

Example 20.7. Power Line 輸電線 By symmetry, E has only y- component. y A long electric power line running along the x-axis carries a uniform charge density  [C/m]. 一條沿x-軸延伸的長輸電線上帶有均勻電荷密度  [C/m]. Find E on the y-axis, assuming the wire to be infinitely long. 設線為無限長，求y-軸上的E。 By symmetry, E has only y- component. 由對稱性可知 E 祇有 y-軸分量。 y dEy dE dE P r r y x dq x dq Perpendicular to an infinite wire 垂直於無限長線

20.5. Matter in Electric Fields 電場內的物質 Point Charges in Electric Fields 電場內的點電荷 Newton’s 2nd law  牛頓第二定律 (point charge in field E) (點電荷在場 E 內) Trajectory determined by charge-to-mass ratio q/m. 軌跡由電荷對質量比 q/m 决定。 Constant E  constant a. 定E  定a. E.g., CRT, inkjet printer, …. 例：陰極射線管，噴墨印表機，…. Uniform field between charged plates (capacitors). 電板(電容器)間的均勻場。

Example 20.8. Electrostatic Analyzer 靜電分析(檢偏)器 Two curved metal plates establish a field of strength E = E0 ( b/r ), where E0 & b are constants. 兩塊彎曲的金屬板建立了一個強度為 E = E0 ( b/r ) 的場，其中 E0 和 b 為常數。 E points toward the center of curvature, & r is the distance to the center. E 指向曲度中心，而 r 是至中心的距離。 Find speed v with which a proton entering vertically from below will leave the device moving horizontally. 如一質子要從下面垂直進入，再從水平方向離開，它的速率必需為何？ Too fast, hits outer wall 太快，撞外壁 For a uniform circular motion: 等速圓周運動  分析器 Too slow, hits inner wall 太慢，撞內壁 質子束

Dipoles in Electric Fields 電場內的雙極 力距使雙極順時轉 Uniform 均勻 E: Total force 總受力: Torque about center of dipole: 繞雙極中點的力距 = dipole moment 雙極距 Work done by E to rotate dipole: 轉動雙極 E 所作功 t // tangent 切線 Potential energy of dipole in E (i = /2) 雙極在 E 中的位能 ( U = 0 for p  E )

Non-uniform field 非均勻場 : Total force 總力: Example: dipole-dipole interaction 例: 雙極-雙極作用 | F | > | F+ | c.f. Van der Waals interaction, long range part. 比較: 范德瓦作用，長程部份。 Force on  end of B is stronger; hence net force is toward A 在 B 負端的力比較強；故淨力朝 A 。

Application: Microwave Cooking & Liquid Crystals 應用：微波烹飪和液晶 Microwave oven 微波爐: GHz EM field vibrates (dipolar) H2O molecules in food 百萬赫電磁場振動食物內的(雙極)水分子  heats up 熱起來. Liquid Crystal Display 液晶顯示器 (LCD): 正常液體 液晶 隨外場對齊 dipolar molecules aligned but positions irregular 雙極分子的方向對齊但位置不規則

Exploded view of a TN (Twisted Nematic) liquid crystal cell showing the states in an OFF state (left), and an ON state with voltage applied (right) 超扭轉向列型液晶單元的放大圖：關閉狀態(左圖) ，開放(加電壓後)狀態(右圖) 。

Conductors, Insulators, & Dielectrics 導體，絕緣體，和電介體 Bulk matter consists of point charges: e & p. 大塊的物質由點電荷組成 Conductors: charges free to move (  electric currents ), 導體: 電荷自由移動 (電流 ) e.g., e (metal), ion ( electrolytes ), e+ion (plasma). 例： e(金屬) ，離子(電解質) ，e+離子(電漿) Insulators: charges are bounded. 絕緣體: 電荷受縛 Dielectrics: insulators with intrinsic / induced dipoles. 電介體: 有內建或感應雙極的絕緣體 internal field from dipoles 雙極所生內場 Induced dipole 感應雙極 Alignment of intrinsic dipoles. 內建雙極的對齊