Short Version : 27. Electromagnetic Induction 短版 : 27.電磁感應

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 27.4. 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

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 ) 。

Application. Electric Generators 發電機 World electricity generation ~ 2TW. 全世界的發電量 ~ 2TW 。 TW ~ 2.7GW 台灣 ~ 2.7GW 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

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

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

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)

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

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 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