1. Magnets

2. Gauss' Law for Magnetic Fields

3. The Magnetism of Earth

4. Magnetism and Electrons

5. Magnetic Materials

6. Diamagnetism

7. Paramagnetism

8. Ferromagnetism

9. Induced Magnetic Fields

10. Displacement Current

11. Maxwell's Equations

Magnets

•Magnets have been known since antiquity in the form of small pieces (lodestones) of a naturally occurring magnetic iron ore called magnetite

•The Greeks eventually noticed that these pieces tended to align themselves pointing north

•Magnets were used for navigation for many centuries

•Magnets are magnetic dipoles whose properties are due to the arrangement of electrons in their atoms

•Magnetic monopoles are NOT known to exist

Gauss' Law for Magnetic Fields

•Gauss' Law for magnetic fields is just a formal way of stating that magnetic monopoles do not exist:

•If magnetic monopoles existed, they should have a net flux out of any area that completely encloses them

The Magnetism of Earth

•The earth forms a giant magnet

•The earth's magnetic field is thought to be due to the movement of molten iron in the earth's core

•The earth's magnetic field is not aligned with the earth's axis

•The average strength of the earth's magnetic field is about half a Gauss (50 µT)

•The strength and direction of the earth's magnetic field varies from point to point on the earth.

•The field declination is the deviation of the magnetic field from true north

•The field inclination is the deviation of the magnetic field from horizontal

•The northern pole of the earth's magnetic field is located near Baffin Bay in Canada

•The "north" pole of the earth's magnetic field is actually a south magnet pole as conventionally defined

•The earth's magnetic field gets stronger and weaker over a period of centuries and has reversed itself many times over millions of years

•The earth's magnetic field extends into space and traps charged particles from the sun in the Van Allen belts

•This protects the earth from higher radiation exposure

Magnetism and Electrons

•An electron has an intrinsic angular momentum (called spin) S

•Associated with this angular momentum is an intrinsic spin magnetic dipole moment µ

•The magnetic moment µ is related to S by:

•S is different from a macroscopic angular momentum in that it cannot be directly measured; its component along any axis can be measured

•S is also quantized which means it is limited to certain values (in this case, the same value, either positive or negative)

•In a magnetic field the energy associated with the spin magnetic moment is:

•In addition to the spin magnetic moment, the electron also has a magnetic moment associated with its orbit around the nucleus

•The orbital angular momentum is symbolized by L

•The orbital magnetic dipole moment µ is:

•The component along the z axis is:

•The orbital magnetic dipole moment behaves as the spin moment does:

Loop Model for Electron Orbits

•The complete explanation for the behavior of electrons in atoms is given by quantum mechanics

•But semi-classical models yield similar results in most cases

•The semi-classical model of an electron orbiting about the nucleus gives a magnetic moment:

•The current i is that of the electron circling the nucleus:

where the period is the circumference divided by the speed

•The angular momentum for a circular orbit is L = mvr

•So, including the sign due to opposing directions of L and µ:

Loop Model in a Non-Uniform Field

•It has previously been shown that the net force exerted by a uniform magnetic field on a current loop is zero

•So how are the magnetic forces between atoms created?

•It should be noted that the magnetic dipoles within atoms do not produce uniform magnetic fields

•In fact, the dipole magnetic fields associated with atomic current loops spread out and drop off rapidly with distance

•Under these circumstances, current loops near atoms are subject to a net magnetic force

Magnetic Materials

•Each electron in an atom has both a spin and an orbital magnetic moment

•These combine vectorially to produce a total magnetic moment for each electron

•The magnetic moments of the electrons within an atom combine vectorially to produce a total magnetic moment of the atom

•And finally, the magnetic moments of the atoms within a substance combine vectorially to produce an overall magnetic moment for the substance

•Substances react to magnetic fields in three general ways:

Diamagnetism

Paramagnetism

Ferromagnetism

•Diamagnetism is similar to electrostatic polarization of a dielectric in that the induced magnetic moment tends to oppose the field that creates it.

