8.022 (E&M) – Lecture 22Topics: Magnetic properties of materials Magnetic dipole of electrons macroscopic behavior of matter Properties of Diamagnetic, Paramagnetic and Ferromagnetic materials 1 Final Exam When and where? Tue Dec. 14, 9:00 – 11:00 AM Please arrive 10 min early: no extra time given if you are late! Format of the exam Similar to quiz 1 and 2: 4 problems, 2 hours Same difficulty, more time: you must do a better job! Topics 1 problem on Quiz 1 material (Electrostatics,…) 1 problem on Quiz 2 material (Currents, Relativity, Induction,…) 2 problems on post-Quiz 2 material (RCL, AC circuits, waves,…) G. Sciolla – MIT 8.022 – Lecture 22 2 2 Final: FAQ When will the final be graded? Immediately after the exam: by 6 PM Tue we expect to be done What is the passing grade? Freshmen: C Everybody else: D What is the passing score? We have not decided yet. It depends on how hard the final will be Be assured you will be graded fairly and consistently. G. Sciolla – MIT 8.022 – Lecture 22 3 3 How to prepare? Read and understand all lecture notes Fast and (hopefully) easy to digest. (Almost) All you need to know Go back to section notes Solve problems done in class by yourselves and check answers Go back to homework problems Solve them again and compare answers Solve old exam problems (posted on Tue) Useful to understand how fast you can solve the problems Attend review session on Sat And office hours if you have last minute questions Read Purcell If you have time left. You should have done this already… G. Sciolla – MIT 8.022 – Lecture 22 4 4 How to simplify your life (and ours): A few tips to a high score Remember: 35 points assigned to final Quiz 1: 20 points, Quiz 2: 20 points, Make up: 7 points Sleep at least 6 hours the night before Being able to THINK is your most important asset! Not sure how to interpret a question/figure? ASK!!! That’s why we are in the exam room! Read all the problems and start working on what you know best Don’t spend 80% of your time on the one problem you cannot solve: 3 perfect problems will give you 75 points Partial credit: if you are unable to solve part a) see if you are able to solve b) Make sure you answer ALL the questions: When you are done, go back to the text and make sure answers are complete (vector direction, etc) G. Sciolla – MIT 8.022 – Lecture 22 5 5 Back to physics… Last time: end of 8.022 official program Energy and momentum carried by EM waves Poynting vector and some of its applications Transmission lines Scattering of light through matter Today: beyond scope of 8.022, just enjoy! Magnetic properties of materials Where do they come from? G. Sciolla – MIT 8.022 – Lecture 22 6 6 ll7 Magnetic properties of materials ☺ l liproperties of electrons and atoms + -G. Scioa – MIT 8.022 – Lecture 22 We went through the whole E&M course without even understanding how a magnet bar works? Yes, so far. Let’s try to make up for this… Very qualitative description: as far as we can go without quantum since In the discussion I assume we are alfamiar with some basic Nucleus at the center, electrons rotating on orbits Electron is negatively charged Electron has an intrinsic angular momentum (spin) Magnetic properties of materials are totally determined by quantum mechanical nature of their molecular structure 78 llEffects due to electron orbits It’s usuall cancellation ld created -vI B Bext BG. Scioa – MIT 8.022 – Lecture 22 Electrons in atoms produce magnetic field Electrons rotate around the nucleus in orbits This is same as having a loop of current Currents produce magnetic fields (Ampere) y a small effect… There are lots of electrons, orbits are randomly oriented: What happens when we put the material in an external B? Lentz’s law: the orbits rearrange so that the magnetic fieby the orbits opposes the external magnetic field Net effect: the total magnetic field will be weaker Lentz 8Magnetic moments of electrons Current due to electron in orbit of radius r: I = ev 2πr The magnetic moment µ of the loop is 2IA πr I evr µ≡= = cc 2c The magnetic moment µ is related to the angular momentum L: −eL rL=× p⇒ µ= 2mce In addition to the standard angular momentum L electrons have intrinsic angular momentum (spin) intrinsic magnetic moment Will this contribute to macroscopic magnetic properties of material? G. Sciolla – MIT 8.022 – Lecture 22 9 9 Effects due to electron’s spin The intrinsic magnetic moment behaves very differently from the standard magnetic moment No Lentz’s law type behavior because this field is associated with the electron itself What happens when we put the material in an external B? A magnetic moment µ placed in an external filed B feels a torque See Purcell 6.22 τ= µ× B τ tends to line up the electron magnetic