Transcript 電路 25. Electric Circuits

25. Electric Circuits
1.
2.
3.
4.
5.
Circuits, Symbols, & Electromotive Force
Series & Parallel Resistors
Kirchhoff’s Laws & Multiloop Circuits
Electrical Measurements
Capacitors in Circuits
電路
電路，符號，和電動勢
串聯和並聯電阻器
基爾霍夫定律和多環路電路
電的測量
電路中的電容器
Festive lights decorate a city.
If one of them burns out, they all go out.
Are they connected in series or in parallel?
慶祝的燈泡點綴了一個城市。
如果其中一個燒了，全部都會熄掉。
它們是串聯還是並聯？
Electric Circuit = collection of electrical components connected by conductors.
電路 = 一些電元件由導體連在一起。
Examples 例 :
Man-made circuits: flashlight, …, computers.
人做的電路：手電筒，… ，電腦。
Circuits in nature: nervous systems, …, atmospheric circuit (lightning).
大自然中的電路：神經系統，… ，大氣電路(閃電) 。
25.1. Circuits, Symbols, & Electromotive Force
電路，符號，和電動勢
Common circuit symbols
常見的電路符號
All wires ~ perfect conductors  V = const on wire
全部接線 ~ 完美導體
 線上 V = 常數
Electromotive force (emf) = device that maintains fixed V across its terminals.
電動勢(emf) = 端點維持着固定 V 的元件。
E.g.,
batteries (chemical),
電池(化學作用) ，
generators (mechanical),
發電機(機械性) ，
photovoltaic cells (light),
光電壓電池(光)，
cell membranes (ions).
細胞薄膜(離子) 。
m~q
Collisions ~ resistance
碰撞 ~ 電阻
g~E
Ideal emf 理想電動勢：
no internal energy loss.
無內部能量消耗。
Lifting ~ emf
抬起 ~ 電動勢
Energy gained by charge transversing battery = q E
( To be dissipated as heat in external R. )
電荷在橫渡電池後獲得的能量 = q E
( 將在外部 R 內當熱來消耗掉。)
Ohm’s law: I  E
歐姆定律
R
GOT IT? 25.1.
The figure shows three circuits.
圖示三個電路。
Which two of them are electrically equivalent?
那兩個的電性相當？
25.2. Series & Parallel Resistors 串聯和並聯電阻器
Series resistors 串聯電阻器 :
I = same in every component
每個元件的 I 都一樣
Same q must go
thru every element.
通過每個元件的 q
都必需要一樣。
E  V1  V2  I R1  I R2

Rs  R1  R2
For n resistors in series:
n 個串聯電阻器
I
n=2:
V1 
R1
R1  R 2
E
 I Rs
E
Rs
Rs   R j
Vj  I Rj 
V2 
R2
R1  R 2
E
n
j 1
Rj
Rs
E
Voltage divider
分壓器
分壓器
Example 25.1. Voltage Divider
A lightbulb with resistance 5.0  is designed to operate at a current of 600 mA.
一個電阻為 5.0  的燈泡是為了在 600 mA 的電流中運作而設計。
To operate this lamp from a 12-V battery, 如果要用一個 12 V 電池來操作這盞燈，
what resistance should you put in series with it?
12 V
 2.4 A  0.6 A
5.0 
lightbulb
燈泡
你應該用甚麼電阻和它串聯？
E  I R1  I R2
R1 
E
 R2
I

 20   5 
12 V
5
0.6 A
 15 
Voltage across lightbulb = V2   600  103 A   5    3 V
燈泡兩端的電壓

1
E
4
Most inefficient
很無効率
GOT IT? 25.2.
Rank order the voltages across the identical resistors R at the top of each circuit
shown, and give the actual voltage for each.
替下面各電路上方同樣的電阻器 R 兩端的電壓排序，並列出每個電壓。
In (a) the second resistor has the same resistance R, and
在 (a)中第二個電阻器的電阻也是 R 。
(b) the gap is an open circuit (infinite resistance).
在 (b)中的缺口代表開路 (電阻無限大) 。
3V
0V
6V
Real Batteries
實際的電池
Model of real battery = ideal emf E in series with internal resistance Rint .
實際電池的模型 = 理想 emf E 與內電阻 Rint 串聯。
I means V drop I Rint
I 代表 V 下跌 I Rint
 Vterminal < E
E  I Rint  I R L
I
E
R int  R L
VRL 
RL
Rint  R L

