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26. Magnetism: Force & Field 磁性:力和場

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1 26. Magnetism: Force & Field 磁性:力和場
What is Magnetism? 磁性是甚麽? Magnetic Force & Field 磁力和場 Charged Particles in Magnetic Fields 磁場中的帶電粒子 The Magnetic Force on a Current 電流受到的磁力 Origin of the Magnetic Field 磁場的來源 Magnetic Dipoles 磁雙極 Magnetic Matter 磁性物質 Ampere’s law 安培定律

2 這幅紫外線影像顯出太陽大氣內,由温度達開氏百萬度的電離氣體所形成的細緻環狀氣流。
Magnetic force. 磁力 Gas not ionized. 氣體沒離子化。 This ultraviolet image shows delicate loops of million-kelvin ionized gas in the Sun’s atmosphere. 這幅紫外線影像顯出太陽大氣內,由温度達開氏百萬度的電離氣體所形成的細緻環狀氣流。 What force shapes the gas into such intricate structures, and why don’t we see similar things in Earth’s atmosphere? 是甚麼力把氣體塑成這麼精細的結構? 為甚麼地球大氣中沒有類似的東西?

3 26.1. What is Magnetism? 磁性是甚麽?
Magnets exert (magnetic) forces on each other & materials like irons. 磁鐵會對其他磁鐵或鐵性材料施加磁力。 Field description 場的說法 : A magnet produces a magnetic field B. 一個磁鐵產生了一個磁場 B。 Another interacts with B at its position. 另一個磁鐵與在它位置上的 B 作用。 Ampere 安培: Moving charges can produce B. 動的電荷會產生 B。 Magnetic dipoles ~ current loops. 磁雙極 ~ 電流圈。 Iron filings align with the magnetic field, tracing out the field of a bar magnet. 鐵屑依磁場方向排列,描出一條磁棒的場。 Quantum mechanics 量子力學: Intrinsic magnetic moment related to spin. 內建磁距跟自旋有關。

4 26.2. Magnetic Force & Field 磁力和場
Magnetic force on charge q moving with velocity v in field B: 在場 B 中速度為 v 的電荷 q 所受磁力 [B] = N  s / (C  m) = T (Tesla 特斯拉) = 10,000 G (Gauss 高斯) v  B: greatest F BEarth ~ 1 G Bsmall magnet ~ 100 G BMRI ~ 1 T 磁振造影儀 Bmagnetar ~ 1011 T 磁星 v // B: F = 0

5 GOT IT? 26.1 Figure shows a proton in a magnetic field. 圖示磁場中一質子。
For which of the three proton velocities shown will the magnetic force be greatest? 圖中質子的三種速度,何者磁力最大? What will be the direction of the force in all three cases? 這三個力的方向為何? & into paper 進入紙張

6 Example 26.1. Steering Protons 引導質子
Figure shows 3 protons entering a 0.10-T magnetic field. 圖示三質子進入一 0.10 T 磁場。 All three are moving at 2.0 Mm/s. 三個的速率都是 2.0 Mm/s 。 Find the magnetic force on each. 求每個受到的磁力。 B 1 3 2 FB + FE = 0 when v = E/B Electromagnetic force 電磁力 velocity selector. 速度選擇器

7 26.3. Charged Particles in Magnetic Fields 磁場中的帶電粒子
 WB = 0 For v  B, FB = q v B. Circular motion 圓周運動 CW about B for q > 0. 正電荷繞 B 順時針轉 v = const

8 GOT IT? 26.2 A uniform magnetic field points out of this page.
一個均勻磁場從這張紙冒出。 Will an electron that’s moving in the plane of the page circle 從紙的上方看時,一個在紙上走動的電子會朝 clockwise or (a) 順時針或 counterclockwise (b) 逆時針走? as viewed from above the page?

