27. Electromagnetic Induction 電磁感應 Induced Currents 感應電流 Faraday’s Law 法拉第定律 Induction & Energy 感應與能量 Inductance 電感 Magnetic Energy 磁能 Induced Electric Fields 感應電場

It takes fourteen 110-car trainloads of each week to fuel this power plant. 這發電廠每個星期要十四列110部卡車的火車供應燃料。 What feature of the equation ε = －dΦB/dt demands this prodigious fuel consumption? 方程式 ε = －dΦB/dt 中那一部份要求消耗這麼驚人的燃料。 The minus sign, which denotes energy conservation in electromagnetic induction. 負號，這代表電磁感應的能量守恆。

27.1. Induced Currents 感應電流 4 results from Faraday / Henry (1831) 法拉第 / 亨利 (1831) 的四個結果 v = 0, I = 0 v > 0, I > 0 Current induced in coil by moving magnet bar. 移動磁鐵會在線圈引發(感應)電流。 v >> 0, I >> 0 v < 0, I < 0 Moving the coil instead of the magnet gives the same result. 不移磁鐵，改移線圈有同樣結果。

changing B induces currents (electromagnetic induction) An induced current also results when a current-carrying circuit replaces the magnet. 把磁鐵換成線圈也會引發感應電流。 A current is also induced when the current in an adjacent circuit changes. 旁邊線圈的電流改變也會引發感應電流。 changing B induces currents (electromagnetic induction) 改變 B 會引發電流 (電磁感應)

27.2. Faraday’s Law 法拉第定律 Magnetic flux 磁通量 Flux & Induced EMF 通量和感應電動勢

Magnetic flux 磁通量 Magnetic flux: 磁通量： Reminder: 提醒你： For a uniform B on a flat surface: 均勻 B 在平面上： Move magnet right  more lines thru loop 磁鐵右移  多些場線穿過廻路：

Example 27.1. Solenoid 螺線管 A solenoid of circular cross section has radius R, 一螺線管，其圓形截面的半徑為 R， consists of n turns per unit length, and carries current I. 其捲線為每單位長度 n 轉，其電流則為 I 。 Find the magnetic flux through each turn of the solenoid. 求通過管中每轉的磁通量。 I B B out of plane 側面圖 正面圖

Example 27.2. Nonuniform Field 非均勻場 A long, straight wire carries current I. 一長而直的電線上有電流 I 。 A rectangular wire loop of dimensions l by w lies in a plane containing the wire, with its closest edge a distance a from the wire, and its dimension l parallel to the wire. 另一 l 乘 w 大的長方形線圈和電線在同一平面上，其最近的邊與電線相距 a ， 其 I 長那邊與電線平行。 Find the magnetic flux through the loop. 求通過線圈的磁通量。 Area element for integration 積分的面積元素

Flux & Induced EMF 通量和感應電動勢 Faraday’s law of induction 法拉第的感應定律: The induced emf in a circuit is proportional to the rate of change of magnetic flux through any surface bounded by that circuit. 一個線路的感應電動勢，與穿透任何由這線路圍住的表面的磁通量成正比。 C is CCW about S. C 繞 S 逆時針轉 Note: dB/dt can be due to 註： dB/dt 可能源自 changing B caused by 下列因素所引起的 B 變化 relative motion between circuit & magnet, 線路與磁鐵的相對運動 changing current in adjacent circuit, 相隣線路內的電流變化 changing area of circuit, 線路面積的變化 changing orientation between B & circuit. B 與線路間角度的變化

Example 27.3. Changing B B在變 S C I CCW 逆時針 A wire loop of radius 10 cm has resistance 2.0 . 一個半徑為 10 cm 的線圈的電阻是 2.0 。 The plane of the loop is perpendicular to a uniform B that’s increasing at 0.10 T/s. 線圈的面垂直於一以 0.10 T/s 增加的均勻 B。 Find the magnitude of the induced current in the loop. 求線圈內感應電流的大小。 S C I CCW 逆時針

