22. Electric Potential 電位 Electric Potential Difference 電位差 Calculating Potential Difference 計算電位差 Potential Difference & the Electric Field 電位差和電場 Charged Conductors 帶電導體

This parasailer landed on a 138,000-volt power line. 這名滑翔傘手落在一條138,000伏的電力線上 Why didn’t he get electrocuted? 為甚麼他沒有被電死？ He touches only 1 line – there’s no potential differences & hence no energy transfer involved. 他祇碰到一條線 ：沒有電位差，所以沒有涉及能量轉移。

22.1. Electric Potential Difference 電位差 Conservative force 守恆力: ( path independent ) 與路徑無關 Electric potential difference  electric potential energy difference per unit charge 電位差  每單位電荷的電位能差 [ V ] = J/C = Volt 伏特 = V 伏 if reference potential VA = 0. 若參考電位 VA = 0. For a uniform field 均勻場: rAB E E points at direction of most rapidly decreasing V. E 指向 V 遞減最快的方向。

Table 22. 1. Force & Field, Potential Energy & Electric Potential 表22 Quantity 量 Symbol / Equation 符號 / 公式 Units 單位 Force 力 F N Electric field 電場 E = F / q N/C or V/m Potential energy difference 位能差 J J/C or V Electric potential difference 電位差

Potential Difference is Path Independent 電位差與路徑無關 Potential difference VAB depends only on positions of A & B. 電位差 VAB 祇與 A 和 B 的位置有關。 Calculating along any paths (1, 2, or 3) gives VAB = E r. 沿着任何一條路徑 (1, 2, 或 3) 來算都得到 VAB = E r 。

GOT IT? 22.1 What would happen to VAB in the figure if E were doubled; E 加倍； r were doubled; r 加倍； the points were moved so the path lay at right angles to E; 移動點使路徑與 E 正交； the positions of A & B are interchanged. A 與 B 的位置互換 doubles 加倍 doubles 加倍 becomes 變成 0 reverses sign 變符號

The Volt & the Electronvolt 伏特和電子伏特 [ V ] = J/C = Volt = V E.g., for a 12V battery, 12J of work is done on every 1C charge that moves from its negative to its positive terminals. 例， 12V 電瓶：每次把 1C 電荷從負極移至正極，就要作功 12J 。 Voltage = potential difference when no B(t) is present. 電壓 = 無 B(t) 時的電位差 Electronvolt (eV) = energy gained by a particle carrying 1 elementary charge when it moves through a potential difference of 1 volt. 電子伏特 (eV) = 帶有 1 基本電荷的粒子通過 1 伏電位差後所增加的能量。 1 elementary charge 基本電荷 = 1.61019 C = e 1 eV = 1.61019 J

Table 22.2. Typical Potential Differences 表22.2. 典型電位差 Between human arm & leq due to 1 mV heart’s electrical activity 因心臟的電性活動而產生在手與腿之間的 Across biological cell membrane 80 mV 生物細胞膜的兩邊 Between terminals of flashlight battery 1.5 V 手電筒電池兩極之間 Car battery 汽車電瓶 12 V Electric outlet (depends on country) 100-240 V 電源插座(因國別而異) Between long-distance electric 365 kV transmission line & ground 長程輸電線與地之間 Between base of thunderstorm cloud & ground 100 MV 雷雨雲層底與地之間

International Voltage 國際電壓 Afghanistan 220 V Australia 240 V Bahamas 120 V Brazil 120/220 V Canada 120 V China 220 V Finland 230 V Guam 110 V HongKong 220 V Japan 100 V Mexico 127 V Spain 230 V United Kingdom 230 V United States 120 V Vietnam 127/220 V

GOT IT? 22.2 A proton ( charge e ), 一粒質子 ( 電荷為 e ) ， 10 eV an  particle ( charge 2e ), and 一粒  粒子( 電荷為 2e ) ，和 a singly ionized O atom 一粒 O 離子 each moves through a 10-V potential difference. 每粒都通過 10-V電位差。 What’s the work in eV done on each? 對每粒所作的功有幾 eV？ 10 eV 20 eV 10 eV

Example 22.1. X Rays In an X-ray tube, a uniform electric field of 300 kN/C extends over a distance of 10 cm, from an electron source to a target; the field points from the target towards the source. 在一X光管中，從電子源到標靶之間的 10 cm 距離內，有一 300 kN/C 均勻電場。 Find the potential difference between source & target and the energy gained by an electron as it accelerates from source to target ( where its abrupt deceleration produces X-rays ). 求電子源到標靶間的電位差和電子在電子源到標靶間加速時所增加的能量 (電子在碰上標靶時因突然減速而產生 X光) 。 Express the energy in both electronvolts & joules. 能量同時以電子伏特和焦耳表示。 電子源 標靶

Example 22.2. Charged Sheet 帶電片 An isolated, infinite charged sheet carries a uniform surface charge density . 一張孤立的無限寬帶電片上有一均勻面電荷密度。 Find an expression for the potential difference from the sheet to a point a perpendicular distance x from the sheet. 求電片至與其垂直距離為 x 的一點間的電位差。 E