9 Example 26.2. Mass Spectrometer 質譜儀
A mass spectrometer separates ions according to their ratio of charge to mass. 質譜儀按離子的電荷對質量比來把它們分開。 Such devices are widely used to analyze unknown mixtures, 這種儀器廣泛用於分析成分不詳的混合物, and to separate isotopes of chemical elements. 也用來分離化學元素的同位素。 Figure below shows ions of charge q & mass m being 圖示一電荷為 q ,質量為 m 的離子 first accelerated from rest through a potential difference V, 先從靜止由電位差 V 加速 then entering a region of uniform B pointing out of the page. 再進入一從紙面冒出的均勻 B 。 Find an expression for the horizontal distance x. 求水平位移 x。 偵測器 離子源

10 The Cyclotron Frequency 迴磁頻率
Period of particle in circular orbit in uniform B: 粒子在均勻 B 中作圓周運動時的週期: Cyclotron frequency 迴磁頻率 Motion // B not affected by it // B 的運動不受影响 Charged particles frozen to B field lines. 帶電粒子凍結在 B 的場線上。 Trajectory in 3-D 3-D 軌跡

11 Application: The Cyclotron 應用: 迴旋加速器
Whole device in vacuum chamber. 整個儀器都在真空中。 Small V across the dee’s, which alternates polarities at the cyclotron frequency. D型片之間有小 V ,其極性以迴磁頻交替。 Particle injected at center of gap & spirals out. 粒子從縫隙中心注入,然後向外縲旋而出。 E ~ MeVs. D型片 往靶 Applications 應用: Manufacture of radioactive isotopes. 制造放射性同位素。 e.g., PET (Positron Emission Tomography). 例:正電子斷層造影 (PET) 。 磁鐵 接交流電壓 Higher energies 更高能量: Relativity effects  Synchrotron (B also varies) 相對論動應  同步輻射器 ( B 也變 )

12 26.4. The Magnetic Force on a Current 電流受到的磁力
Force on carrier in wire: 電線內載子所受力: Force on straight wire of cross section A & length L: 截面為 A ,長為 L 的畢直電線所受力: F out of paper F 從紙出來 e moving left deflected upward … 往左走的 e 朝上偏移… Prob 58 Fmag on + charge is also upward 正電荷受到的 Fmag 也是朝上 … resulting charge separation leads to upward force on whole wire. … 做成的電荷分離使得整條電線受到朝上的力。

13 GOT IT? 26.3 Figure shows a flexible wire passing through a magnetic field that points out of page. 圖示一條彈性電線穿過一個從紙面冒出的磁場。 The wire is deflected upward, as shown. 電線被推往上方 (見圖) 。 Is the current flowing 電流是往 to the left or (a) 左,或 to the right? (b) 右 ?

14 Conceptual Example 26.1. Power Line 輸電線
A power line runs along Earth’s equator, where B points horizontally from south to north; 一條輸電線沿地球赤道走,該處的 B 在水平面上由南指向北 ; the line carries current flowing from west to east. 線上有電流,由西向東流。 What’s the direction of the magnetic force on the power line? 輸電線所受磁力的方向為何 ? F upward往上 L

15 Making the Connection 連起來
Earth’s equatorial field strength is 30 T, and the power line carries 500 A. 地球赤道處的場約 30 T,且輸電線上流着 500 A. What’s the magnetic force on a kilometer of the line. 一公里長的線上所受的磁力為何? upward 朝上 c.f. weight of wire is ~ 10 kN 可與線的重量 (~ 10 kN) 比較

16 The Hall Effect 霍爾効應 e moves to left & deflected upward e 往左,被推往上 Direction of FB depends on I, not on sign of charged carriers. FB 的方向由 I 而定,與電載子的正負號無關。 Carriers of both signs are deflected upwards 兩種符號的載子都往上偏 EH is upward; EH 朝上 Direction of E due to built-up charges depends on signs of charged carriers: Hall effect. 堆積電荷的 E 的方向由電載子的正負號决定:霍爾効應 EH is downward; EH 朝下 y Steady state 穩態 , Fz = 0 : z x Hall potential: 霍爾電位: p moves to right & deflected upward p 往右,被推往上 Hall coefficient: 霍爾常數:

17 26.5. Origin of the Magnetic Field 磁場的來源
Biot-Savart Law: 畢歐沙瓦定律: permeability constant 導磁率常數 Curl of finger gives B. 手指捲的就是 B 的方向 C.f.比較 Thumb // I 大拇指 // I

18 Example 26.4. Current Loop 電流圈
Find the magnetic field at an arbitrary point P on the axis of a circular loop of radius a carrying current I. 求半徑為 a ,電流為 I 的圓形電流圈軸上任一點 P 的場。 dL  r cos = a / r By symmetry, only Bx  0. 由對稱性可知,衹有 Bx  0 。

19 Example 26.5. Straight Line 直線
Find the magnetic field produced by an infinitely long straight wire carrying current I. 一無限長直線,上帶電流 I ,求其所生磁場。

20 The Magnetic Force Between Conductors 導體間磁力
Field of I1 at I2 : I1 所生在 I2 的場 Force on length L of I2 : 在 I2 上 L 長一段所受的力 points toward I1 指向 I1 Force per unit length on I2 : I2 上單位長度所受的力: Hum from electric equipments are vibrations of transformers in response to AC. 電氣設備的嗡嗡聲就是變壓器因交流電而產生的振動。 Definition: 1 A is the current in two long, parallel wire 1 m apart & exerting 2107 N per meter of length. 定義:兩條長而平行的導線,相距 1m ,若受力各為每單位長度 2107 N ,其上的電流就是 1A。 1C is the charge passing in 1s through a wire carrying 1A. 1C 就是 1A 電流在 1s 內通過一條導線的電荷。

21 GOT IT? 26.4 A flexible wire is wound into a flat spiral as shown in figure. 一條輭線如圖扭成一個平的螺旋線。 If a current flows in the direction shown, will the coil 如是通上圖示方向的電流,螺旋會 tighten or 收緊或 become looser? 放鬆? Does your answer depend on the current direction? 你的答案跟電流方向有沒有關係? No

22 26.6. Magnetic Dipoles 磁雙極 Field on axis of current loop of radius a : 半徑為 a 的電流圈,其軸上的場為: C.f. electric dipole: 對比:電雙極 Setting Far away, fields look similar … 遠處,兩個場很像 … … but close in, they’re different. … 但靠近時,它們就不一樣。 magnetic dipole 磁雙極 N-turn current loop: N-轉電流圈:

23 Detailed calculations show 詳細計算可得:
 = I A valid for arbitrary loop  = I A 適用於任何迴圈。 Vector behavior of  similar to that of p for r >> a. r >> a 時, 的向量行為與 p 的類似 Multi-turn loops = electromagnets 多轉電流圈 = 電磁鐵 Very strong field requires superconducting wires, e.g., MRI scanners. 極強的場需用超導電線,如 MRI 掃瞄器。 Orbiting e in atom  . 原子中軌道上的 e  

24 Application: Magnetic Fields of Earth & Sun 應用 : 地球與太陽的磁場
Currents flows in Earth liquid-iron outer core (due to rotation) 地球的液態鐵外地殼內流着電流 (因自轉) dipole field with  = 8.0  1022 Am2 ( direction not exactly S-N ) 双極場  = 8.0  1022 Am2 (方向與南-北軸稍異) Field deflects harmful high E solar particles. 場把有害的高能太陽粒子擋開 Magnetic reversal 磁極逆轉 Earth (period ~ millions of yrs) : map for sea-floor spreading. 地球(週期 ~ 數百萬年) :海底擴展圖。 Sun (period ~ 11 yrs.) : ~ sunspots. 太陽(週期 ~ 11 年) :~ 日斑。