Example 27.3. Changing B B在變 S C I CCW 逆時針 A wire loop of radius 10 cm has resistance 2.0 . 一個半徑為 10 cm 的線圈的電阻是 2.0 。 The plane of the loop is perpendicular to a uniform B that’s increasing at 0.10 T/s. 線圈的面垂直於一以 0.10 T/s 增加的均勻 B。 Find the magnitude of the induced current in the loop. 求線圈內感應電流的大小。 S C I CCW 逆時針

Example 27.4. Changing Area 面積在變 Two parallel conducting rails a distance l apart are connected at one end by a resistance R. 兩條相距 l 的平行導電桿，一端連上電阻 R 。 A conducting bar completes the circuit, joining the two rails electrically but free to slide along. 迴路由另一可滑動的導電棒湊成。 The whole circuit is perpendicular to a uniform B, as show in figure. 整個電路垂直於一均勻 B。 (如圖) Find the current when the bar is pulled to the right with constant speed v. 求棒以等速 v 向右移時的電流。 Let x = 0 be at the left end of rail. 把 x = 0 點放在桿的左端。 S C I CCW 逆時針 x

Example 27.4. Changing Area 面積在變 Two parallel conducting rails a distance l apart are connected at one end by a resistance R. 兩條相距 l 的平行導電桿，一端連上電阻 R 。 A conducting bar completes the circuit, joining the two rails electrically but free to slide along. 迴路由另一可滑動的導電棒湊成。 The whole circuit is perpendicular to a uniform B, as show in figure. 整個電路垂直於一均勻 B。 (如圖) Find the current when the bar is pulled to the right with constant speed v. 求棒以等速 v 向右移時的電流。 Let x = 0 be at the left end of rail. 把 x = 0 點放在桿的左端。 S C I CCW 逆時針 x

27.3. Induction & Energy 感應與能量 m I Direction of emf is to oppose magnet’s motion. 電動勢的方向是要抵擋磁鐵的移動。 RH rule: thumb // m. 右手法則：拇指 // m 。 Loop ~ magnet with N to left. 廻路 ~ N在左端的磁鐵 Magnet moving right 磁鐵往右 Lenz’s law 楞次定律: Direction of induced emf is such that B created by the induced current opposes the changes in  that created the current. 電動勢的方向是要感應電流所生的 B ，會抵消引發該電流的  變化。 I m RH rule: thumb // m. 右手法則：拇指 // m 。 Loop ~ magnet with S to left. 廻路 ~ S在左端的磁鐵 Magnet moving left 磁鐵往左

GOT IT? 27.1 You push a bar magnet towards a loop, with the north pole toward the loop. 你把一條磁鐵以北極朝向一個線圈推過去。 If you keep pushing the magnet straight through the loop, 如果你一直把磁鐵推， what will be the direction of the current as you pull it out the other side? 當你把它從線圈的另一邊拉出來時，電流的方向為何？ Will you need to do work, or will work be done on you? 你要作功，還是被作功？ Reversed 逆轉 I I m m

Motional EMF & Len’s Law 運動電動勢和楞次定律 Motional emf: induced emf due to motion of conductor in B. 運動電動勢：因導在 B 中運動而誘發的電動勢。 Square loop of sides L & resistance R pulled with constant speed v out of uniform B. 邊長為 L，電阻為 R 的方形線圈以等速 v 離開均勻 B。 Force on e: e 所受力： downward force upward I (CW) 力朝下  I 朝上(順時) Force on current carrying wire: 有電流的電線所受力：

Work done is used to heat up circuit ( E conservation ). > 0 S I C CW 順時 x Work done is used to heat up circuit ( E conservation ). 所作功都用來把線路加熱 ( E 守恆 ) 。