Curved Paths & Nonuniform Fields 彎曲路徑和非均勻場 Staight path, uniform field: 直路徑，均勻場： Curved path, nonuniform field: 彎曲路徑，非均勻場：

GOT IT? 22.3 The figure shows three straight paths AB of the same length, each in a different electric field. 圖示三條在不同電場中的等長直路徑 AB 。 The field at A is the same in each. 在各A點的場都是一樣。 Rank the potential differences ΔVAB. 請為各電位差 ΔVAB 排序。 Smallest ΔVAB . 最小 Largest ΔVAB . 最大

22.2. Calculating Potential Difference 計算電位差 Potential of a Point Charge 點電荷的電位 For A,B on the same radial A,B 皆在同一徑線上 For A,B not on the same radial, break the path into 2 parts,1st along the radial & then along the arc. 若 A, B 不在同一徑線上，應把路徑拆成兩段： 前一段沿徑線走，後一段沿弧線走。 Since, V = 0 along the arc, the above equation holds. 因弧線上 V = 0 ，上列式子還是對的。

The Zero of Potential 電位的零點 Only potential differences have physical significance. 祇有電位的差別才有物理意義。 Simplified notation: 簡化的符號 R = point of zero potential 電位原(零)點 VA = potential at A. A 的電位 Some choices of zero potential 一些電位零點 Power systems / Circuits 電力系统 / 電路 Earth ( Ground ) 地球(地) Automobile electric systems 汽車的電力系统 Car’s body 車身 Isolated charges 孤立電荷 Infinity 無限遠處

GOT IT? 22.4 You measure a potential difference of 50 V between two points a distance 10 cm apart in the field of a point charge. 在一個點電荷的場中，你量得相距 10 cm 的兩點之間的電位差為 50 V 。 If you move closer to the charge and measure the potential difference over another 10-cm interval, will it be 如果你往電荷靠近，再量另一相距 10 cm 的兩點之間的電位差。結果會 greater, 比較大， less, or 比較小，還是 the same? 一樣？

Example 22.3. Science Museum The Hall of Electricity at the Boston Museum of Science contains a large Van de Graaff generator, a device that builds up charge on a metal sphere. 波士頓科學博物館的電力廳用一部大型范得格拉夫起電機將一個金屬球起電。 The sphere has radius R = 2.30 m and develops a charge Q = 640 C. 球的半徑是 R = 2.30 m ，總電荷可達 Q = 640 C 。 Considering this to be a single isolate sphere, find 把它當成一個孤立的球，求 the potential at its surface 它表面上的電位， the work needed to bring a proton from infinity to the sphere’s surface, 把一質子從無限遠處帶到球面上所需的功， the potential difference between the sphere’s surface & a point 2R from its center. 球面與離球心 2R 處的電位差。 (a) (b) (c) 徑距離

Example 22.4. High Voltage Power Line 高壓電力線 A long, straight power-line wire has radius 1.0 cm & carries line charge density  = 2.6 C/m. 一條半徑為 1 cm 的長而直的電力線上帶有線電荷密度  = 2.6 C/m 。 Assuming no other charges are present, what’s the potential difference between the wire & the ground, 22 m below? 假定沒有其他電荷，電力線與在它下面 22 m 處的地面之間的電位差為何？

Finding Potential Differences Using Superposition 以叠加求電位差 Potential of a set of point charges: 一組點電荷的電位 Potential of a set of charge sources: 一組電荷的電位

Example 22.5. Dipole Potential 雙極電位 An electric dipole consists of point charges q a distance 2a apart. 一個雙極由相距 2a 的點電荷 q 組成。 Find the potential at an arbitrary point P, and approximate for the casewhere the distance to P is large compared with the charge separation. 求任一點 P 的電位，並取得到 P 的距離比電荷間距離大很多時的近似值。 +q: hill 丘 r >> a  V = 0 q: hole 洞 p = 2qa = dipole moment 雙極距

GOT IT? 22.5 The figure show 3 paths from infinity to a point P on a dipole’s perpendicular bisector. 圖示從無限遠處到一雙極的中垂線上一點 P 的三條路徑。 Compare the work done in moving a charge to P on each of the paths. 比較在這些路徑上把一個電荷移至 P 所需做的功。  V is path independent V與路徑無關 work on all 3 paths are the same. 三條路徑上要做的功都一樣。 Work along path 2 is 0 since V = 0 on it. 路徑 2 上 V = 0 ，所以要做的功為 0。 Hence, W = 0 for all 3 paths. 故三條路徑上都是 W = 0 。  2 3 P  q q 1

Continuous Charge Distributions 連續電荷分佈 Superposition: 叠加

Example 22.6. Charged Ring 帶電環 A total charge Q is distributed uniformly around a thin ring of radius a. 一半徑為 a 的幼環上均勻地分佈了 Q 電荷。 Find the potential on the ring’s axis. 求環軸上的電位。 Same r for all dq 所有 dq 的 r 都一樣