moments with external field Net effect: the total magnetic field will be stronger Bext Bspin G. Sciolla – MIT 8.022 – Lecture 22 10 10 What effect is stronger? Summary of the situation so far: Lentz’s law on the orbit of the electrons opposes B fields from entering material Magnetic torque acting on individual electrons augments the B field in the material Opposite behaviors! Who wins? It depends on the properties of the material (chemical structure, how free electrons are, etc) 3 categories: Diamagnetic materials Paramagnetic materials Ferromagnetic materials G. Sciolla – MIT 8.022 – Lecture 22 11 11 Diamagnetic materials Diamagnetic materials defined as materials in which the magnetization opposes the external magnetic field When material is immersed in external B field, magnetic field inside the material is weaker than external B Lentz’s law wins out on effect of spin Diamagnetism is usually very weak and hard to see Lentz’s law plays a role in all materials. Spin effect (if present) are stronger if preset it usually covers completely diamagnetic behavior Examples of diamagnetic materials Typically orbits filled with paired electron orbit has no net magnetic moment Most substances: H20, Cu, NaCl, etc Consequence: diamagnetic substances will be expelled from B field G. Sciolla – MIT 8.022 – Lecture 22 12 12 Paramagnetic materials Paramagnetic materials are defined as materials in which the magnetization augments the external magnetic field When material is immersed in B field, magnetic field inside the material is stronger than outside Effect of Spin wins out on Lentz’s law Examples of diamagnetic materials Typically have several electron orbits that contain unpaired electrons orbit has a net magnetic moment Exception: Oxygen O2 is paramagnetic. To see this property need to cool it to a liquid state, or random motion will wipe out effect Example: Na, Al, NiSO4, etc Consequence: Paramagnetic materials are pulled into magnetic fields If paramagnetic behavior is “extra strong”: Ferromagnetic material G. Sciolla – MIT 8.022 – Lecture 22 13 13 ll14 Diamagnetic and paramagnetic materials (J4 and J6) What if we put Al? Al is paramagnetic orients //to B 2 2O? 2 liTube containing substances to test G. Scioa – MIT 8.022 – Lecture 22 What happen when we put bismuth in the tube? Remember: Bismuth is diamagnetic! What happen if we pour liquid Obetween magnets? What if we pour HRemember: Oquid is paramagnetic, water is diamagnetic Magnet Magnet 14Magnetization and H field Magnetization M is defined as the magnetic dipole moment of a substance per unit volume Magnetic moment of a material with volume V and magnetization M µ=MV Dimension analysis ⎡⎤ magnetic moment = current× area/velocity current /velocity BM= = =⎡⎤⎣⎦ volume volume length ⎣⎦ M has same dimensions as magnetic field B Define a new kind of magnetic field H B =H+4πM B is the total magnetic field; H is the “normal field” due to currents, M is the magnetization, component of B due to material’s properties In vacuum, B=H G. Sciolla – MIT 8.022 – Lecture 22 15 15 H and Maxwell’s equations Let’s quickly look at something that you will study in 8.07 The curl of H defines the free electrical currents 4π ∇×H = Jfree with Jfree= density of free electrical currentc The curl of the magnetization defines the bound currents 1 ∇×M = Jbound with Jbound= density of bound electrical current c Plug into Ampere’s law 4π ∇×B =∇× H+4π∇× M = 4π(Jfree +Jbound )= J c c ⇒ J =Jfree +Jbound Total current density is due to sum of current that we can control Jfree and current due to the material Jbound G. Sciolla – MIT 8.022 – Lecture 22 16 16 Magnetic susceptibility Many substances exhibit linear magnetization, e.g. the magnetization depends linearly on the external field applied H =χ Mm Where χ m= magnetic susceptibility Since B = H+ 4π Mit follows that B = H(1 + 4πχ m) Classification of material based on magnetic susceptibility χ m<0: magnetic field B decreases when we immerse the substance in an external magnetic field B: diamagnetism χ m>0: magnetic field B increases when we immerse the substance in an external magnetic field B: paramagnetism G. Sciolla – MIT 8.022 – Lecture 22 17 17 Magnetic properties of materialsClassification of materials based on magnetic susceptibility Paramagnetic: χm>0 Diamagnetic: χm<0 Material χm(10-5) Uranium 40 Aluminum 2.2 Oxygen gas 0.2 H20 -0.9 Lead (Pb) -1.8 Carbon (diamond) -2.1 Bismuth -16.6 G. Sciolla – MIT 8.022 – Lecture 22 18 18 Ferromagnetism “Ferromagnetism is paramagnetism on steroids” Prof. S. Hughes – 8.022 S-2004 Nonlinearity distinguishes it from paramagnetism M and H do not have a simple linear relation Magnetization remains after external field is turned off This is how permanent