Maxwell’s Equations Gauss’ Law for Magnetic Fields Induced Magnetic Fields Displacement Current Maxwell’s Equations Earth’s Magnetic Field Magnetism and Electrons Magnetic Materials pps by C Gliniewicz

One can observe the shape of the magnetic field of a bar magnet by covering it with a piece of paper and then sprinkling iron filings on the paper. The iron particles align themselves with the magnet field, showing its shape. The filed points from the north pole toward the south pole. This pair of north and south poles is a magnetic dipole. No matter how many pieces into which one breaks the magnet, there is always a north and south pole. One does not see a magnetic monopole. As far as is known, magnetic monopoles do not exist. Gauss’ Law for magnetic fields can be compared to Gauss’ Law for electric fields. The law for electric fields states that the flux is proportional to the charge enclosed by the Gaussian surface. The law for magnetic fields states that the flux is always zero, whether the Gaussian surface encloses all or part of an object. One also knows that Faraday’s Law of Induction states that a changing magnetic field can induce an electric field.

pps by C Gliniewicz The obvious next question is whether a changing electric field can create a magnetic field. James Clerk Maxwell determined an almost symmetric law of induction. A magnetic field is induced by a changing electric field in this equation. The coefficient is the product of the permeability of free space and the permittivity of free space which happens to be the inverse of the speed of light squared. One can combine Maxwell’s Law with Ampere’s Law to determine the magnetic field under any circumstances. This law works when either there is an enclosed charge, a changing magnetic field or both. Using this information, one can calculate a fictitious displacement current due to the changing electric field.

pps by C Gliniewicz One can determine the induced magnetic field inside and outside a circular capacitor. James Clerk Maxwell grouped all the equations for electric and magnetic fields together and they have been called Maxwell’s equations since. The first magnets which were discovered were formed from the magnetic mineral magnetite. The rock was called lodestone. Later these lodestones were used to make the first compasses. The earth has electric currents moving in the pliable rock of the mantle creating a natural magnetic field.

pps by C Gliniewicz The north magnetic pole of the earth lies in northwestern Greenland and the south magnetic pole is in Antarctica. Since the magnetic poles are not aligned with the axial poles, the magnetic field does not point toward the northern axis of the earth. The declination of the field is the angle east or west of the actual direction of north. Since the field lines curve out of the earth and back into it, the field is inclined to the surface. It is near vertical at the magnetic pole and is horizontal at the magnetic equator. Sometimes this inclination is called the magnetic dip. The magnetic field tends to move about from place to place. Over a lifetime, it may move as much as 30 or more degrees. In fact over longer periods, it has been discovered that the magnetic field has flipped, changing south and north. This was first found by measuring the magnetic field in rocks on either side of the Mid- Atlantic Ridge. Electrons have a spin. That spin is associated with a spin angular momentum, S, and thus also a spin magnetic dipole moment, .

pps by C Gliniewicz Spin is quantized, meaning it has discrete values. The quantity on the right is called the Bohr magneton, When an electron is placed in a magnetic field, a potential energy can be associated with the orientation of the electron. When an electron is in orbit in an atom, it has an orbital angular momentum, L, and an orbital magnetic dipole moment, . In general, one can discuss three types of magnetic materials: diamagnetism, paramagnetism and ferromagnetism.

pps by C Gliniewicz A material which is diamagnetic when placed in an external magnetic field will develop a magnetic moment opposite that of the field. If the field is non-uniform, the diamagnetic material is repelled from a region of greater field toward a region of lesser field. A paramagnetic material when placed in an external magnetic field will develop a magnetic moment in the direction of the field. If the field is non-uniform, the diamagnetic material is attracted toward a region of greater field away from a region of lesser field. In 1895 Pierre Curie discovered that the magnetization of a paramagnetic material is directly proportional to the external magnetic field and inversely proportional to the temperature in Kelvins. The proportionality constant is called the Curie constant. This law is true when the ratio of the magnetic field to the temperature is not too large. Ferromagnetic materials, including iron, nickel, gadolinium, dysprosium and their alloys can exhibit strong, permanent magnetism, unlike the temporary magnetism of diamagnetic and paramagnetic materials.

pps by C Gliniewicz Ferromagnetic materials can lose their magnetism above a certain temperature called the Curie Point. If the metal is heated above that temperature all the magnetic properties cease and when the metal cools, the properties reappear. He magnetization of ferromagnetic materials is not retraced as the external field is increased and then decreased. This lack or retraceability is called a hysteresis loop.