Part 4. Electromagnetism 電學 Bio-EM 生物-電 : Heartbeat pacing 心跳規律 Nerves 神經衝脈 Osmosis thru cell membranes 細胞膜滲透 Applications 應用: Computer microchips 電腦微片 Cell phones 行動電話 Motors & generators 馬達&發電機 4 fundamental laws: 四個基本定律： Maxwell’s eqs. 麥克斯韋方程。

Electromagnetism Electric Charge, Force, & Fields 電荷，力，和場 Gauss’s Law 高斯定律 Electric Potential 電位 Electrostatic Energy & Capacitors 靜電能和電容器 Electric Current 電流 Electric Circuits 電路 Magnetism: Force & Field 磁：力和場 Electromagnetic Induction 電磁感應 Alternating-Current Circuits 交流電路 Maxwell’s Equations & EM Waves 麥克斯韋方程和電磁波

20. Electric Charge, Force, & Field 電荷，力，和場 Coulomb’s Law 庫倫定律 The Electric Field 電場 Fields of Charge Distributions 電荷分佈的場 Matter in Electric Fields 電場中的物質

What holds your body together? 甚麼東西把你的身體結合在一起？ What keeps a skyscraper standing? 甚麼東西維持一幢摩天大樓直立不倒？ What keeps a car on the road as it turns? 甚麼東西把一輛在轉彎中的車子保持在路上？ What governs the electronics in computers? 甚麼東西支配了電腦內的電子行為？ What provides the tension in a climbing rope? 甚麼東西提供了一條爬山繩索的張力？ What enables the photosynthesis of plants? 甚麼東西讓植物的光合作用得以進行？ Ans: electric forces. 答案： 電力。 All macroscopic phenomena are governed by gravity & EM forces. 一切的巨觀現象都由重力和電磁力支配。 Notable effective forces of electrical origin: 幾個出名的，源自電力的有効力： Tension 張力 Normal Forces 正力 Compression 壓縮 Friction 摩擦力 Most forces in chemistry & biology 化學和生物學中大部份的力

What’s the fundamental criterion for initiating a lightning strike? 引起閃電的基本條件是甚麼？ Ans. 答案： E > 3 MV/m

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。

20.2. Coulomb’s Law 庫倫定律 “Insulators” can be charged by rubbing. “絕緣體”可經摩擦生電。 Examples 例: Rubbed balloon sticks to clothing. 摩擦過的氣球會粘著衣服。 2 rubbed balloons repel each other. 兩個摩擦過的氣球會互相排斥。 Socks from dryer cling to clothings. 剛從乾衣機拿出來的襪子粘著衣服。 Bits of styrofoam cling to hand. 保麗龍顆粒粘在手上。 Walk across carpet & feel shock touching doorknob. 走過地毯後被門的把手電到。 Ground or low energy state of matter tends to be charge neutral. 物質的底或低能態都傾於中性。 … but the electric force is responsible for its cubical shape … 但它的立方形源自電力 A salt gain is electrically neutral … 一顆鹽粑是中性的 …

Triboelectric Series 摩擦電系列 Steel (No charge) 鋼(不帶電) Wood (Small negative charge) 木(小量負電) Lucite 螢光樹脂 Amber 琥珀 Sealing wax 封蠟 Acrylic 丙烯酸 Polystyrene 聚苯乙烯 Rubber balloon 橡膠氣球 Resins 樹脂 Hard rubber 硬橡膠 Nickel, Copper 鎳，銅 Sulfur 硫磺 Brass, Silver 黃銅，銀 Gold, Platinum 金，鉑 Acetate Rayon (醋酸鹽)人造纖維 Synthetic rubber 人造橡膠 Polyester 聚酯纖維 Styrene (Styrofoam) 苯乙烯(保麗龍) Orlon 奧綸(聚丙烯腈絲) Plastic wrap 保鮮膜 Polyurethane 聚氨酯 Polyethylene (like Scotch tape) 聚乙烯(透明膠帶) Polypropylene 聚丙烯 Vinyl (PVC) 乙烯基(黑膠) Silicon 硅 (矽) Teflon 鐵氟龍 (不粘鍋) Silicone rubber 硅樹脂 Ebonite 硬橡膠(保齡球)   Most negatively charged 最帶負電 Most positively charged 最帶正電 + + Air 空氣 Human skin 人的皮膚 Leather 皮革 Rabbit's fur 兔毛皮 Glass 玻璃 Quartz 石英 Mica 雲母 Human hair 人的頭髮 Nylon 尼龍 Wool 羊毛 Lead 鉛 Cat's fur 貓毛皮 Silk 絲 Aluminium 鋁 Paper (Small positive charge) 紙(小量正電) Cotton (No charge) 棉花 (不帶電) 0 0

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

GOT IT? 20.1. Charge q1 is at x = 1 m, y = 0. 電荷 q1 位於 x = 1 m, y = 0. What is the unit vector in F12 if q2 is located at 求 F12 中的單位向量 ，設 q2 位於 the origin, (a) 原點， x = 0, y = 1 m ? (b) x = 0, y = 1 m。 Explain why you can answer without knowing the sign of either charge. 解釋為甚麽你可以不用知道這些電荷的正負號也可以回答問題。 (a) (b)

Example 20.1. Force Between Two Charges 兩電荷間的力 A 1.0 C charge is at x = 1.0 cm, & a 1.5 C charge is at x = 3.0 cm. 一個 1.0 C 電荷位於 x = 1.0 cm，而另一個 1.5 C 電荷則位於 x = 3.0 cm. What force does the positive charge exert on the negative one? 正電荷加於負電荷的力為何？ How would the force change if the distance between the charges tripled? 如果電荷間的距離變成原來的三倍，其間的力有何變化？ Distance tripled  force drops by 1/32. 距離變三倍  力掉為 1/32。

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:

Example 20.3. Thunderstorm 雷雨 A charged raindrop carrying 10 C experiences a force of 0.30 N in the +x direction. 一滴帶電 10 C 的雨點承受一個朝 +x 方向， 0.30 N 的力。 What’s the electric field at its location? 在它位置上的電場為何？ What would the force be on a 5.0 C drop at the same point? 在這一點上，一滴 5.0 C 的雨點受力為何？

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)

Example 20.4. Two Protons 兩個質子 Two protons are 3.6 nm apart. 兩個質子相距 3.6 nm 。 Find the electric field at a point between them, 1.2 nm from one of them. 求位於它們之間，離其中一個 1.2 nm 處的電場。 Find the force on an electron at this point. 求在此處的一個電子所受的力。 3.6 nm P x 1.2 nm 2.4 nm

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

GOT IT? 20.3. e p 2H 3He 4He C13 16O q/m 1800 1/1 1/2 2/3 2/4 6/13 An electron, a proton, a deuteron (1p, 1n), a 3He nucleus (2p, 1n), a 4He nucleus (2p, 2n), a 13C nucleus (6p, 7n), & an 16O nucleus (8 p, 8 n) all find themselves in the same electric field. 茲有電子，質子，氘核(1p, 1n)，3氦核(2p, 1n) ，4氦核(2p, 2n) ，13碳核(6p, 7n) ，和16氧核(8p, 8n)各一處於同一電場中。 Rank order their accelerations from lowest to highest assuming 請為其加速度從小至大排序。 p & n have the same mass. p 和 n質量相同。 The mass of a composite particle is the sum of the masses of its constituents. 組合粒子的質量為其成份的質量和。 e p 2H 3He 4He C13 16O q/m 1800 1/1 1/2 2/3 2/4 6/13 8/16 Ans: 13C, (16O, 4He, deuteron), 3He, p, e.

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