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by: Norberto Miller


Norberto Miller
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This 6 page Class Notes was uploaded by Norberto Miller on Thursday October 22, 2015. The Class Notes belongs to GEOL 100 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/227003/geol-100-university-of-california-santa-barbara in Geology at University of California Santa Barbara.




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Date Created: 10/22/15
Materials 100A Class 15 Magnetic Properties I Ram Seshadri MRL 2031 x6129 seshadri mrlucsbedu httpWWWmrlucsbeduNseshadriteachhtml Magnetism Some basics A magnet is associated With magnetic lines of force and a north pole and a south pole The lines of force come out of the north pole the source and are pulled in to the south pole the sink A current in a ring or coil also produces magnetic lines of force The magnetic dipole a northsouth pair is usually represented by an arrow Magnetic elds act on these dipoles and tend to align them The magnetic eld strength H generated by N closely spaced turns in a coil of Wire carrying a current I for a coil length ofl is given by The units of H are amp res per meter Am l in SI units or oersted Oe in CGS 1 Am 1 47f X 10 3 Oe If a coil or solenoid encloses a vacuum then the magnetic ux density B generated by a eld strength H from the solenoid is given by BM0H Where 0 is the vacuum permeability ln SI units 0 47f X 10 7 Hm If the solenoid encloses a medium of permeability it instead of the vacuum then the magnetic ux density is given by B H and M MMO MT is the relative permeability Materials respond to a magnetic eld by developing a magnetization M Which is the number of magnetic dipoles per unit volume The magnetization is obtained from B MOH MOM The second term MOM is re ective of how certain materials can actually concentrate or repel the magnetic eld lines The response of a material to the magnetic eld is called the magnetic susceptibility X de ned Materials 100A Class 15 Magnetic Properties I XHM or MXH X is unitless and is related to the relative permeability according to X MT 7 1 The origins of magnetism Magnetism arises from the magnetic moment associated With single electrons and their quantum mechanical behavior electrons have a magnetic and spin quantum numbers The magnetic moment of individual electrons is iMB or 7 depending on Whether the spin is up or down Where MB is the Bohr magnetoni 1 MB 927 X 10 24 Am li In a normal paramagnet all the spins are noninteracting At any nite temperature there is a tendency for the spins to misalign even in the presence of a magnetic eld Only at 0 K do the spins all line up in the presence of a magnetic eld and the magnetic susceptibility X MH diverges M is the magnetization the number of magnetic dipoles formed per unit volume of sample and H is the magnetic eld Divergence of the susceptibility arises When even an in nitessimally small eld aligns all the spins The system obeys the Curie law and is called a Curie paramagnet With X T The Curie constant C is indicative of the number of magnetic spins per atomi Good examples of paramagnets are those containing magnetic ions in insulating solids1 Where the ions are far from one anotheri An example is the alum NH4CrSO4212HQO Where the large number of water molecules in the crystal help isolate the magnetic Cr3 ionsi Ferromagnetic interactions between spins arise as a result of an internal eld the Weiss eld A system that is normally a paramagnet With ferromagnetic interactions displays as a result of the internal eld CurieWeiss behavior in the paramagnetic regime 7 0 Tea As a result of the internal eld tending to align the spins the susceptibility diverges even at nite temperatures The temperature at Which the divergence occurs is called the Weiss constant 9 and it is often but not necessarily the Curie temperature To for the paramagnetic to ferromagnetic phase transition 1In metallic solids spins can communicate With oneianother over long distances through the mediation of conduction electrons Materials 100A Class 15 Magnetic Properties I Curie Weiss Paramagnet ferro Curie Paramagnet a x vie Systems showing such CurieWeiss behavior in the magnetic susceptibility are typical ferromagnets such as Fe Co Ni CrOg COSQ etc In materials that are ferromagnetic magnetic domains Which are regions of aligned spins tend to form below the Curie temperature To The domains themselves need not be all aligned The application of an external magnetic eld however tends to ensure that the net spin of each domain is aligned With the magnetic eld One of the indicators of the presence of