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 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 :

AB B AU U U ABWB

Ad F r

Electric potential difference electric potential energy difference per unit charge電位差 每單位電荷的電位能差B

Ad E rAB

ABUVq

BV if reference potential VA = 0.若參考電位 VA = 0.

[ V ] = J/C = Volt 伏特 = V 伏

For a uniform field 均勻場 :

AB ABV E r B A E r r

( path independent ) 與路徑無關

rAB EE 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 Symbol / Equation符號 / 公式 Units單位

B

AB AU d F r

UVq

B

AB AV d E r

Force 力Electric field 電場

Potential energy difference位能差

Electric potential difference電位差

F N

E = F / q N/C or V/m

J

J/C or V

U F

V E

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 0

reverses sign變符號

The Volt & the Electronvolt 伏特和電子伏特[ V ] = J/C = Volt = V U q 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 = 1.61019 C = e

1 eV = 1.61019 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 kVtransmission 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 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 V

GOT 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。What’s the work in eV done on each? 對每粒所作的功有幾 eV ？

10 eV20 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。

ABV E r 300 / 0.10kN C m 30 kV

AB ABU e V 30 keV

30AB ABK U keV 154.8 10 J 4.8 fJ

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。

0 0xV E x 02

x

E

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

Curved path, nonuniform field:彎曲路徑，非均勻場：0

limAB i ii

V

rE r B

Ad E r

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。

GOT IT? 22.3

Largest ΔVAB .最大Smallest ΔVAB .最小

22.2. Calculating Potential Difference 計算電位差Potential of a Point Charge點電荷的電位

AB B AV V V B

Ad E r 2

ˆB

A

k q dr

r r

2

B

A

r

AB r

k qV d rr

ˆ ˆ ˆd d r r r r r d r

1 1

B A

k qr r

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 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 RA A R AV V V V

Some choices of zero potential 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’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(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 sphere’s surface,把一質子從無限遠處帶到球面上所需的功，(c) 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。

QV R kR

69

640 109.0 10 /

2.30

CVm C

m

(a) 2.50 MV

W e V R(b) 2.50 MeV 191.6 10 2.50C MV 134.0 10 J

,2 2R RV V R V R (c)2Q Qk kR R

2QkR

1.25 MV 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？

B

A

r

AB rV d E r

0

ˆ ˆ2

B

A

r

rd r

r

r r

0

12

B

A

r

rd r

r

0

ln2

B

A

rr

9 6 222 9.0 10 / 2.6 10 / ln0.01

mV m C C mm

360 kV

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 i

i P i

qV P k r r

Potential of a set of charge sources:一組電荷的電位 ii

V P V P

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。

1 2

qqV P k kr r

1 2

1 1kqr r

2 2 21 2 cosr r a r a

2 2 22 2 cosr r a r a

2 22 1 4 cosr r r a

2 1 1 2r r r r

r >> a 2 1 2 cosr r a

2 12

r rV P k qr

2

2 cosqakr

2

cospkr

p = 2qa = dipole moment 雙極距

2 1

2 1

r rkqr r

+q: hill 丘

q: hole 洞V = 0

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 。

P

1

23

q q

Continuous Charge Distributions 連續電荷分佈Superposition:叠加

V dV d qkr

dVkr

3d rV k

r

rr r

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。

dqV x kr

2 2

k dqx a

2 2

k Qx a

Same r for all dq所有 dq 的 r 都一樣

Qk x ax

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。

V x dV 2 2

k dqx r

2 22

2k Q x a xa

22 202

a k Q r drax r

2 2 20

2 aQ rk dra x r

2 22

0

2 aQk x ra

sheet 片

point charge 點電荷disk 盤

盤軸上的距離 x

電位 2

0

2 22

k Q kQa x x x aa a

k Q x ax

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 = const.等電位 = V 為定值的面

V > 0V < 0 V = 0

Steep hill 丘峭Close contour 線密Strong E E 強

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B

A

r

AB rV d E r

dV d E r i ii

E dx ii i

V d xx

ii

VEx

V E = ( Gradient of V ) V 的陡 (梯，坡 )度

V V Vx y z

E i j k

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: 2 2

2

2k QV x x a xa

2 2 2

2 1k Q xa x a

xVEx

x > 0x < 0

0y zE E 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。

xVEx

V falling, Ex > 0V 在降， Ex > 0

V flat, Ex = 0

V rising, Ex < 0V 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。

11

1

QV kR

22

2

QV kR

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

1 2

1 2

Q QR R

24j

jj

QR

In terms of the surface charge densities用表面電荷密度來算1 1 2 2R R we have可得

Smaller sphere has higher field at surface.比較小的球，表面上的場比較強。

1 1 2 2E R E R

Same VV 相同

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。

Conceptual Example 22.1. An Irregular Condutor 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。

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。

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。

7 mm12 mm

Strongest field 最強的電場 :

3

707 10

VEm

10 /kV m

Weakest field 最弱的電場 :

3

7012 10

VEm

5.8 /kV m

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 )

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%) 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。