Short Version : 22. Electric Potential 短版 : 22. 電位

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 。

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

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

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

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

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。

Conceptual Example 22.1. An Irregular Conductor 不規則導體 Sketch some equipotentials & electric field lines for an isolated egg-shaped conductor. 為一個蛋形導體畫幾條等位線和電場線。 Ans. 答 Surface is equipotential  | E | is larger where curvature of surface is large. 表面是等位線  | E | 在曲度較大的表面處較大。 More field lines emerging from sharply curved regions. 表面較尖銳處有較多電場線冒出來。 From afar, conductor is like a point charge. 在遠處，導體就像一個點電荷。

Making the Connection 連起來 The potential difference between the conductor and the outermost equipotential shown in figure is 70 V. 圖中導體和最外面那條等位線之間的位勢差為 70 V 。 Determine approximate values for the strongest & weakest electric fields in the region, assuming it is drawn at the sizes shown. 求區域內最強和最弱的電場，假設相關距離如圖示。 Strongest field 最強的電場 : Weakest field 最弱的電場 : 7 mm 12 mm

Conductor in the Presence of Another Charge 有另一電荷在場時的導體 帶電的球與外界隔絕，場是對稱的 …. …但旁邊的電荷把對稱性破壞了。

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。