magnetic domains is the characteristic hysteresis loop of ferromagnetsl HH CHM HH HH MN MW WM HH In some systems the internal eld tends to antiallign spinsl These systems are antiferromagnetsl The CurieWeiss plot is shifted so that 9 is negative In other words even at 0 K the spins cannot be lined up by applying a magnetic eld Only at some ctitious negative temperature Will the spins all line up Materials 100A Class 15 Magnetic Properties I CurlerWeiss Paramagne1amiferro x M s TN 9 T The intercept of the lX trace With the temperature axis is at some 9 lt 0 In a wellbehaved antiferromagnet the phase transition from a paramagnet to an antiferromagnet takes place at the Neel temperature TN and TN 79 Examples of antiferromagnets are Cr Mn 01 203 CoO aFegOg H l The origins of cooperative magnetism Valence d and f orbitals are more contracted than valence s and p orbitalsl As a result systems With d of f valence electrons are often magneticl Please see the handoutl Also please read a book like Kittel if you Wish to learn more In a magnetic d electron compound spinpolarization splits the densities of state unpolarized spin polarized ltE minority majority F The states are usually called spinup and spin down or alternately majority and minority Up and down are not be thought of as actual directions in Which the spins pointl n a ferromagnet all the spins on the magnetic d ions point in the same direction On the d ion site the DOS might look like Phil 100A Section Handout 3 Utilitarianism The principle of utility The right action is the one that maximizes total aggregate happiness Objections to Utilitarianism Objection 1 Too weak Utilitarianism allows too much 0 Slavery injustice 0 ChopupChuck Possible Responses 1 Bite the bullet Common sense morality is simply wrong in these cases 2 We can explain our common sense intuitions in a utilitarian framework a F7 Our utilitarian evaluation was incorrect since it only accounted for the short term consequences If we account for the long term consequences then the utilitarian evaluations will usually coincide with our common sense evaluations Problem What about cases where we have correctly taken into account the longterm consequences and our intuitions still diverge eg ChopupChuck The Surgeon Returns Our intuitions stem from the adoption of a deontological decision procedure ie live in accordance with our common sense moral principles But this decision procedure is justified on utilitarian grounds Given our limitations as finite fallible selfinterested human beings the attempt to always make decisions based on the utilitarian principle will lead to less aggregate happiness than would making decisions based on common sense moral principles 3 Rule Utilitarianism 1 Your actions should conform to a set of moral principles and 2 You should act only on those rules the adoption of which would lead to the greatest aggregate happiness Illustration ofthe difference between Act UT and Rule UT Collapse Problem Rule UT does not maximize the good and so is not a consequentialist theory Objection 2 Too Strong UT demands too much 0 Canoe example Possible reply The intuitions stem from the plausible decision procedure rule parents should look out for their own kids 0 Spending on luxury items movies dining out nice clothing etc Objection 3 related to objection 2 The distinction between acts that are obligatory and acts that are supererogatory collapses under utilitarianism Possible Reply This moral distinction is wrong and is based off of our selfinterest Wouldn t it be convenient if sacrificing some comfort in order to help the starving was not morally obligatory but only supererogatory Utilitarianism and lustice Taxonomy of Right Actions Expedient actions are those that Expedient actions maximize total aggregate happiness 0 Obligatory actions are expedient Obligatory actions which it is optimal to compel people to perform Just actions are obligatory actions requiring compliance with a perfect duty Le a duty that implies a corresponding right UT explanations ofjustice 1 Mill Security is indispensible for human happiness So the provision of security is extremely important for the maximization of total aggregate happiness Just actions are those that are required for the provision of security 2 An alternative utilitarian explanation We can usually promote our own happiness more effectively than others can Justice is then derived from the principle of utility in the following manner a Often you ought to promote your own happiness b You ought to refrain from interfering with the efforts of others to follow rule a for themselves


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