•Paramagnetism is created when atoms with permanent dipole moments tend to align with an external magnetic field; the resulting field is greater that of the external magnetic field

•Ferromagnetism is similar to paramagnetism except much stronger and occurs primarily in iron, nickel, cobalt

Diamagnetism

•Diamagnetic materials do not have permanent dipole moments

•Diamagnetism is the weakest of the three types of magnetism in materials

•All materials exhibit diamagnetism but it is overwhelmed in paramagnetic and ferromagnetic materials

•Diamagnetism arises because the dipole moments induced in atoms tend to line up opposite of the direction of the external magnetic field that created them

•Diamagnetic materials thus weaken the external field

Magnetization Vector and Magnetic Field Strength

•To describe the magnet field of materials a new symbol, M, is used

•M is the magnetization vector of a material (dia-, para-, or ferro-magnetic)

•To describe the external magnetic in the presence of materials a new symbol, H, is used

•H is the magnetic field strength due to some source of external magnetic field

•B remains the overall total magnetic field and is often called the magnetic induction or magnetic flux density to avoid confusion with H

•H and M have different units than B to account for the magnetic properties of materials; both have units of Amps/meter

•Including the magnetic susceptability in the B equation

•And the magnetic permeability for a substance is

Paramagnetism

•In paramagnetic materials, the various electron magnetic moments in an atom do not cancel and this leaves a permanent magnetic dipole moment

•When an external magnetic field is applied, the dipole moments tend to line up in the direction of the external field

•The reason that the susceptabilities are so small is that the alignment of the magnetic dipole must compete with the random thermal motion of the dipole

•So it is expected that the magnetization M should increase linearly as the external B field is increased until the dipoles are mostly aligned

•This deduction was proven experimentally by Pierre Curie in 1895 and is called Curie's Law

where C is a constant for each material

Ferromagnetism

•Ferromagnetism is a much stronger effect than para- or dia-magnetism

•Ferromagnetism is a result of the combining in the same direction of the many parallel spin electrons in iron, nickel, or cobalt

•The effect (called exchange coupling) extends beyond any single atom and links thousands to millions of ferromagnetic atoms

•If the temperature is raised above several hundred °C, the exchange coupling effect is overwhelmed and the atoms cease to couple in domains

•Under these conditions, ferromagnetism ceases to operate and the material becomes weakly paramagnetic

•Normally, however, those atoms will have a commonly oriented magnetic moment in a region called a magnetic domain

•Any piece of ferromagnetic material will normally consist of thousands of microscopic domains that are randomly oriented with respect to each other

•When a magnetic field is applied, some of the domains will shift their dipole alignments toward that of the external field

•As the field is strengthened, more of the domains will be realigned and the resulting magnetization will be much more intense than in non-ferromagnetic materials

•Even when the external field is removed, some of the domains will remain at the alignment they held, creating a permanent magnetic moment, i.e. a permanent magnet

•Lodestone is created when lightning strikes a surface deposit of iron ore and the high currents involved create a strong, though temporary, magnetic field

•Once a ferromagnetic material is magnetized, it is difficult to remove the magnetization

•Even if the external field is reversed in direction, some of the original magnetization will persist

•This effect is called hysteresis and makes it difficult to demagnetize any ferromagnetic material

•The usual way demagntization is done is to place the material in a magetic field that reverses itself many times a second and then to slowly withdraw the material from the field

Induced Magnetic Fields

•Faraday's Law of Induction can be written:

•James Clerk Maxwell reasoned that, if a varying magnetic field creates an electrical field, might not a varying electric field create a magnetic field

•It turns out this conjecture is correct, a changing electric field between the plates of a capacitor does create a magnetic field

•This is one of Maxwell's contributions to electromagnetic theory

•This is generalized to include the more typical sources of magnetic field, static currents

•This is called the Ampere-Maxwell Law

Displacement Current

•Since magnetic fields are generally created by currents, it is natural to think of Maxwell's contribution as a current

•This is a fictitious current called the displacement current (for historical current)

•It is easy to show that the displacement current arises naturally from the changing polarization of a charging capacitor

•The charge on the plates is

•And, using the Ampere-Maxwell Law, the B field for the capacitor will be

Maxwell's Equations

•Our study of electromagnetics is now complete

•We have looked at electrostatics, magnetics, magnetic induction, and displacement current

•Each of these is summarized in an equation

•Collectively, these 4 equations are known as Maxwell's Equations and they provide a complete, though compact, summary of electromagnetics

•In integral form, Maxwell's Equations are: