Lesson 9 Dipoles and Magnets

Class 27 Today we will: learn the definitions of electric and magnetic dipoles. find the forces, torques, and energies on dipoles in uniform fields. learn what happens when we put dipoles in nonuniform fields.

Lesson 9 Dipoles and Magnets

Section 1 Force on a Current-carrying Wire

Force on a Wire in a Magnetic Field There is a force on charge carriers in a current-carrying wire. This force is transferred to the wire itself. i L

Force on a Wire in a Magnetic Field N charge carriers are in length L of the wire. T is the time it takes a charge to go the distance L. The force is: i L

Force on a Wire in a Magnetic Field More generally

Section 2 Force and Torque on Wire Loops

Force on a Wire Loop in a Uniform Magnetic Field The net force on a wire loop is zero. i

Torque on a Wire Loop in a Uniform Magnetic Field The net torque on a wire loop is not zero. i

Torque on a Wire Loop in a Uniform Magnetic Field We define an area vector for the loop, by using the right hand rule. i

Right-hand Rule for Current Loops The direction of an area for a current loop is 1)normal to the plane of the loop and 2)In the direction of your thumb if your fingers loop in the direction of the current. i

Right-hand Rule for Current Loops Note that the directed area of the loop is the same as the direction of the magnetic field produced by the loop! i

Now calculate the torque about an axis through the center of the loop going into the screen. i Torque on a Wire Loop in a Uniform Magnetic Field

The magnitude of the torque is the product of the force and the moment arm.

Torque on a Wire Loop in a Uniform Magnetic Field The magnitude of the torque is the product of the force and the moment arm.

Torque on a Wire Loop in a Uniform Magnetic Field

Section 3 Magnetic Dipoles

Define the Magnetic Dipole Moment Since appears in a number of formulas, we give it a name: the magnetic dipole moment. Note that it is a vector with the direction given by the right-hand rule.

Torque on a Wire Loop in a Magnetic Field In terms of the magnetic dipole moment:

Direction of the Torque A dipole feels a torque that tends to align the dipole moment with the external field.

Potential Energy of a Dipole The maximum potential energy is when the dipole is opposite the field. The minimum potential energy is when the dipole is in the direction of the field.

Potential Energy of a Dipole The work done by a force is

Potential Energy of a Dipole The work done by a force is Similarly, the work done by a torque is

Potential Energy of a Dipole The work done by a force is Similarly, the work done by a torque is Since the change in potential energy is the work it takes to rotate the dipole, we have:

Potential Energy of a Dipole We can choose the constant of integration to be anything we want.

Potential Energy of a Dipole It’s simplest if we choose it to be zero.

Potential Energy of a Dipole Caution!!! U=0 is not the minimum energy. It is the energy when the dipole is perpendicular to the field!

Section 4 Electric Dipoles

The Electric Dipole An electric dipole is a charge +q and a charge -- q held apart a distance apart.

The Electric Dipole Moment An electric dipole moment is The direction of goes from the  charge to the + charge.

Torque and Potential Energy of an Electric Dipole Electric dipoles work just the same way as magnetic dipoles. In uniform fields, there is no net force on the dipole. Torque: Potential energy:

An Electric Dipole in a Nonuniform Field First, the dipole feels a torque that aligns the dipole with the field. (  end toward the source of the field.)

An Electric Dipole in a Nonuniform Field Then, the dipole feels a net force in the direction of the stronger field.

A Magnetic Dipole in a Nonuniform Field Magnetic dipoles behave in much the same way. They first experience a torque that aligns them with external field. I

A Magnetic Dipole in a Nonuniform Field Then, they experience a net force that pulls them in the direction of the stronger field. I

Permanent Magnets Permanent magnets have magnetic dipole moments much as current loops. N S

Permanent Magnets In a nonuniform external field, permanent magnets experience a torque… N S N S

Permanent Magnets …then a net force in the direction of stronger magnetic field. N S N S

Permanent Magnets …then a net force in the direction of stronger magnetic field. N S S N

Class 28 Today we will: define magnetization and magnetic susceptiblity learn about paramagnetic, diamagnetic, and ferromagnetic materials learn about the opposing effects of domain alignment and thermal disalignment learn how to understand hysteresis curves characterize ferromagnetic materials in terms of residual magnetization and coercive force

Section 5 Paramagnetism and Diamagnetism

Permanent Magnets 1)Magnetite or loadstone was known from antiquity.

Permanent Magnets "Magnetism" comes from the region called Magnesia, where loadstone (magnetite) was found.

Permanent Magnets 1)Magnetite or loadstone was known from antiquity. 2)Loadstone floating on wood rotates so one end always points north.

Permanent Magnets 1)Magnetite or loadstone was known from antiquity. 2)Loadstone floating on wood rotates so one end always points north. This is the north pole.

Permanent Magnets 1)Magnetite or loadstone was known from antiquity. 2)Loadstone floating on wood rotates so one end always points north. This is the north pole. 3)If two magnets are placed near other, like poles attract and unlike poles repel.

Permanent Magnets William Gilbert in 1600 publushed De Magnete – where he described magnetism as the “soul of the earth.”

