Magnetism March 10, 2014 Physics for Scientists & Engineers 2, Chapter 27 1
Notes! Homework is due on We night! Exam 4 next Tuesday Covers Chapters 27, 28, 29 in the book Magnetism, Magnetic Fields, Electromagnetic Induction Material from the week before spring break and this week Homework sets 7 and 8! I will be out of town next week Prof. Joey Huston will teach lectures March 10, 2014 Chapter 29 2
Force on a Current Carrying Wire! The magnitude of the magnetic force on a wire of length L carrying a current i is F = il B B F = ilbsinθ! θ is the angle between the current and the magnetic field! Force is perpendicular to both current and magnetic March 10, 2014 Physics for Scientists & Engineers 2, Chapter 27 3
Magnetic Field from a Long, Straight Wire! Magnitude: B = µ 0i 2πr! Direction: March 10, 2014 Physics for Scientists & Engineers, Chapter 28 4
Magnetic Field due to a Wire Loop! Using more advanced techniques and with the aid of a computer we can determine magnetic field from a wire loop at other points in space March 10, 2014 Physics for Scientists & Engineers, Chapter 28 5
Ampere s Law! We have calculated the magnetic field from an arbitrary distribution of current elements using d B = µ 0 4π! We may be faced with a difficult integral! In cases where the distribution of current elements has cylindrical or spherical symmetry, we can apply Ampere s Law to calculate the magnetic field from a distribution of current elements with much less effort than using a direct integration! Ampere s Law is B d s id s r r 3 = µ 0 i enc March 10, 2014 Physics for Scientists & Engineers, Chapter 28 6
Ampere s Law! As an example of Ampere s Law, consider the five currents shown below:! The currents are perpendicular to the plane.! We can draw an Amperian loop represented by the red line.! This loop encloses i 1, i 2, and i 3 and excludes i 4 and i 5.! Ampere s Law tells us that: B i d s = µ 0 i 1 + i 2 + i 3 ( ) March 10, 2014 Chapter 28 7
Magnetic Fields of Solenoids! Current flowing through a single loop of wire produces a magnetic field that is not very uniform! Applications often require a uniform field! A first step toward a more uniform magnetic field is the Helmholtz coil! This coil consists of 2 sets of coaxial wire loops March 10, 2014 Physics for Scientists & Engineers, Chapter 28 10
Magnetic Fields of Solenoids! Let s look at the magnetic field created by 4 coaxial coils! Let s go from 4 coils to many coils March 10, 2014 Physics for Scientists & Engineers, Chapter 28 12
Magnetic Fields of Solenoids! Let s look at the magnetic field created by 600 coaxial coils! 600 coils produce an excellent field inside the coils March 10, 2014 Physics for Scientists & Engineers, Chapter 28 13
Magnetic Fields of Solenoids! An ideal solenoid has a magnetic field of zero outside and of a uniform constant finite value inside.! To determine the magnitude of the magnetic field inside an ideal solenoid, we can apply Ampere s Law to a section of a solenoid far from its ends.! To do so, we first choose an Amperian loop over which to carry out the integration. March 10, 2014 Chapter 28 14
Magnetic Fields of Solenoids! The left hand side of Ampere s Law is: b c d a B i ds = B i ds + B i ds + B i ds + B i ds = Bh a b c d 0 0 Bh! The right hand side of Ampere s Law is: 0 µ0 ienc = µ0 nhi n is the number of turns per unit length! So we get: Bh = µ0 nhi B = µ0 ni March 10, 2014 Chapter 28 15
Atoms as Magnets! 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.! Consider a highly simplified model of an atom. March 10, 2014 Chapter 28 16
Atoms as Magnets! Imagine that an atom consists of an electron moving with a constant speed v in a circular orbit with radius r.! 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 10, 2014 Chapter 28 17
Atoms as Magnets! The magnetic moment of the orbiting electron is given by: µ orb = ia = ve 2πr πr 2 = ver 2! We can define the orbital angular momentum of the electron to be: L orb = rp = rmv! Solving and substituting gives us: L orb = rm 2µ orb er = 2mµ orb e March 10, 2014 Chapter 28 18
Atoms as Magnets! Rewriting and remembering that the magnetic dipole moment and the angular momentum are vector quantities, we can write: µ orb = e 2m L orb! 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 10, 2014 Chapter 28 19