Work done is used to heat up circuit ( E conservation ). < 0 S I C CW 順時 x Work done is used to heat up circuit ( E conservation ). 所作功都用來把線路加熱 ( E 守恆 ) 。

GOT IT? 27.2 What will be the direction of the current when the loop first enters the field from the left side? 當線圈開始從左邊進來磁場時，電流的方向為何？  S Current is CCW 電流的方向為逆時針 C

GOT IT? 27.2 What will be the direction of the current when the loop first enters the field from the left side? 當線圈開始從左邊進來磁場時，電流的方向為何？  S Current is CCW 電流的方向為逆時針 C

Application. Electric Generators 發電機 World electricity generation ~ 2TW. 全世界的發電量 ~ 2TW 。 Rotating loop changes  & induces emf. 旋轉的線圈改變  而引發電動勢。 Rotating slip rings. 旋轉的套環。 Sinusoidal AC output 正弦形交流電輸出。 Work required due to Lenz’s law. 因楞次定律而需要作功。 Electric load 電載 Stationary brushes 固定的電刷 Rotating conducting loop 旋轉的導電圈 Hand-cranked generator ~ 100W 手搖發電機 ~ 100W

GOT IT? 27.3 If you lower the electrical resistance connected across a generator while turning the generator at a constant speed, 如果你把發電機外接的電阻降低，同時維持發電機的轉速不變， will the generator get easier or harder to turn? 發電機會變得容易些還是難些轉？ constant speed  fixed peak emf 轉速不變  emf 峯值不變 lower R  larger power 較低 R  功率較大

Example 27.5. Designing a Generator 設計一台發電機 An electric generator consists of a 100-turn circular coil 50 cm in diameter. 一台發電機內含一個直徑為 50 cm 的 100-轉 圓形線圈。 It’s rotated at f = 60 rev/s to produce standard 60 Hz alternating current. 它以 f = 60 rev/s 轉動來產生標準的 60 Hz 交流電。 Find B needed for a peak output voltage of 170 V, 求輸出峰壓為 170 V 時所需 B ： which is the actual peak in standard 120 V household wiring. 這也是住宅標準 120 V 線路的峰壓。 Loop rotation 迴路轉動 線圈面積

電磁感應是磁性錄製的根源 ( 如錄音，錄影，電腦磁碟，… ) EM induction is basis of magnetic recording ( audio, video, computer disks, …). 電磁感應是磁性錄製的根源 ( 如錄音，錄影，電腦磁碟，… ) Iron 鐵 Coil 線圈 Card motion 卡片動向 Modern hard disks: Giant magnetoresistance. 近代硬碟：巨大磁致電阻。 Magnetic strip 磁帶 Information stored in magnetization pattern 資料存在磁化的式樣中 Swiping a credit card. 刷一張信用卡。 Patterns of magnetization on the strip induce currents in the coil. 磁帶的磁化式樣令線圈產生感應電流。

Eddy Currents 渦流 Application: Metal Detectors 應用：金屬檢測器 Eddy current : current in solid conductor induced by changing . 渦流： 因  變動而在固態導體內產生的電流。 Usage: non-frictional brakes for rotating saw blades, train wheels, … 用途：非摩擦性剎車，用於轉動形電鋸，火車輪子，… Application: Metal Detectors 應用：金屬檢測器 Nothing between coils 線圈間無物 Induced Current 感應電流 Current detector 電流檢測器 AC 交流電 Strong 強 I Transmitter coil 發送線圈 Receiver coil 接收線圈 Weak I : alarm. 弱 I ：警訊 Metal between coils 線圈間有金屬品

GOT IT? 27.4 A copper penny falls on a path that takes it between the poles of a magnet. 一個銅板下墜的路徑讓它通過一個磁鐵的兩極之間。 Does it hit the ground going 它掉到地上時會比無磁鐵時 faster, (a) 快些， slower, (b) 慢些， at the same speed as if the magnet weren’t present? (c) 還是速度一樣？ Eddy current dissipates KE. 渦流會耗掉 KE 。