magnets work! Why nonlinear behavior? Way beyond scope of 8.022 But it’s easy enough to describe qualitatively how it works… G. Sciolla – MIT 8.022 – Lecture 22 19 19 G. Sciolla – MIT 8.022 – Lecture 22 20 Ferromagnetic domains l l i in small regi lli Curiosity: you cannot see can hear them: Ferromagnetic materiabefore B is applied Ferromagnetic materiaafter B is applied Ferromagnetism is conceptually simlar to paramagnetism Difference: magnetic moments of many atoms are tend to be aligned ons (domains) Paramagnetic materials: moments are randomly arranged until external B aligns them Since domains are small (0.1 mm – few mm) and randomly oriented: overall M=0 When material is put into externa B, domains re-agn //to B When external B is removed they stay aligned: permanent magnets! domains flipping but you Barkhausen effect (J3) Sizes of domains range from a 0.1 mm to a few mm. When an external magnetic field is applied, the domains already aligned in the direction of this field grow at the expense of their neighbors. If all the spins were aligned in a piece of iron, the field would be about 2.1 Tesla. A magnetic field of about 1 T can be produced in annealed iron with an external field of about 0.0002 T, a multiplication of the external field by a factor of 5000! Barkhausen effect: Domains are well modeled by the compass table, an array of about one hundred small compass needles used for showing fields of bar magnets, etc. When there is no strong external Bfield, sections of the array line up in different directions, each individual compass needle aligning itself with the local field. When the array is tapped sharply, it will be seen that the needles on the boundaries of the domains are the least stable (vibrate the most), and some of them realign causing one domain to grow at the expense of another. In the Barkhausen effect, a large coil of fine wire is connected through an amplifier to a speaker. When an iron rod is placed within the coil and stroked with a magnet, an audible roaring sound will be produced from the sudden realignments of the magnetic domains within the rod. A copper rod, on the other hand, produces no effect. 20 ll21 Magnetization of iron (J2) Initially off Fe pll Domains line up i Domains still lined up late will When it falls: domains break Fe plate will fall Iron plate 5 Kg G. Scioa – MIT 8.022 – Lecture 22 Electromagnet is hanging from a support structure ate fals off Turn on electromagnet Fe plate will stck Turn off electromagnet Fe pstick because it’s now magnetized Add up to 5 Kg or so! Electromagnet with switch 21ll22 Magnetization and demagnetization of iron rod (J7) Iron rod ~ Solenoid G. Scioa – MIT 8.022 – Lecture 22 Use DC to magnetize iron rod Current in solenoid creates B; Fe rod’s domains align with B rod becomes magnetic: it attracts paper clips etc Demagnetize Fe rod with AC current Run it slower and slower, flipping the direction of the domains slower and slower 2223 llNonlinearity and hysteresis ferromagnetic material 00 led . B G. Scioa – MIT 8.022 – Lecture 22 In ferromagnetic materials B and H have a nonlinear dependence Let’s find out experimentally what that is Apply external field H (x axis) and measure total field B (y axis) in the Start with value of H (H), decrease to 0, flip the direction and reach –HThe curve describing relationship between H and B is calhysteresis curve When H=0, B.ne.0 What value will it take? +H? –H? It depends on the magnetization history 23Histeresis curve of Fe core transformer (J11)B Measure hysteresis curve using the Fe core of a transformer Send to an oscilloscope: Channel 1 (x): I into primary winding of the transformer (H) Channel 2 (y): I from secondary winding (B) G. Sciolla – MIT 8.022 – Lecture 22 24 24 ll25 Curie Temperature Why does it happen? At T>TC i For Fe: TCo C C ill lies. G. Scioa – MIT 8.022 – Lecture 22 Curie temperature is the temperature above which ferromagnetic materials stop acting as such NB: transition is very sudden! the random motion of the magnetic moments becomes so strong that they cannot algn anymore to form domains =770Demo J10 Iron nut sticks to permanent magnet Heat up the nut until it reaches TNut wtemporariy loose its ferromagnetic propert 25. ll26 Summary Today: ☺ Final Exam: If you enjoyed 8.022, you will love Physics is cool! G. Scioa – MIT 8.022 – Lecture 22 Magnetic properties of materials Diamagnetism, paramagnetism, ferromagnetism Qualitative description Quantum will answer deeper questions: SOON!!! On Tue Dec. 14, 9:00 – 11:00 AM Please arrive 10 min early! Quantum Mechanics! 26Merry Christmas… the 8.022 way!⎧∇iE = 4πρ ⎪ ⎪∇iB= 0 ⎪ ⎨∇× E⎪ ⎪ ⎪∇× B ⎩ 1 ∂B= − ∂ct 4π 1 = J + cc ∂E ∂t Merry Christmas! G. Sciolla – MIT 8.022 – Lecture 22 27 27