1February 24, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 24February 24, 2005 Physics for Scientists&Engineers 2 2ReviewReview The force between two current-carrying wires is given by The torque exerted by a magneticfield on a current-carrying loop isgiven byF12=µ0i1i2L2!d!= iABsin"February 24, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2) We define the magnitude of the magnetic dipole momentof a coil to be We can express the torque on a coil in amagnetic field as The magnetic potential energy of a magnetic dipole in amagnetic field is given byµ= NiA !!=!µ"!B !µ !ni U = !!µ"!B = !µB cos#February 24, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3) The magnetic field inside an ideal solenoid is givenby The magnetic field inside an ideal toroidal magnetis given byB =µ0inB =µ0Ni2!r2February 24, 2005 Physics for Scientists&Engineers 2 5Atoms as MagnetsAtoms as Magnets The atoms that make up all matter contain moving electronsthat form current loops that produce magnetic fields In most materials, these current loops are randomlyoriented and produce no net magnetic field Some materials naturally have some fraction of thesecurrent loops aligned and produce a net magnetic field andare called magnetic Other materials can have these current loops aligned by anexternal magnetic field and become magnetized. Let’s construct a very much-simplified model of the atomFebruary 24, 2005 Physics for Scientists&Engineers 2 6Atoms as Magnets (2)Atoms as Magnets (2) Consider an electron moving at a constantspeed v in a circular orbit with radius r asillustrated to the right We can think of the moving charge of theelectron as a current i Current is defined as the charge per unittime passing a particular point For this case the charge is the charge of the electron eand the time is related to the period of the orbiti =eT=e2!r( )/ v=ve2!rFebruary 24, 2005 Physics for Scientists&Engineers 2 7Atoms as Magnets (3)Atoms as Magnets (3) The magnetic moment of the orbitingelectron is given by We can define the orbital angularmomentum of the electron to be• where m is the mass of the electron Solving and substituting gives usµorb= iA =ve2!r!r2=ver2Lorb= rp = rmvLorb= rm2µorber!"#$%&=2mµorbeFebruary 24, 2005 Physics for Scientists&Engineers 2 8Atoms as Magnets (4)Atoms as Magnets (4) Rewriting and remembering that themagnetic dipole moment and the angularmomentum are vector quantities wecan write• The negative sign arises because of thedefinition of current as the flow of positive charge This result can be applied to the hydrogen atom, and thecorrect result is obtained However, other predictions of the properties of atomsbased on the idea that electrons exist in circular orbits inatoms disagree with experimental observations !µorb= !e2m!Lorb3February 24, 2005 Physics for Scientists&Engineers 2 9FerromagnetismFerromagnetism The elements iron, nickel, cobalt, gadolinium, anddysprosium and alloys containing these elements exhibitferromagnetism Ferromagnetic materials show long-range ordering at theatomic level, which causes the dipole moments of atoms toline up with each other in a limited region called a domain Within this domain, the magnetic field can be strong However, in the bulk these domains are randomly orientedleaving no net magnetic field An external magnetic field can align these domains andproduce magnetic fieldsFebruary 24, 2005 Physics for Scientists&Engineers 2 10Ferromagnetism (2)Ferromagnetism (2) A ferromagnetic material will retain all or some ofthis induced magnetism when the externalmagnetic field is removed In addition, the magnetic field produced by acurrent in a device like a solenoid or toroid will belarger if a ferromagnetic Demo• Insert a ferromagnetic material in the core of a solenoidand see how much the magnetic field is increasedFebruary 24, 2005 Physics for Scientists&Engineers 2 11DiamagnetismDiamagnetism Most materials exhibit diamagnetism However diamagnetism is weak compared with the othertwo types of magnetism and is thus masked by those formsif they are present in the material In diamagnetic materials, a weak magnetic dipole moment isinduced by an external magnetic field in a directionopposite the direction of the external field The induced magnetic field disappears when the externalfield is removed If the external field is non-uniform, in interaction of theinduced dipole moment of the diamagnetic material with theexternal field creates a force directed from a region ofgreater magnetic field to a region of lower magnetic fieldFebruary 24, 2005 Physics for Scientists&Engineers 2 12Diamagnetism (2)Diamagnetism (2) An example of a live frog exhibiting diamagnetism is shown below In this picture diamagnetic forces induced by a non-uniform externalmagnetic field of 16 T are levitating a live frog The normally negligible diamagnetic force is large enough in this case toovercome gravityA live frog being levitatedby a strong magnetic fieldat the High Field MagnetLaboratory, RadboudUniversity Nijmegen, TheNetherlands.4February 24, 2005 Physics for Scientists&Engineers 2 13ParamagnetismParamagnetism Materials containing certain transition elements, actinides, and rareearths exhibit paramagnetism Each atom of these elements has a permanent magnetic dipole, butthese dipole moments are randomly oriented and produce no netmagnetic field In the presence of an external magnetic field, some of thesemagnetic dipole moments align in the same direction as the externalfield When the external field is removed, the induced magnetic dipolemoment disappears If the external field is non-uniform, this induced magnetic dipolemoment interacts with the external field to produce a forcedirected from a region of lower magnetic field to a region of highermagnetic field.February 24, 2005 Physics for Scientists&Engineers 2 14Nuclear Magnetic ResonanceNuclear Magnetic Resonance Elementary particles such as protons have an intrinsicmagnetic dipole moment Consider the case in which we place protons in a strongmagnetic field Because of quantum mechanical reasons, the magneticdipole moment of a proton can only have two directions,parallel or anti-parallel with the external field The difference in energy between the two states is givenby
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