Closed & Open Circuits 閉路和開路 B of induced I points out of page 感應 I 的 B 從紙出來 Setting n // Bin  C is CCW &  < 0 Bin   d  /d t < 0  E > 0  I is CCW RH rule gives CCW I 右手法則得逆時 I Bin  C +_ B  E > 0

GOT IT? 27.5 A long wire carries a current I as shown. 一長電線有電流 I (圖示) 。 What’s the direction of the current in the circular conducting loop when I is 當 I 在下列情况時，圓形廻路中的電流方向為何？ increasing and (a) 增加中 decreasing? (b) 減小中。 逆時 CCW 順時 CW Setting n // B  C is CW &  > 0 I   B   d  /d t > 0  E < 0  Iind is CCW B

GOT IT? 27.5 A long wire carries a current I as shown. 一長電線有電流 I (圖示) 。 What’s the direction of the current in the circular conducting loop when I is 當 I 在下列情况時，圓形廻路中的電流方向為何？ increasing and (a) 增加中 decreasing? (b) 減小中。 逆時 CCW 順時 CW Setting n // B  C is CCW &  < 0 I   B   d  /d t < 0  E > 0  Iind is CCW B

27.4. Inductance 電感 [ L ] = T  m2 / A = Henry Mutual Inductance 互感: Changing current in one circuit induces an emf in the other. 因一電路中的電流發生變化，而在另一電路產生電動勢。 Large inductance 大電感： two coils are wound on same iron core. 兩線圈繞在同一個鐵芯上。 Applications 應用: Transformers, ignition coil, battery chargers, … 變壓器，點火線圈，電池充電器，… Self-Inductance 自感 : Changing current induces emf in own circuit & opposes further changes. 因電路中的電流發生變化，而在本電路產生電動勢，並抗拒進一步的變化。 Applications 應用: Inductors  frequency generator / detector … 電感器  頻率產生器 / 偵察器 … [ L ] = T  m2 / A = Henry

Example 27.6. Solenoid 螺線管 求其自感。 A long solenoid of cross section area A and length l has n turns per unit length. 一長螺線管，截面積為 A ，長度為 l ，每單位長度有 n 轉。 Find its self-inductance. 求其自感。 B of solenoid: 螺線管的 B：

+E direction = V  along I. +E 的方向 = 沿 I 朝 V  back emf 反電動勢 電流方向 Rapid switching of inductive devices can destroy delicate electronic devices. 快速開關電感元件會損毁精密電子設備。 d I /d t < 0

 (lower voltage) (較低壓)  (lower voltage) (較低壓) Inductor 電感器  (lower voltage) (較低壓)  (lower voltage) (較低壓) E E + (higher voltage) (較高壓) + (higher voltage) (較高壓)

GOT IT? 27.6 V  along I  E < 0  dI/dt > 0 Current flows left to right through the inductor shown. 電流從左往右通過圖中電感器。 A voltmeter connected across the inductor gives a constant reading, and shows that the left end of the inductor is positive. 一個接在電感器兩端的電位表的讀數保持一定，且顯示電感器的左端為正。 Is the current in the inductor 電感器內的電流是在 increasing, (a) 增加中， decreasing, or (b) 減少中，還是 steady? Why? (c) 保持穩定？為甚麽？ V  along I  E < 0  dI/dt > 0

Example 27.7. Dangerous Inductor 危險的電感器 A 5.0-A current is flowing in a 2.0-H inductor. 一個 2.0-H 的電感器內流着 5.0 A 的電流。 The current is then reduced steadily to zero over 1.0 ms. 然後電流在 1.0 ms 內平穩地減少至零。 Find the magnitude & direction of the inductor emf during this time. 求這段時間內電感器的電動勢的大小和方向。 + here ( E > 0 ) helps keep I flowing + 在這 ( E > 0 ) 讓 I 繼續流 I ，遞減 外部線路

