Review The force between two currentcarrying wires is given by F12 Physics for Scientists Engineers 2 0i1i2 L 2 d The torque exerted by a magnetic field on a current carrying loop is given by Spring Semester 2005 Lecture 24 iABsin February 24 2005 Physics for Scientists Engineers 2 1 February 24 2005 Physics for Scientists Engineers 2 Review 2 Review 3 We define the magnitude of the magnetic dipole moment of a coil to be NiA We can express the torque on a coil in a magnetic field as B The magnetic field inside an ideal solenoid is given by B 0in n The magnetic field inside an ideal toroidal magnet is given by i The magnetic potential energy of a magnetic dipole in a magnetic field is given by U B B cos February 24 2005 Physics for Scientists Engineers 2 2 B 3 February 24 2005 0 Ni 2 r Physics for Scientists Engineers 2 4 1 Atoms as Magnets Atoms as Magnets 2 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 We can think of the moving charge of the electron as a current i Some materials naturally have some fraction of these current loops aligned and produce a net magnetic field and are called magnetic 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 Other materials can have these current loops aligned by an external magnetic field and become magnetized Let s construct a very much simplified model of the atom February 24 2005 Physics for Scientists Engineers 2 Consider an electron moving at a constant speed v in a circular orbit with radius r as illustrated to the right i 5 February 24 2005 Atoms as Magnets 3 ve ver r2 2 r 2 The negative sign arises because of the definition of current as the flow of positive charge Lorb rp rmv This result can be applied to the hydrogen atom and the correct result is obtained where m is the mass of the electron Solving and substituting gives us February 24 2005 However other predictions of the properties of atoms based on the idea that electrons exist in circular orbits in atoms disagree with experimental observations 2 2m orb rm orb er e Physics for Scientists Engineers 2 6 Rewriting and remembering that the magnetic dipole moment and the angular momentum are vector quantities we can write e orb Lorb 2m We can define the orbital angular momentum of the electron to be Lorb Physics for Scientists Engineers 2 Atoms as Magnets 4 The magnetic moment of the orbiting electron is given by orb iA e e ve T 2 r v 2 r 7 February 24 2005 Physics for Scientists Engineers 2 8 2 Ferromagnetism Ferromagnetism 2 The elements iron nickel cobalt gadolinium and dysprosium and alloys containing these elements exhibit ferromagnetism A ferromagnetic material will retain all or some of this induced magnetism when the external magnetic field is removed Ferromagnetic materials show long range ordering at the atomic level which causes the dipole moments of atoms to line up with each other in a limited region called a domain In addition the magnetic field produced by a current in a device like a solenoid or toroid will be larger if a ferromagnetic Within this domain the magnetic field can be strong Demo However in the bulk these domains are randomly oriented leaving no net magnetic field Insert a ferromagnetic material in the core of a solenoid and see how much the magnetic field is increased An external magnetic field can align these domains and produce magnetic fields February 24 2005 Physics for Scientists Engineers 2 9 February 24 2005 Diamagnetism An example of a live frog exhibiting diamagnetism is shown below However diamagnetism is weak compared with the other two types of magnetism and is thus masked by those forms if they are present in the material A live frog being levitated by a strong magnetic field at the High Field Magnet Laboratory Radboud University Nijmegen The Netherlands In diamagnetic materials a weak magnetic dipole moment is induced by an external magnetic field in a direction opposite the direction of the external field The induced magnetic field disappears when the external field is removed If the external field is non uniform in interaction of the induced dipole moment of the diamagnetic material with the external field creates a force directed from a region of greater magnetic field to a region of lower magnetic field Physics for Scientists Engineers 2 10 Diamagnetism 2 Most materials exhibit diamagnetism February 24 2005 Physics for Scientists Engineers 2 11 In this picture diamagnetic forces induced by a non uniform external magnetic field of 16 T are levitating a live frog The normally negligible diamagnetic force is large enough in this case to overcome gravity February 24 2005 Physics for Scientists Engineers 2 12 3 Paramagnetism Nuclear Magnetic Resonance 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 Consider the case in which we place protons in a strong magnetic field In the presence of an external magnetic field some of these magnetic dipole moments align in the same direction as the external field When the external field is removed the induced magnetic dipole moment disappears If the external field is non uniform this induced magnetic dipole moment interacts with the external field to produce a force directed from a region of lower magnetic field to a region of higher magnetic field February 24 2005 Physics for Scientists Engineers 2 13 Nuclear Magnetic Resonance 2 The difference in energy between the two states is given by the difference in magnetic potential energy which is 2 B where is the component of the proton s magnetic moment along the direction of the external field B February 24 2005 Physics for Scientists Engineers 2 14 A magnetic resonance imaging device uses the physical principle of nuclear magnetic resonance just described Because the magnetic potential energy can only have two possible values the energy required to flip the direction is a discrete value depending on the magnitude of the external field Thus only one given oscillation frequency will cause the dipole moment to flip If the time varying electric field is switched off the protons in the
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