Physics 24100 Electricity & Optics Lecture 15 Chapter 27 sec. 3-5 Fall 2016 Semester Professor Koltick
Magnetic Fields B = μ 0 4π I dl r r 2 = μ 0 4π I dl r r 3 B = μ 0 2I 4π R B = μ 0 2 IR 2 R 2 + z 2 3/2 B = μ 0 n I
Forces on Current Carrying Wires Two wires carrying currents I 1 and I 2 will exert forces on each other: Magnetic field from I 1 is B = μ 0 4π Force on I 2 is df 12 = I 2 dl 2 B Conversely Magnetic field from I 2 is B = μ 0 4π Force on I 1 is df 21 = I 1 dl 1 B I 1 dl 1 r r 2 I 2 dl 2 r r 2
Forces on Parallel Wires B = μ 0 4π 2I 1 R df 12 = I 2 dl 2 B Force per unit length: df 12 dl 2 = μ 0 4π 2I 1 I 2 R
Force between Two Parallel Current Carrying Wires
Force on Parallel Wires I 1 I 2 I 1 I 2 attraction repulsion Will this loop expand or contract?
Magnetic Pressure Normal dipole magnets B ~ 1 T Superconducting dipole magnets B ~ 5 T Magnets need to withstand about 5 tons of internal forces without distortion.
Remember Gauss s Law? Electric field: E = 1 4πε 0 r r 2 dq Gauss s Law: n E da = Q inside S ε 0 If E is constant over the surface then we can bring it outside the integral The integral is just the surface area This works only when there is sufficient symmetry
Remember Gauss Law for Electric Fields E da Enet Q enclosed o 3/1/2016 12
Gauss Law for Magnetism B da Bnet 0 Since all lines of B are closed loops, any B line leaving a closed surface MUST reenter it somewhere. TRUE IN GENERAL, not just for this dipole example 3/1/2016 13
Gauss s Law Applied Magnetism In magnetism we can have dipoles or currents but no magnetic monopoles Gauss s law: S n B da = Q inside ε 0 = 0 One of Maxwell s Equations: B = 0 Always zero!
Ampere s Law But can we do something similar to calculate the magnetic field in cases with lots of symmetry? Yes: B dl = μ 0 I C I C C C
I C C Example What is the magnetic field around a long, straight wire? From symmetry, we expect that the magnetic field is always azimuthal: B = Bφ The path length element is also azimuthal: dl = dlφ r C B dl 2πBr = μ 0 I C = B dl C = μ 0 I C B = μ 0I C 2πr
Magnetic Field Inside a Long Straight Wire I C = I r2 R 2 B(r) = μ 0I C 2πr B = μ 0Ir 2πR 2 B = μ 0I 2πr Ratio of areas inside the wire (Inside) (Outside)
Magnetic Field Inside a Solenoid Symmetry principles: The magnetic field always points along the axis of the solenoid: B = Bk It is independent of z, except at the ends. Outside the solenoid, we expect B 0 as r Inside the solenoid, does B depend on r? z
Magnetic Field Inside a Solenoid C B dl b = B dl a c + B dl b d + B dl c a + B dl d = μ 0 I C These will all be zero!
Magnetic Field Inside a Solenoid n is the number of turns per unit length. Enclosed current: I C = n I h Make the path cd very far away, where B 0. C B dl B = μ 0 n I = B h = μ 0 I C Independent of r inside the solenoid.
Magnetic Field Inside a Toroid
Lecture 15-13 Toroids We assume an Amperian Loop in form of a circle with radius r such that r 1 < r < r 2 The magnetic field is always directed tangential to the Amperian Loop, so we can write B ds 2 rb The enclosed current is the number of turns N in the toroid times the current i in each loop, so Ampere s Law becomes: 2 rb 0 Ni Bds or B 0enc i 0Ni Note the r dependence! 2 r Direction: given by right-hand rule 13
When Ampere s Law doesn t Help B can t be factored out of the integral. insufficient symmetry finite length current segment is (unphysical) current is not continuous (time dependent)
Magnetic Properties of Materials Atoms in many materials act like magnetic dipoles. Magnetization is the net dipole moment per unit volume: M = dμ dv In the presence of an external magnetic field, these dipoles can start to line up with the field: Net current inside the material is zero. We are left with a surface current and therefore a magnetic moment
Magnetization and Bound Current Magnetic dipole for a current loop: μ = A I n Magnetic moment per unit length: dμ = A di Magnetization: M = dμ dv = dμ A dl = di dl This is the surface current per unit length. Magnetic field due to the surface current is the same as in a solenoid: B = μ 0 n I = μ 0 M current per unit length
Magnetization and Magnetic Susceptibility How well do the microscopic magnetic dipoles align with an external applied magnetic field? Simplest model: linear dependence on B app Magnetization: M B app Magnetic field due to surface current: B m = μ 0 M χ m B app Magnetic susceptibility: χ m Total magnetic field: B = B app + B m = (1 + χ m )B app K m B app Relative permeability: K m
Magnetic Susceptibility Different materials react differently to external magnetic fields: χ m > 0 small χ m Paramagnetism aluminum, tungsten χ m < 0 small χ m Diamagnetism bismuth, copper, silver χ m > 0 large χ m Ferromagnetism iron, cobalt, nickel Dipoles in paramagnetic materials align with B app Dipoles in diamagnetic materials align opposite B app Ferromagnetic materials align strongly even in weak B app
Magnetic Susceptibility Material χ m Type Bi 1.66 x 10 5 diamagnetic Ag 2.6 x 10 5 diamagnetic Al 2.3 x 10 5 paramagnetic Fe (annealed) 5,500 ferromagnetic Permalloy 25,000 ferromagnetic mu-metal 100,000 ferromagnetic superconductor 1 diamagnetic (perfect)