Inductors in Circuits 線路中的電感器 Current through inductor can’t change instantaneously. 電感器內的電流不能馬上改變。 Switch just closed : 開關剛關上： I = 0, dI/dt  0; EL = E0 ; L ~ open circuit. Long after switch closing: 開關關上後很久： I  0, dI/dt = 0; EL = 0; L ~ wire. Switch open: 開關打開： I = 0

+_ EL < 0 ; | EL |  Inductive time constant = L / R 電感時間常數 I   V = IR  But rate is 但速率  EL < 0 ; | EL |  電感器的電動勢 線路的電流 Inductive time constant = L / R 電感時間常數 c.f. capacitive time constant = RC 對比：電容時間常數

Example 27.8. Firing Up a Electromagnet 起動一個電磁鐵 A large electromagnet used for lifting scrap iron has self-inductance L = 56 H. 一個用來吸起廢鐵的大電磁鐵的自感是 L = 56 H。 It’s connected to a constant 440-V power source; 它接在一個定壓 440-V 的電源上； the total resistance of the circuit is 2.8 . 整個線路的電阻是 2.8  。 Find the time it takes for the current to reach 75% of its final value. 求電流達到其最後值的 75% 所需時間。 

Switch at B, battery’s shorted out. I  exponentially. 開關在 B，電池短路掉。 I 以指數函數  Switch at 開關在 A, I . Short times: IL can’t change instantaneously. 短時間： IL 不能馬上改變。 Long times: EL = 0 ; inductor  wire. 長時間： EL = 0 ; 電感器  電線

Conceptual Example 27.1. Short & long Times 長和短時間 The switch in figure is initially open. 圖中開關本來是開的。 It is then closed and, a long time later, reopened. 之後它關上，很久之後又再打開。 What’s the direction of the current in R2 after the switch is reopened? 在開關再打開之後， R2 中的電流方向為何? right after switch closed : L ~ open circuit. 開關關上當時 : L ~ 開路 Long after switch closed : L ~ short circuit. 開關關上很久之後 : L ~ 短路 right after switch reopened : I in L ~ continues. 開關再打開當時 : L 內 I ~ 不變

Making the Connection 連起來 Verify that the current in just after the switch is reopened has the value indicated . 証明在開關再打開當時，電流值如圖所示。 Immediately after the switch is closed: 開關剛關上時： L ~ open circuit. 開路 Long time after the switch is closed: 開關關上很久之後： Immediately after 2nd switch opening : 剛在第二次打開開關時： L ~ short circuit. 短路 I in L ~ continues. 繼續流

27.5. Magnetic Energy 磁能 Any B contains energy. 任何都蘊涵能量。 This eruption of a huge prominence from the sun’s surface releases energy stored in magnetic fields. 這個從太陽表面噴發的巨型日珥把儲存在磁場中的能量釋放出來。

Magnetic Energy in an Inductor 電感器內的磁能 RL circuit : 阻感線路：  Power from battery 從電池來的功率 Power dissipated 消耗掉的功率 Power taken by inductor 電感器拿走的功率 Energy stored in inductor 儲存在電感器的能量：

Example 27.10. MRI Disaster 磁振造影儀災難 Superconducting electromagnets like solenoids in MRI scanners store a lot of magnetic energy. 超導電磁鐵，如磁振造影儀所用的螺線管，儲存了龐大的磁能。 Loss of coolant is dangerous since current quickly decays due to resistance. 冷媒流失很危險，因為電流會遇電阻而快速消失。 A particular MRI solenoid carries 2.4 kA and has a 0.53 H inductance. 某個磁振造影儀的螺線管流着 2.4 kA ，而且電感為 0.53 H 。 When it loses superconductivity, its resistance goes abruptly to 31 m. Find 當它失去超導性時，它的電阻驟升至 31 m 。求 the stored magnetic energy, and 儲存的磁能，和 the rate of energy release at the instance the superconductivity is lost. 在失去超導性當時，釋出能量的速率。 In practice, Cu / Ag are incorporated into the superconducting wires to reduce R. 實際上，超導線中會滲有 Cu / Ag 以降低 R。