25 Dipoles & Monopoles 雙極和單極
Some elementary particle theories suggest existence of magnetic monopoles. 有些基本粒子理論認為磁單極應該存在。 But none was ever observed. 但從來沒有被量到。 Microscopic origin of B: B 的微觀來源: Charged current. 1. 電流。 Intrinsic spin 內建自旋。 No magnetic charges : B lines always form loops, either encircling moving charges, or joining the 2 poles of a magnet. 沒有磁荷:場線永遠都成圈狀, 是繞着運動中的電荷,便是連起磁鐵的兩個極。 Gauss Law 高斯定律

26 GOT IT? 26.5 The figure shows two fields. 圖中兩個場。
Which could be a magnetic field? 那一個有可能是場?

27 Torque on a Magnetic Dipole 磁雙極的力距
Forces on top & bottom cancel. 頂和底的力相互扺消。 Forces on sides also cancel; but give net torque. 兩邊的力也扺消掉;不過有淨力距。 above plane 在紙面之上 beneath plane 在紙面之下 Torque on dipole: 雙極上的力距: Associated energy: 所附能量:

28 Application: Electric Motors 應用:電動馬達
Rotating loop 轉動的迴路 CW順時 Commutator 換向器 CCW 逆時 Brushes Battery 電池 Always CCW 永遠逆時

29 Example 26.6. Hybrid Car Motor 油電混合車馬達
Toyota’s Prius gas-electric hybrid car uses a 50-kW electric motor that develops a maximum torque of 400 Nm. 豐田的 Prius 油電混合車使用的 50-kW 電動馬達,最大扭力可達 400 Nm 。 Suppose you want to produce this same torque in the motor just discussed, with a 950-turn rectangular coil of wire measuring 30 cm by 20 cm, in a uniform field of 50 mT. 假設你想用一個 950-轉,30 cm 乘 20 cm 的長方形線圈,在 50 mT 均勻場中產生上述扭力。 How much current does the motor need? 這馬達需用多少電流? Max torque at sin = 1 : 最大扭力在 sin = 1 : Rectangular coil : 長方形線圈: 950-turn coil 950-轉線圈

30 26.7. Magnetic Matter 磁性物質 Ferromagnetism 鐵磁性:
e.g., Fe, Ni, Co, & their alloys. 如:鐵,鎳,鈷,和它們的合金。 Material divided into magnetic domains in which ’s are aligned. 材料內部會分割成多個磁疇,每個疇內的  都同向。 T > TC : orientations of  in different domains random 不同疇的  方向無規律  <  >  B (paramagnetic 順磁物質) T < TC : orientations of  in different domains aligned 不同疇的  方向相連  <  >  0 even when B = 0 (ferromagnetic 鐵磁物質) Atomic current loops are CCW. 原子電流圈都是反時針的。 Adjacent loops cancel 相隣電流圈相互抵消 No cancellation on boundaries: net I on surface. 邊界上無抵消: 表面有淨 I 。

31  // Bapplied p // Eapplied Field inside dipole > Bapplied
Distant field similar. 遠處的場相似 Internal field opposite. 內部的場反向。 Field inside dipole < Eapplied 雙極內的場 < Eapplied p // Eapplied

32 Paramagnetism 順磁性: Materials with randomly oriented permanent  and very weak - interaction. 材料含方向隨機的永久  且 - 間幾無作用。 <  > =  B with  > 0  = magnetic susceptibility 磁化率 Diamagnetism 抗磁性: Materials with no intrinsic . 材料無內建  。 <  > =  B with  < 0 ( induced 感應  )

33 26.8. Ampere’s law 安培定律 Field around long wire carrying current I :
from Biot-Savart law 由畢奧薩瓦特定律 True for arbitrary closed paths & steady currents: 適用於任何閉路和穩電流: Bdr = 0 Ampere’s law 安培定律 net field from ALL sources 包括所有來源的淨場 c.f. Gauss’ law 對比:高斯定律

34 Example 26.7. Solar Currents 太陽的電流
The long dimension of the rectangular loop in figure is 400 Mm, and B near the loop has a constant magnitude of 2 mT. 圖中的長方形迴路長 400 Mm ,且迴路附近的 B 大小是一常數 2 mT 。 Estimate the total current enclosed by the rectangle. 估計長方形所圍的總電流。 B  dr on the short segment 短的一截 B  dr = 0 there. B // dr on the long segment 長的一截  Direction: into the page 方向:朝紙內 3D: around equator 3D: 繞着赤道