Atoms as Magnets (orbital momentum, classical description) The atoms that make up all matter contain moving electrons that form current loops that produce magnetic fields In most materials, these current loops are randomly oriented and produce no net magnetic field In magnetic materials, some of these current loops are naturally aligned Other materials can have these current loops aligned by an external magnetic field and become magnetized Simple model of an atom: Consider an electron moving at a constant speed v in a circular orbit with radius r as illustrated to the right March 2, 2015 49
Atoms as Magnets (2) We can think of the moving charge of the electron as a current i Current is defined as the charge per unit time passing a particular point For this case the charge is the charge of the electron e and the time is related to the period of the orbit i = e T = e ( 2πr)/ v = ve 2πr March 2, 2015 50
Atoms as Magnets (3) The magnetic moment of the orbiting electron is given by! ve 2 ver µ orb = ia = π r = 2π r 2! We can define the orbital angular momentum of the electron to be! L = rp = rmv orb where m is the mass of the electron Solving and substituting gives us! 2 orb 2m Lorb rm µ " = # $ = µ % er & e orb March 2, 2015 51
Atoms as Magnets (4) Rewriting and remembering that the magnetic dipole moment and the angular momentum are vector quantities we can write! e! µ orb = L orb! 2m The negative sign arises because of the definition of current as the flow of positive charge This result can be applied to the hydrogen atom, and the correct result is obtained However, other predictions of the properties of atoms based on the idea that electrons exist in circular orbits in atoms disagree with experimental observations March 2, 2015 52
Lecture 15-20 Diamagnetism Zero net magnetic moment, but As B is turned on, this pair of electrons develop magnetic moments opposite to B. Superconductor B (1 ) B 0 m m 1 app Meissner effect
Diamagnetism (2) An example of a strawberry exhibiting diamagnetism can be seen on the site for the High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands at: http://www.ru.nl/hfml/research/levitation/diamagnetic/ In this movie, diamagnetic forces induced by a non-uniform external magnetic field of 16 T are levitating the strawberry. The normally negligible diamagnetic force is large enough in this case to overcome gravity. Levitating a live Flying Dutchfrog An example of a frog exhibiting diamagnetism can also be seen on the site for the High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands at: http://www.ru.nl/hfml/research/levitation/diamagnetic/! March 2, 2015 56
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Lecture 15-21 Paramagnetism Materials containing certain transition elements, actinides, and rare earths exhibit paramagnetism Each atom of these elements has a permanent magnetic dipole, but these dipole moments are randomly oriented and produce no net magnetic field In the presence of an external magnetic field, some of these magnetic dipole moments align with the external field in the same direction as the external field Resulting in an attractive force When the external field is removed, the induced magnetic dipole moment disappears m 0 21
Ferromagnetism (1) The elements iron, nickel, cobalt, gadolinium and dysprosium and alloys containing these elements exhibit ferromagnetism Ferromagnetic materials: Long-range ordering at the atomic level Dipole moments of atoms are lined up with each other Typically in a limited region called a domain Within a domain, the magnetic field can be strong Domains are small and randomly oriented leaving no net magnetic field An external magnetic field can align these domains and magnetize the material Tilden mine, UP Iron Ore March 2, 2015 53
Electron spin electron: small magnet (spin). Quantum effect!!! Spin of electron: either up or down: s=+1/2, s= -1/2 Two electrons: total spin =0 or 1. (total spin 1 can have three projections: -1,0,1). Spin =0 is a bit preferred energetically (like two magnets) Even # of electrons: typically spin =0 in an atom There is a spin of the nucleus, and orbital momentum of elections The electron spin is the main contribution to ferromagnetism March 2, 2015 54
Lecture 15-22 Ferromagnetism A ferromagnetic material will retain all or some of this induced magnetism when the external magnetic field is removed The induced magnetic field is in the same direction as the external magnetic field Amplification of magnetic fields 22
Magnetic hysteresis B ext Start with domains aligned Impose B-field against Spins want align with each other and with external B-field Stronger external B-field - stronger alignment March 2, 2015 54
HYSTERESIS FOR A FERROMAGNET Lack of retraceability shown is called hysteresis. Memory in magnetic disk and tape Alignment of magnetic domains retained in rock (cf. lodestones) Area enclosed in hysteresis loop Energy loss per unit volume hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, higher memory) soft magnet: narrow hysteresis loop (easy to demagnetize, )