Magnetic Energy Density 磁能密度 Solenoid with length l & cross-section area A : 長 l ，截面積 A 的縲線管： (Eg. 27.6) Magnetic Energy Density : 磁能密度： c.f. electric energy density : 對比：電能密度：

27.6. Induced Electric Fields 感應電場 EMF acts to separate charges 電動勢使電荷分離： Battery 電池： chemical reaction 化學反應 Motional emf 運動電動勢： F = v  B. Stationary loop in changing B : induced E 變化中的 B 內靜止的廻路： 感應電場 Faraday’s law 法拉第定律 區域內是變化中的場 Induced E forms loop. 感應 E 必成廻路 Static E begins / ends on charge. 靜 E 從電荷開始 / 結東

Example 27.11. Solenoid 縲線管 S CCW S in  C cw A long solenoid has circular cross section of radius R. 一個長縲線管的截面是圓形，其半徑為 R。 The solenoid current is increasing, & as a result so is B in solenoid. 縲線管的電流在增加中，故管內的 B 亦然。 The field strength is given by B = b t, where b is a constant. 場的強度是 B = b t ，其中 b 是個常數。 Find the induced E outside the solenoid, a distance r from the axis. 求縲線管外離軸心 r 處的感應 E。 Loop for Faraday’s law法拉第定律的廻路 Symmetry  E lines are circles. 對稱性  E 線都是圓。 S CCW S in  C cw

Example 27.11. Solenoid 縲線管 S CCW S out  C ccw A long solenoid has circular cross section of radius R. 一個長縲線管的截面是圓形，其半徑為 R。 The solenoid current is increasing, & as a result so is B in solenoid. 縲線管的電流在增加中，故管內的 B 亦然。 The field strength is given by B = b t, where b is a constant. 場的強度是 B = b t ，其中 b 是個常數。 Find the induced E outside the solenoid, a distance r from the axis. 求縲線管外離軸心 r 處的感應 E。 Loop for Faraday’s law法拉第定律的廻路 Symmetry  E lines are circles. 對稱性  E 線都是圓。 S CCW S out  C ccw

Conservative & Nonconservative Electric Fields 守恆和非守恆電場 For stationary charges (electrostatics) : 靜止電荷 (靜電學) ： E W against E 對 E 作功 E is conservative E 是守恆的 Induced fields (electromagnetics) : 感應場 (電磁學) ： E does W E 作功 E is non-conservative E 是非守恆的

GOT IT? 27.7 The figure shows three resistors in series surrounding an infinitely long solenoid with a changing magnetic field; 圖示三個成串連的電阻圍住一個無限長的縲線管，管內的磁場則在改變中； the resulting induced electric field drives a current counterclockwise, as shown. 因此而產生的感應電場推動一個逆時針方向電流 (如圖示) 。 Two identical voltmeters are shown connected to the same points A and B. 兩個同樣的伏特計都接在 A 和 B 點上 (如圖示) 。 What does each read? 它們的讀數為何？ Explain any apparent contradiction. 解釋任何表面上的矛盾。 Hint: this is a challenging question! 題示：這是個棘手問題。 VA  VB = 2IR VA  VB = IR

Diamagnetism 反磁性 net  0 Classical model of diamagnetism (not quite right) 反磁性的古典模型 (不太對) Superconductor is a perfect diamagnet (Meissner effect). 超導體是個完美的反磁鐵 (麥士那效應) B = 0: net = 0 net  0 B  0 This e slows down. 這 e 慢下來 This e speeds up. 這 e 快起來