General Physics 32 Maxwells Equations Models of Magnetism

普通物理 General Physics 32 - Maxwell's Equations, Models of Magnetism 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc. ncue. edu. tw 網頁: http: //ykuo. ncue. edu. tw 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授

Outline n n n 32 -1 What Is Physics? 32 -2 Gauss’ Law for Magnetic Fields 32 -3 Induced Magnetic Fields 32 -4 Displacement Current 32 -5 Maxwell’s Equations 32 -6 Magnets 32 -7 Magnetism and Electrons 32 -8 Magnetic Materials 32 -9 Diamagnetism 32 -10 Paramagnetism 32 -11 Ferromagnetism 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 2

32 -1 What Is Physics? n This chapter reveals some of the breadth of physics because it ranges from the basic science of electric and magnetic fields to the applied science and engineering of magnetic materials. n We conclude our basic discussion of electric and magnetic fields, finding that most of the physics principles can be summarized in only four equations, known as Maxwell’s equations. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 3

32 -2 Gauss’ Law for Magnetic Fields n The simplest magnetic structure that can exist is a magnetic dipole. n Magnetic monopoles do not exists as far as we know. n Instead new pole faces appear on the newly cut faces of the pieces and the net result is that we end up with three smaller magnets, each of which is a magnetic dipole. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 4

32 -2 Gauss’ Law for Magnetic Fields Magnetic Flux For each element we calculate the magnetic flux through it: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 5

32 -2 Gauss’ Law for Magnetic Fields SI magnetic flux unit: T‧m² known as the “Weber” (Wb) For any closed surface: ФB=0. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 6

32 -3 Induced Magnetic Fields Maxwell’s law of induction : 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 7

Example 32 -1 n. A parallel-plate capacitor with circular plates of radius R is being charged. (a) Derive an expression for the magnetic field at radius r for the case r ≤ R. (b) Evaluate the field magnitude B for r = R/5 = 11. 0 mm and d. E/dt = 1. 50 × 1012 V/m · s. (c) Derive an expression for the induced magnetic field for the case r ≥ R. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 8

32 -4 Displacement Current B O 2021/10/26 R r 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 14

Example 32 -2 n The circular parallel-plate capacitor is being charged with a current i. (a) Between the plates, what is the magnitude of , in terms of μ 0 and i, at a radius r = R/5 from their center? (b) In terms of the maximum induced magnetic field, what is the magnitude of the magnetic field induced at r = R/5, inside the capacitor? 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 16

32 -5 Maxwell’s Equations Gauss’s law for E: Gauss’s law for B: Faraday’s law: Ampere’s law: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 18

32 -6 Magnets The magnetism of earth: The dipole axis does not coincide exactly with the rotation axis but the two axed form an angle of 11. 5˚. The direction of the earth’s magnetic field at any location is described by two angles: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 19

32 -6 Magnets Geographic North N Compass needle S Fig. a : Top view S Compass needle horizontal Fig. b : Side view N Field declination θ is defined as the angle between the geographic north and the horizontal component of the earth’s magnetic field. Field inclination ψ is defined as the angle between the horizontal and the earth’s magnetic field. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 20

32 -7 Magnetism and Electrons Orbital magnetic dipole moment. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 21

32 -7 Magnetism and Electrons Spin magnetic dipole moment: In addition to the orbital angular momentum, an electron has what is known as “intrinsic” or “spin” angular momentum. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 22

32 -7 Magnetism and Electrons Spin quantization: The quantized values of the spin angular momentum are: The constant “Planck’s constant”. is known as It is the yardstick by which we can tell whether a system is described by classical or by quantum mechanics. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 23

32 -7 Magnetism and Electrons The term ms can take the values +1/2 or -1/2. The constant is known as the electron “Bohr magneton” (symbol μB). 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 25

32 -8 Magnetic Materials can be classified on the basis of their magnetic properties into three categories: Diamagnetic, paramagnetic, and ferromagnetic. SI unit for M: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 26

32 -9 Diamagnetism e C . C When the magnetic field the magnetic moments for each orbit are antiparallel and thus the net magnetic moment. . . 2021/10/26 Two electrons move on identical orbits radius r with angular speed ω0 e 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 27

32 -9 Diamagnetism The negative sign indicates that 2021/10/26 are antiparallel. 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 28

32 -9 Diamagnetism e C . C Top electron: . . 2021/10/26 e 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 29

32 -9 Diamagnetism Bottom electron: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 30

32 -10 Paramagnetism When the ratio B/T is below 0. 5 the magnetization M of a paramagnetic material follows Curie’s law: 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 31

32 -10 Paramagnetism The atoms of paramagnetic materials have a net magnetic dipole moment in the absence of an external magnetic field. This moment is the vector sum of the electron magnetic moments. n For very high magnetic fields and/or low temperatures, all magnetic moments are parallel to B field and the magnetization. n 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 32

Example 32 -3 n. A paramagnetic gas at room temperature (T = 300 K) is placed in an external uniform magnetic field of magnitude B = 1. 5 T; the atoms of the gas have magnetic dipole moment μ = 1. 0μB. Calculate the mean translational kinetic energy K of an atom of the gas and the energy difference ∆UB between parallel alignment and antiparallel alignment of the atom’s magnetic dipole moment with the external field. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 33

32 -11 Ferromagnetism Magnetic domains 2021/10/26 Hysteresis 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 35

32 -11 Ferromagnetism is exhibited by Iron, Nickel, Cobalt, Gadolinium, Dysprosium and their alloys. 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 36

Example 32 -4 (a) n. A compass needle made of pure iron (density 7900 kg/m 3) has a length L of 3. 0 cm, a width of 1. 0 mm, and a thickness of 0. 50 mm. The magnitude of the magnetic dipole moment of an iron atom is μFe = 2. 1 × 1023 J/T. If the magnetization of the needle is equivalent to the alignment of 10% of the atoms in the needle, what is the magnitude of the needle’s magnetic dipole moment ? 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 37

Example 32 -3 (b) n If the compass needle is jarred slightly from its (horizontal) north–south equilibrium position, it oscillates about that position. If the period of oscillation is 2. 2 s, what is the horizontal component of the local magnetic field? 2021/10/26 普通物理講義-32/國立彰化師範大學物理系/郭艷光教授 38