Chapter 9 Magnetic Forces, Materials and Inductance The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge: F = qv x B 1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B. 2. The magnitude of the force is F = qvB sin where is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. 3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product. From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb meter) or Newton per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss. Force On A Moving Charge Lorentz Force Law Both the electric field and magnetic field can be defined from the Lorentz force law: The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule. Force On A Moving Charge Force on a Differential Current dF = dQv x B J v v dF J Bdv dF v dv v B dF J Bdv Jdv dQ v dv F J B dv vol F K B dS S KdS IdL dF K BdS F I dL B I B_x_dL dF IdL B F IL B Example 9.1 3 3 6 1 Fy1 2 10 3 10 dx x 1 H I az 2 x az ax Fy1 6.592 10 2 3 6 1 Fx1 2 10 3 10 dy 1 3 0 az ay Fx1 4 10 1 3 6 1 Fy2 2 10 3 10 dx x 3 az ax Fy2 6.592 10 0 Fx2 2 10 3 10 1 dy 1 2 az ay Fx2 1.2 10 3 B o H 6 F Fx1 Fx2 Fy1 Fy2 F I BxdL 9 9 9 8 9 F 8 10 The net force on the loop is in the -ax direction Force Between Differential Current Elements Example 9.2 D9.4 Force And Torque On A Closed Circuit F I B_x_dL F IB 1 dL T R F Force And Torque On A Closed Circuit dT IdS B Magnetic Dipole Moment dm dm IdS dT T dm B IS B m B Force And Torque On A Closed Circuit Build a DC Motor – Hands On Explain Operation Force And Torque On A Closed Circuit DC Motor - Illustration Example 9.3 and 9.4 The Nature of Magnetic Materials Magnetic Materials Magnetic Materials may be classified as diamagnetic, paramagnetic, or ferromagnetic on the basis of their susceptibilities. Diamagnetic materials, such as bismuth, when placed in an external magnetic field, partly expel the external field from within themselves and, if shaped like a rod, line up at right angles to a non-uniform magnetic field. Diamagnetic materials are characterized by constant, small negative susceptibilities, only slightly affected by changes in temperature. Paramagnetic materials, such as platinum, increase a magnetic field in which they are placed because their atoms have small magnetic dipole moments that partly line up with the external field. Paramagnetic materials have constant, small positive susceptibilities, less than 1/1,000 at room temperature, which means that the enhancement of the magnetic field caused by the alignment of magnetic dipoles is relatively small compared with the applied field. Paramagnetic susceptibility is inversely proportional to the value of the absolute temperature. Temperature increases cause greater thermal vibration of atoms, which interferes with alignment of magnetic dipoles. Ferromagnetic materials, such as iron and cobalt, do not have constant susceptibilities; the magnetization is not usually proportional to the applied field strength. Measured ferromagnetic susceptibilities have relatively large positive values, sometimes in excess of 1,000. Thus, within ferromagnetic materials, the magnetization may be more than 1,000 times larger than the external magnetizing field, because such materials are composed of highly magnetized clusters of atomic magnets (ferromagnetic domains) that are more easily lined up by the external field. Magnetization and Permeability Magnetization and Permeability Example 9.5 Magnetic Boundary Conditions The Magnetic Circuit The Magnetic Circuit The Magnetic Circuit Inductance and Mutual Inductance A self-induced electromotive force opposes the change that brings it about. Consequently, when a current begins to flow through a coil of wire, it undergoes an opposition to its flow in addition to the resistance of the metal wire. On the other hand, when an electric circuit carrying a steady current and containing a coil is suddenly opened, the collapsing, and hence diminishing, magnetic field causes an induced electromotive force that tends to maintain the current and the magnetic field and may cause a spark between the contacts of the switch. The self-inductance of a coil, or simply its inductance, may thus be thought of as electromagnetic inertia, a property that opposes changes both in currents and in magnetic fields.