Example 22.7. Charged Disk 帶電盤 A charged disk of radius a carries a charge Q distributed uniformly over its surface. 一半徑為 a 的盤上均勻地分佈了 Q 電荷。 Find the potential at a point P on the disk axis, a distance x from the disk. 求盤軸上離盤 x 遠的 P 點 的電位。 point charge 點電荷 disk 盤 電位 sheet 片 盤軸上的距離 x

22.3. Potential Difference & the Electric Field 電位差和電場 Equipotential = surface on which V = const. 等電位 = V為定值的面 W = 0 along a path  E  E的路徑上 W = 0 V = 0 between any 2 points on a surface  E.  E 的面上任兩點的 V = 0 Equipotential  Field lines. 等電位  場線 Steep hill 丘峭 Close contour 線密 Strong E E 強 V > 0 V < 0 V = 0

GOT IT? 22.6 The figure show cross sections through 2 equipotential surfaces. 圖示兩等位面的橫切面。 In both diagrams, the potential difference between adjacent equipotentials is the same. 兩圖中相隣的等位線的電位差都一樣。 Which could represent the field of a point charge? Explain. 那一個可能代表一點電荷的場？ 解釋。 (a). Potential decreases as r 1 , so the spacings between equipotentials should increase with r. 電位以 r 1 遞減，故等位線間隔應隨 r 而增。

Calculating Field from Potential 從電位算電場   =  ( Gradient of V ) V的陡(梯，坡)度 E is strong where V changes rapidly ( equipotentials dense ). V 變化較急(等位較密)處 E 較強。

Example 22.8. Charged Disk 帶電盤 Use the result of Example 22.7 to find E on the axis of a charged disk. 用例 22.7 的結果來求一帶電盤軸上的 E。 Example 22.7: x > 0 x < 0 dangerous conclusion 危險的結論

Tip: Field & Potential 祕笈：場和電位 Values of E and V aren’t directly related. E 和 V 的數值無直接關係。 V flat, Ex = 0 V falling, Ex > 0 V 在降， Ex > 0 V rising, Ex < 0 V 在升， Ex < 0

22.4. Charged Conductors 帶電導體 In electrostatic equilibrium, 在靜電平衡時， 導體內 E = 0 inside a conductor. 導體表面 E// = 0 on surface of conductor. W = 0 for moving charges on / inside conductor. 導體上或內移動電荷時 W = 0 。 The entire conductor is an equipotential. 整個導體是一個等位。 Consider an isolated, spherical conductor of radius R and charge Q. 有一個孤立的球狀導體，其半徑為 R ，電荷為 Q。 Q is uniformly distributed on the surface Q 均勻地分佈在它的表面 E outside is that of a point charge Q. 球外的 E 跟點電荷 Q 的一樣。 V(r) = k Q / R. for r  R.

Consider 2 widely separated, charged conducting spheres. 茲有二相距甚遠的帶電導體球。 Their potentials are 其電位為 If we connect them with a thin wire, 如果我們用一條幼線把他們連起來， there’ll be charge transfer until V1 = V2 , i.e., 電荷便會流動，直至 V1 = V2 為止，亦即 Same V V 相同 In terms of the surface charge densities 用表面電荷密度來算 we have 可得  Smaller sphere has higher field at surface. 比較小的球，表面上的場比較強。

Isolated conductor with irregular shape. 孤立的不規則形狀導體。 Surface is equipotential  | E | is larger where curvature of surface is large. 表面是等位的  | E | 在曲度大處較大。 More field lines emerging from sharply curved regions. 從曲度大的地方出來的場線也比較多。 From afar, conductor is like a point charge. 從遠處來看，導體就像一顆點電荷。

Conductor in the Presence of Another Charge 有另一個電荷存在時的導體 孤立的帶電導體，場是對稱的… …旁邊有另一個電荷時，對稱性就毀了。

Application: Corona Discharge, Pollution Control, and Xerography 應用：電暈放電，汚染控制，和靜電複印(影印) Air ionizes for E > MN/C. E > MN/C時空氣就會離子化。 Recombination of e with ion e 和離子重組 Corona discharge ( blue glow ) 電暈放電 (發藍光) Electrostatic precipitators 靜電沉澱器： Removes pollutant particles (up to 99%) using gas ions produced by Corona discharge. 用電暈放電所產生的氣態離子來清除(多至99%的) 汚染粒子。 Corona discharge across power-line insulator. 跨越電力線絕緣器的電暈放電。 Laser printer / Xerox machines 雷射印表機 / 影印機: Ink consists of plastic toner particles that adhere to charged regions on light-sensitive drum, which is initially charged uniformy by corona discharge. 墨水內含有塑料調色劑粒子，能粘在光敏滾筒的帶電部份。滾筒則預先由電暈放電而均勻帶電。