35 GOT IT? 26.6 The figure shows three parallel wires carrying current of the same magnitude I, but in one of them the current is opposite to that of the other two. 圖示三條平行電線,其上電流大小都是 I ,但其中一條的電流方向與其他兩條相反。 If around loop 2, 若廻路 2 的 (a) what is around loop 1, and (a) 廻路 1 的 是多少 A (b) which current is the opposite one? (b) 那個電流是反向的?

36 Using Ampere’s Law 使用安培定律
STRATEGY 26.1 Ampère’s Law : 策略 安培定律: Base on symmetry, choose the amperian loop such that B is either // or  to it. 以對稱性為考量,安培廻路應選成與 B // 或  。

37 Example 26.8. Outside & Inside a Wire 電線內外
A long, straight wire of radius R carries a current I distributed uniformly over its cross section. 一條半徑為 R 的筆直長電線,其電流 I 均勻地分佈在它的截面上。 Find the magnetic field 求 outside and 線外和 inside the wire. 線內磁場。 By symmetry, B is azimuthal. 由對稱性可知, B 在方位角的方向。 Amperian loop is a circle. 安培廻路是個圓。 (a) (b)

38 Example 26.9. Current Sheet 電流片
An infinite flat sheet carries current out of this page. 一片無限大的平片,其電流從紙面流出。 The current is distributed uniformly along the sheet, with current per unit width given by JS . 電流在片上均勻分佈,每單位寬度的電流是 JS 。 Find the magnetic field of this sheet. 求此片的磁場。 By symmetry, B is // to sheet &  I. 由對稱性可知, B 與片 // 而  I 。 Amperian loop is a rectangle. 安培廻路是個長方形。 Far out, B lines nearly circular 遠處,B線近乎圓形 Close in, B lines ~ infinite sheet 近遠處,B線 ~ 無限大平片值

39 Fields of Simple Current Distributions 一些單純的電流分佈的場
For arbitrary distributions 隨便一個分佈: Far away from any loop ~ dipole. 離任何廻路甚遠 ~ 雙極 Very near any wire ~ infinite straight wire. 靠任何線甚近 ~ 無限長直線 Very near any current sheet ~ infinite flat sheet. 靠任何電流面甚近 ~ 無限寬平片

40 Solenoids 線圈(螺線管) Solenoid: a long, tightly wounded coil.
線圈:一束長而旋得很緊的電線。 n turns per unit length. 單位長度 n 轉。 n L loops in L. L 內有 n L 圈。 encircled current = nLI. 圍到的電流 = nLI 。 solenoid field 線圈場 Boutside = 0 Application: magnetic switches, valves, …. 應用: 開關, 閥,…

41 Table 26. 1. Fields of Some Simple Charge & Current Distributions 表 26
Field場(r) Q E J B Dipole 雙極 Dipole 雙極 r 3 Point / spherical 點 / 圓形 NA r 2 r 1 Line Line Const 常數 Sheet Sheet

42 Example 26.10. MRI Scanner 磁共振造影掃描器
The solenoid used in an MRI scanner is 2.4 m long & 95 cm in diameter. 一部磁共振造影掃描器內的螺線管長為 2.4 m ,直徑為 95 cm 。 It’s wounded from superconducting wire 2.0 mm in diameter, 它繞的是直徑為 2.0 mm 的超導線, with adjacent turns separated by an insulating layer of negligible thickness. 相隣的線圈由厚度可以忽略的絕緣層分開。 Find the current that will produce a 1.5-T magnetic field inside the solenoid. 求可在管內產生 1.5-T 磁場的電流。 wire diameter 線直徑 = 1/500 m n = 500 turns/m 轉 / 米

43 Field of a solenoid is very similar to that of a bar magnet.
螺線管的場和磁鐵的十分相似。


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