Permanent Magnets Gilbert: A perfectly spherical magnet spins without stopping – because the earth is a perfect sphere and it’s a magnet and it spins without stopping. N S

Permanent Magnets From last time: Permanent magnets behave like current loops. In a nonuniform external field, permanent magnets experience a torque… N S N S

Permanent Magnets …then a net force in the direction of stronger magnetic field. N S N S

Permanent Magnets …then a net force in the direction of stronger magnetic field. N S S N

Permanent Magnets If we don’t allow magnets to rotate: SNSN SNSN SNSN SNSN

Permanent Magnets When dipole moments align, magnets attract. When dipole moments are opposite, magnets repel. SNSN SNSN SNSN SNSN

Atoms as Magnets If we throw a magnet really fast (so it doesn’t have time to rotate) through a non-uniform field, what happens? SNSN

Atoms as Magnets If we throw a magnet really fast through a non- uniform field, what happens? SNSN

Atoms as Magnets If we throw a magnet really fast through a non- uniform field, what happens? SNSN

Atoms as Magnets If we throw a magnet really fast through a non- uniform field, what happens? SNSN

Atoms as Magnets Now take Ag atoms and do the same thing: Stern- Gerlach experiment gerlach/stern-gerlach_en.html (PhET U. of Colo.)

Atoms as Magnets Now take Ag atoms and do the same thing – Stern Gerlach experiment Conclusion: Ag atoms are magnets The magnets seem to all be aligned either with the field or against the field. How could an atom be a magnet?

Atoms as Magnets How could an atom be a magnet? Magnetic fields are caused by moving charges, so what’s moving?

Atoms as Magnets Magnetic fields are caused by moving charges, so what’s moving? Bohr atom:

Electrons as Magnets Electrons do the same thing (sort of). How could an electron be a magnet?

Electrons as Magnets An electron spins… N S

Electrons as Magnets We can measure the magnetic dipole moment of an atom by measuring the force on electrons in a nonuniform field. N S

Electrons as Magnets We can model an electron as a spinning sphere and determine its radius: 2.8 fm N S

Electrons as Magnets But scattering measurements find electrons to be point particles or nearly so. N S

Section 5 Paramagnetism and Diamagnetism

Magnetic Properties of Magnets In most materials, all the “little magnets” are randomly arranged, so the material is affected only slightly by external magnetic fields.

Materials react to external magnetic fields in three different ways 1) Paramagnetic materials are very weakly attracted by external magnetic fields. Most materials are paramagnetic.

Materials react to external magnetic fields in three different ways 1) Paramagnetic materials are very weakly attracted by external magnetic fields. Most materials are paramagnetic. 2) Diamagnetic materials are very weakly repelled by external magnetic fields.

Materials react to external magnetic fields in three different ways 1) Paramagnetic materials are very weakly attracted by external magnetic fields. Most materials are paramagnetic. 2) Diamagnetic materials are very weakly repelled by external magnetic fields. 3) Ferromagnetic materials are strongly attracted or repelled by external magnetic fields.

How do we understand paramagnetism? Paramagnetic atoms are like little magnetic dipoles. They experience a torque which aligns them with the external field, then they feel a net force that pulls them into the field. The magnetic dipole moment results primarily from electron spin and angular momentum. NSNS

How do we understand diamagnetism? Diamagnetism is something that is not adequately explained without resorting to quantum mechanics. SNSN

How do we understand ferromagnetism? Domain alignment: If atoms have large magnetic dipole moments, they tend to align with each other much as a collection of magnets tends to align. NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS NSNS

How do we understand ferromagnetism? Thermal disalignment: Heat causes atoms to vibrate, knocking them around and disaligning the dipoles.

How do we understand ferromagnetism? Domains: Small regions that have aligned dipole moments are called domains. In unmagnetized iron, the domains are randomly oriented.

How do we understand ferromagnetism? Domains: In a permanent magnet, the domains tend to be aligned in a particular direction.

The Curie Point Curie Temperature: When a ferromagnetic material gets hot enough, the domains break down and the material becomes paramagnetic.

Getting Quantitative We define magnetization as the total magnetic dipole moment per unit volume. A magnetized object has an internal magnetic field given by the relation:

Getting Quantitative The internal magnetic field can also be expressed in terms of the external magnetic field: where is called the magnetic susceptibility.

Susceptibilites paramagnetic diamagnetic ferromagnetic

Susceptibilities for Ferromagnetic Materials Ferromagnetic materials have a “memory.” If we know the external field, we can’t predict the internal field, unless we know the previous history of the sample. We describe the relationship between internal and external fields by means of a “hysteresis curve.”

Hysteresis Curve We start with no internal field (unaligned) and no external field.

Hysteresis Curve We increase the external field, causing some of the domains to align.

Hysteresis Curve As the external field increases, the internal field eventually stops growing. Why?

Hysteresis Curve All the domains eventually align.

Hysteresis Curve We then decrease the external field. The domains want to stay aligned, so the internal field remains large.

Hysteresis Curve When the external field goes to zero, some domains remain aligned.

Hysteresis Curve residual magnetization The internal field that remains is called the residual magnetization.

Hysteresis Curve To reduce the internal field, we must apply an external field in the opposite direction.

Hysteresis Curve coercive force To reduce the internal field, we must apply an external field in the opposite direction.

Hysteresis Curve coercive force The external field needed to bring the internal field back to zero is called the coercive force.

Hysteresis Curve As we continue to increase the external field in the negative direction, domains align with the field.

Hysteresis Curve The process continues just as when the external field was in the positive direction.

Soft Iron A nail made of soft iron has domains that align easily, but it can ’ t hold the magnetization.

Soft Iron A nail made of soft iron has domains that align easily, but it can ’ t hold the magnetization. The coercive force and the residual magnetization of soft iron are both small.

Good Permanent Magnet A good permanent magnet has a large coercive force and a large residual magnetization.