Atoms as Magnets! The magnetic dipole moment from the orbital motion of electrons is not the only contribution to the magnetic moment of atoms.! Electrons and other elementary particles have their own, intrinsic magnetic moments, due to their spin.! The phenomenon of spin will be covered thoroughly in the discussion of quantum physics in Chapters 36 through 39, but some facts about spin and its connection to a particle s intrinsic angular momentum have been discovered experimentally and do not require an understanding of quantum mechanics.! Electrons, protons and neutrons all have spin s = ½. March 10, 2014 Chapter 28 20
Atoms as Magnets! The angular momentum of these particles is: ( ) S = s s+1! The z-component of the angular momentum of these particles is: S z = 1 2 or + 1 2! This spin cannot be explained by the orbital motion of some substructure of the particles.! Spin is an intrinsic property, similar to mass or electric charge. March 10, 2014 Chapter 28 21
Magnetic Properties of Matter! Magnetic dipoles do not experience a net force in a constant magnetic field.! Magnetic dipoles do experience a net torque in a constant magnetic field.! This torque drives a single free dipole to an orientation with the lowest magnetic potential energy.! What happens when matter is placed in a magnetic field?! The dipole moment of atoms in a material can point in different directions or in the same direction.! The magnetization of a material is defined as the net dipole moment created by the dipole moments of the atoms in the material per unit volume. March 10, 2014 Chapter 28 22
Diamagnetism and Paramagnetism! The magnetization depends on the magnetic field strength.! For most materials, but not all, the relationship is linear: M = χ m H! The quantity χ m is the magnetic susceptibility.! There are materials that do not obey this relationship.! The most prominent among those are ferromagnets.! Let s first look at materials that do obey this relationship, diamagnetic and paramagnetic materials.! If χ m < 0, the dipoles in the material tend to arrange themselves to oppose an external magnetic field.! Material with χ m < 0 are diamagnetic. March 10, 2014 Chapter 28 23
Diamagnetism! An example of a strawberry exhibiting diamagnetism is shown below! A strawberry being levitated by a strong magnetic field at the High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands.! Levitating a live frog! In this picture diamagnetic forces induced by a non-uniform external magnetic field of 16 T are levitating a strawberry! The normally negligible diamagnetic force is large enough in this case to overcome gravity October 13, 2008 Physics for Scientists & Engineers 2, Lecture 27 24
Paramagnetism! If χ m > 0, the magnetization of the material points in the same direction as the external field.! This property is paramagnetism and materials exhibiting this property are called paramagnetic.! Materials containing certain transition elements, including actinides and rare earths, exhibit paramagnetism.! Each atom of these elements has a permanent magnetic dipole that is normally randomly oriented.! In the presence of an external magnetic field, some of the magnetic dipole moments align with the external field.! When the external field is removed, the induced magnetic dipole moment disappears. March 10, 2014 Chapter 28 25
Ferromagnetism! 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 line 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! Iron Ore! Tilden mine, UP March 10, 2014 Chapter 28 26
Ferromagnetism (2)! 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, for example inside electromagnets, solenoids or toroids! Demo Insert a ferromagnetic material in the core of a solenoid and see how much the magnetic field is increased March 10, 2014 Chapter 28 27
Healing magnets! Can magnets in shoes or worn as bracelets heal people or improve athletic ability?! Superposition principle:! Field is at most 10-4 T at a distance of d=1mm from the magnet center and decreases as 1/d 3! Magnet holder has thickness ~ 5mm, skin thickness ~1mm B field is much less than 10-6 T in muscles and tissue! Much smaller than earth s magnetic field (5*10-5 T)!! And of course the human body is non-magnetic March 10, 2014 Physics for Scientists & Engineers 2, Chapter 27 28