E
E
RL
Example 25.2. Starting a Car
起動車子
Your car has a 12-V battery with internal resistance 0.020 .
你的車子有一個 12-V 電瓶，其內電阻為 0.020 。
When the starter motor is cranking, it draws 125 A.
起動馬達轉動時會抽取 125 A 。
Battery terminals
電瓶兩極
What’s the voltage across the battery terminals while starting?
電瓶兩極間的電壓在起動時是多少？
E  I R int  I R L
RL 
E
12 V
 Rint 
 0.020 
I
125 A
 0.096   0.020 
 0.076 
Voltage across battery terminals = E  V  12 V  125 A  0.020   9.5 V
int
電瓶兩極間的電壓
Typical value for a good battery is 9 – 11 V.
通常一個好電瓶的數值是 9 – 11 V 。
Parallel Resistors
Parallel resistors 並聯電阻器 :
V = same in every component
每個元件的 V 都一樣
I  I1  I 2

E
E
E



Rp
R1 R 2
1
1
1


R p R1 R 2
For n resistors in parallel :
n 個並聯電阻器：
Rp 
R1 R 2
R1  R 2
n
1
1

Rp j  1 R j
GOT IT? 25.3.
The figure shows all 4 possible combinations of 3 identical resistors.
圖示三個完全相同的電阻器的全部四種組合。
Rank them in order of increasing resistance.
按遞增電阻值替它們排序。
3R
R/3
1
4
2R/3
3
3R/2
2
Analyzing Circuits
Tactics 策略 :
• Replace each series & parallel part by their single component equivalence.
把每個串和並聯部份以它們的單一相當元件取代。
• Repeat 重覆。
Example 25.3. Series & Parallel Components
串和並聯元件
Find the current through the 2- resistor in the circuit.
求電路中通過 2- 電阻器的電流。
Equivalent of parallel 2.0- & 4.0- resistors:
2.0- 和 4.0- 並聯電阻器的相當值：
1
1
1
3



R 2.0  4.0 
4.0 

R  1.33 
Equivalent of series 1.0-, 1.33-  & 3.0- resistors:
1.0-, 1.33-  和 3.0- 串聯電阻器的相當值：
RT  1.0   1.33   3.0   5.33 
Total current is
總電流
Voltage across of parallel 2.0- & 4.0- resistors:
跨越 2.0- 和 4.0- 並聯電阻器的電壓：
Current through the 2- resistor:
通過 2- 電阻器的電流：
I 2 
I 5.33 
12 V
E

