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1、22. Electric Potential電位1.Electric Potential Difference電位差2.Calculating Potential Difference計算電位差3.Potential Difference & the Electric Field電位差和電場4.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 didnt he get electrocuted?為甚麼他沒有被電死？He touches onl
2、y 1 line theres 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:Electric potential difference electric potential energy difference per unit charge電位差 每單位電荷的電位能差if reference potential VA = 0.若參考電位 VA =
3、 0. V = J/C = Volt 伏特 = V 伏For a uniform field 均勻場:( path independent ) 與路徑無關rABEE 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22.1.力和場，位能和電位Quantity 量量Symbol / Equation符號符號 / 公式公式Units單位單位Force 力Electric field 電場Po
4、tential energy difference位能差Electric potential difference電位差FNE = F / qN/C or V/mJJ/C or VPotential 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
5、算都得到 VAB = E r 。GOT IT? 22.1What would happen to VAB in the figure if以下情况下，圖中的 VAB 會怎樣？(a) E were doubled;E 加倍；(b) r were doubled; r 加倍；(c) the points were moved so the path lay at right angles to E;移動點使路徑與 E 正交；(d) the positions of A & B are interchanged.A 與 B 的位置互換doubles 加倍doubles 加倍becomes 變成 0r
6、everses sign變符號The Volt & the Electronvolt伏特和電子伏特 V = J/C = Volt = VE.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Elec
7、tronvolt (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 = 1.61019 C = e1 eV = 1.61019 JTable 22.2. Typical Potential Differences表22.2. 典型電位差Between human arm & l
8、eq due to1 mV hearts electrical activity因心臟的電性活動而產生在手與腿之間的Across biological cell membrane80 mV生物細胞膜的兩邊Between terminals of flashlight battery1.5 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 kVtransmission line & ground長程輸電線
9、與地之間Between base of thunderstorm cloud & ground100 MV雷雨雲層底與地之間INTERNATIONAL VOLTAGE國際電壓Afghanistan 220 VAustralia 240 VBahamas 120 VBrazil 120/220 VCanada 120 VChina 220 VFinland 230 VGuam 110 VHongKong 220 VJapan 100 VMexico 127 VSpain 230 VUnited Kingdom 230 VUnited States 120 VVietnam 127/220 VGO
10、T IT? 22.2(a) A proton ( charge e ), 一粒質子 ( 電荷為 e ) ，(b) an particle ( charge 2e ), and一粒 粒子( 電荷為 2e ) ，和(c) 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。Whats the work in eV done on each?對每粒所作的功有幾 eV？10 eV20 eV10 eVExample 22.1. X RaysIn an X-ray tube,
11、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 i
12、t 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 surf
13、ace 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。ECurved 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：T
14、he 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 VAB.請為各電位差 VAB 排序。GOT IT? 22.3Largest VAB .最大Smallest VAB .最小22.2. Calculating Potential Difference 計算電位差Po
15、tential of a Point Charge點電荷的電位For A,B on the same radialA,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 Po
16、tential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Some choices of zero potential 一些電位零點一些電位零點R = point of zero potential電位原(零)點VA = potential at A. A 的電位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
17、s body 車身Isolated charges孤立電荷Infinity無限遠處GOT IT? 22.4You 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 in
18、terval, 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(a) greater,比較大，(b) less, or比較小，還是(c) the same?一樣？Example 22.3. Science MuseumThe 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
19、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(a) the potential at its surface 它表面上的電位，(b) the work needed to bring a proton from infinity to the spheres surface,把一質子從無限遠
20、處帶到球面上所需的功，(c) the potential difference between the spheres 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 。Ass
21、uming no other charges are present,whats 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
22、. 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。r a p = 2qa =
23、 dipole moment 雙極距+q: hill 丘丘 q: hole 洞洞V = 0GOT IT? 22.5The figure show 3 paths from infinity to a point P on a dipoles 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 a
24、ll 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 。P123qqContinuous 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
25、 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 rings 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 dis
26、tance x from the disk.求盤軸上離盤 x 遠的 P 點 的電位。sheet 片片point charge 點電荷點電荷disk 盤盤盤軸上的距離 x電位22.3. Potential Difference & the Electric Field電位差和電場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Equipotential = surface on which V =
27、const.等電位 = V為定值的面V 0V 0x 0V 在降， Ex 0V flat, Ex = 0V rising, Ex 0V 在升， Ex MN/C.E MN/C時空氣就會離子化。Recombination of e with ione 和離子重組 Corona discharge ( blue glow )電暈放電 (發藍光)Corona discharge across power-line insulator.跨越電力線絕緣器的電暈放電。Electrostatic precipitators 靜電沉澱器：Removes pollutant particles (up to 99%
28、) 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。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。