RT 5.33 
 2.25 A
V1.33   2.25 A1.33   2.99 V
2.99 V
2.0 
 1.5A
GOT IT? 25.4.
The figure shows a circuit with 3 identical lightbulbs and a battery.
圖示由三個同樣燈泡和一個電瓶所組成的電路。
(a) Which, if any, of the bulbs is brightest?
如果有的話，那個燈泡最亮？
(b) What happens to each of the other two bulbs if you remove bulb C?
如果你把燈泡 C 拿掉，其他兩個燈泡會怎樣？
dimmer
暗些
brighter
亮些
25.3. Kirchhoff’s Laws & Multiloop Circuits
基爾霍夫定律和多環路電路
Kirchhoff’s loop law 基爾霍夫的環路定律：
 V = 0 around any closed loop 繞任何環路一周。
( energy is conserved 能量守恆 )
Kirchhoff’s node law 基爾霍夫的節點定律:
This circuit can’t be analyzed using series
and parallel combinations.
這電路不能用串聯和並聯的組合來分析
I=0
at any node 每一節點.
( charge is conserved 電荷守恆 )
多環路電路
Multiloop Circuits
Problem Solving Strategy 解題策略:
INTERPRET 分析
■ Identify circuit loops and nodes
找出電路的環路和節點。
■ Label the currents at each node, assigning a direction to each.
為每個節點的電流命名，並設定方向。
DEVELOP 發展
■ Apply Kirchhoff’s node law to all but one nodes. ( Iin > 0, Iout < 0 )
把基爾霍夫的節點定律用於全部扣一個節點
( Iin > 0, Iout < 0 ) 。
■ Apply Kirchhoff’s loop law to all independent loops:
把基爾霍夫的環路定律用於全部獨立的環路：
Batteries: V > 0 going from  to + terminal inside the battery.
電瓶：從  到 + 極 V > 0 。
Resistors: V =  I R going along +I.
電阻器：順 +I 走 V =  I R 。
 Some of the equations may be redundant.
有些方程式可能是多餘的。
Example 25.4. Multiloop Circuit 多環路電路
求圖中 R3 的電流。
Find the current in R3 in the figure below.
Node 節 A:
 I1  I 2  I 3  0
E1  I1R1  I 3 R 3  0
6  2 I1  I 3  0
Loop 環 2: E2  I 2 R 2  I 3 R 3  0
9  4I2  I3  0
Loop 環 1:

1
I1   I 3  3
2
I2 
9
1 1 


1
I

3


 3
4
2 4 

I3 
4 21

 3A
7 4
1
9
I3 
4
4
Application: Cell Membrane
應用：細胞膜
Hodgkin-Huxley (1952) circuit model of cell membrane (Nobel prize, 1963):
霍奇金-赫胥黎 (1952) 細胞膜的電路模型 ( 1963 年諾貝爾獎 ) ：
Resistance of cell membranes
細胞膜的電阻
Membrane
potential
細胞膜電位
Electrochemical effects
電力化學効應
Time dependent effects
與時間有關的効應
25.4. Electrical Measurements 電的測量
A voltmeter measures potential difference between its two terminals.
一個電壓(伏特)計可以量得它兩端點間的電位差。
Ideal voltmeter: no current drawn from circuit  Rm = 
理想電壓計：不從電路抽取電流  Rm = 
Conceptual Example 25.1. Measuring Voltage
What should be the electrical resistance of an ideal voltmeter?
理想電壓計的電阻為何 ?
An ideal voltmeter should not change the voltage
across R2 after it is attached to the circuit.
理想電壓計在接上電路後不應改變 R2 兩端的電壓。
The voltmeter is in parallel with R2.
電壓計與 R2 並聯。
In order to leave the combined resistance, and
hence the voltage across R2 unchanged, RV must
be .
欲要不改變合併電阻，以及 R2 兩端的電壓， RV 必
需為 。
Making the Connection 連起來
What do you get if you measure the voltage across the 40- resistor in fig. with
你用下列工具去量圖中 40- 電阻器上的電壓時，會得到甚麽?
理想電壓計
(a) an ideal voltmeter?
(b) a voltmeter whose resistance is 1000 ? 內阻為 1000  的電壓計。
(a)
(b)


40 
V40  
12 V   4 V

 40   80  
R parallel 
 40  1000  
40   1000 
 38.5 


38.5 
V40  
12 V   3.95V

 38.5   80  
3.95  4.
 1.25%
4.
Example 25.5. Two Voltmeters
兩個電壓計
You want to measure the voltage across the 40- resistor.
你要量 40- 電阻器上的電壓。
(a) What readings would an ideal voltmeter give?
一個理想電壓計上的讀數為何？
(b) What readings would a voltmeter with a resistance of 1000  give?
一個電阻 1000  的電壓計上的讀數為何？
(a)
(b)


40 
V40  
12 V   4 V

 40   80  
R parallel 
 40  1000  
40   1000 
 38.5 


38.5 
V40  
12 V   3.95V

 38.5   80  
GOT IT? 25.5.
If an ideal voltmeter is connected between points A and B in figure, what will it read?
如果圖中A 和B 接了一個理想電壓計，它的讀數為何？
All the resistors have the same resistance R.
所有電阻器都有同樣的電阻 R。
½E
Ammeters 電流(安培)計
An ammeter measures the current flowing through itself.
一個電流(安培)計可以量得通過它本身的電流。
Ideal voltmeter: no voltage drop across it  Rm = 0
理想電流計： 它本身無電位差
 Rm = 0
Ohmmeters & Multimeters
電阻計和萬用電表
An ohmmeter measures the resistance of a component.
( Done by an ammeter in series with a known voltage. )
一個電阻計可以量得一個元件的電阻。
(以一個電流計和一個已知電壓串聯而成。)
Multimeter: combined volt-, am-, ohm- meter.
萬用電表： 電壓，電流，和電阻計合而為一。
25.5. Capacitors in Circuits
電路中的電容器
Voltage across a capacitor cannot change instantaneously.
電容器兩端的電壓不能馬上改變。
The RC Circuit: Charging
阻容電路：充電時
C initially uncharged  VC = 0
開始時無電荷
開關
開關在 t = 0 時關上
Switch closes at t = 0.
VR (t = 0) = E
VR  but rate 

I  but rate 
I (t = 0) = E / R
C charging 充電中:
VC   VR   I 
Charging stops when I = 0. 充電到 I = 0 時停止。
VC  but rate 
E I R

dI
dt

I
RC


Q
0
C
dI
I
R 0
dt
C
I
dQ
dt
t dt
dI


I0 I 0 RC
I
VC ~ 2/3 E
ln
I
t

I0
RC
I  I0 e

t
RC
VC  E  VR
E  RCt
 e
R
t



 E 1  e RC 


Time constant 時間常數 = RC
I ~ 1/3 E/R
The RC Circuit: Discharging
阻容電路：放電時
C initially charged to VC = V0
開始時已充電到
Switch closes at t = 0.
dQ
I 
dt
Q
I R0
C
開關在 t = 0 時關上
VR = VC = V

I 0 = V0 / R
C discharging 放電中 : VC   VR   I 
dI
dt

I
RC
Disharging stops when I = V = 0.
放電到 I = V = 0 時停止。
I  I0 e

t
RC
V  V0 e
V0  RCt
 e
R

t
RC
Example 25.6. Camera Flash 相機閃光燈
A camera flash gets its energy from a 150-F capacitor & requires 170 V to fire.
一個相機閃光燈從一個 150-F 電容器取得能量，而且需要 170 V 才能閃。
If the capacitor is charged by a 200-V source through an 18-k resistor,
如果電容器透過一個 18-k 電阻器從一個 200-V 電源充電，
how long must the photographer wait between flashes?
那個攝影師要隔多久才能再次閃光？
Assume the capacitor is fully charged at each flash.
假定每次閃光時電容器都充滿電。
 V 
t   RC ln  1  C 
E 

 170 V 
  18  103  150  106 F  ln 1 

 200 V 
 5.1 s
RC Circuits: Long- & Short- Term Behavior
阻容電路：長和短時行為
For t << RC: VC  const,

C replaced by short circuit if uncharged.
C 未充電時可以短路取替。

C replaced by battery if charged.
C 已充電時可以電瓶取替。
For t >> RC: IC  0,

C replaced by open circuit.
C 以斷(開)路取替。
Example 25.7. Long & Short Times
長和短時
The capacitor in figure is initially uncharged.
圖中的電容器開始時未充電。
Find the current through R1
求通過 R1 的電流
(a) the instant the switch is closed and
開關關上的一剎那。
(b) a long time after the switch is closed.
開關關上後很長一段時間。
(a)
(b)
I1 
I1 
E
R1
E
R1  R2
GOT IT? 25.6.
A capacitor is charged to 12 V
一個電容器充電到 12V
& then connected between points A and B in the figure,
然後接到圖中 A 和 B 點上
with its positive plate at A.
(正板在 A)。
What is the current through the 2-k resistor
通過 2-k電阻器的電流為何
6 mA
(a)
immediately after the capacitor is connected and
(a) 接上電容器那一刻
2 mA
(b)
a long time after it’s connected?
(b) 接上後很久？