1 EEE 498/598 Overview of Electrical Engineering Lecture 8: Magnetostatics: Mutual And Self-inductance; Magnetic Fields In Material Media; Magnetostatic Boundary Conditions; Magnetic Forces And Torques

Lecture 8 2 Lecture 8 Objectives To continue our study of magnetostatics with mutual and self-inductance; magnetic fields in material media; magnetostatic boundary conditions; magnetic forces and torques. To continue our study of magnetostatics with mutual and self-inductance; magnetic fields in material media; magnetostatic boundary conditions; magnetic forces and torques.

Lecture 8 3 Flux Linkage Consider two magnetically coupled circuits Consider two magnetically coupled circuits C1C1 I1I1 S1S1 S2S2 C2C2 I2I2

Lecture 8 4 Flux Linkage (Cont’d) The magnetic flux produced I 1 linking the surface S 2 is given by The magnetic flux produced I 1 linking the surface S 2 is given by If the circuit C 2 comprises N 2 turns and the circuit C 1 comprises N 1 turns, then the total flux linkage is given by If the circuit C 2 comprises N 2 turns and the circuit C 1 comprises N 1 turns, then the total flux linkage is given by

Lecture 8 5 Mutual Inductance The mutual inductance between two circuits is the magnetic flux linkage to one circuit per unit current in the other circuit: The mutual inductance between two circuits is the magnetic flux linkage to one circuit per unit current in the other circuit:

Lecture 8 6 Neumann Formula for Mutual Inductance

Lecture 8 7 Neumann Formula for Mutual Inductance (Cont’d)

Lecture 8 8 Neumann Formula for Mutual Inductance (Cont’d) The Neumann formula for mutual inductance tells us that The Neumann formula for mutual inductance tells us that L 12 = L 21 L 12 = L 21 the mutual inductance depends only on the geometry of the conductors and not on the current the mutual inductance depends only on the geometry of the conductors and not on the current

Lecture 8 9 Self Inductance Self inductance is a special case of mutual inductance. Self inductance is a special case of mutual inductance. The self inductance of a circuit is the ratio of the self magnetic flux linkage to the current producing it: The self inductance of a circuit is the ratio of the self magnetic flux linkage to the current producing it:

Lecture 8 10 Self Inductance (Cont’d) For an isolated circuit, we call the self inductance, inductance, and evaluate it using For an isolated circuit, we call the self inductance, inductance, and evaluate it using

Lecture 8 11 Generation of Magnetic Field I I iron core air gap with constant B field N S

Lecture 8 12 Equivalent of a Magnetic Dipole I N S Magnetic dipole can be viewed as a pair of magnetic charges by analogy with electric dipole.

Lecture 8 13 Forces Exerted on a Magnetic Dipole in a Magnetic Field N S B

Lecture 8 14 Current Loops (Magnetic Dipoles) in Atoms Electron orbiting nucleus Electron orbiting nucleus Electron spin Electron spin Nuclear spin Nuclear spin negligible  A complete understanding of these atomic mechanisms requires application of quantum mechanics.

Lecture 8 15 Current Loops (Magnetic Dipoles) in Atoms (Cont’d) In the absence of an applied magnetic field, the infinitesimal magnetic dipoles in most materials are randomly oriented, giving a net macroscopic magnetization of zero. In the absence of an applied magnetic field, the infinitesimal magnetic dipoles in most materials are randomly oriented, giving a net macroscopic magnetization of zero. When an external magnetic field is applied, the magnetic dipoles have a tendency to align themselves with the applied magnetic field. When an external magnetic field is applied, the magnetic dipoles have a tendency to align themselves with the applied magnetic field.

Lecture 8 16 Magnetized Materials A material is said to be magnetized when induced magnetic dipoles are present. A material is said to be magnetized when induced magnetic dipoles are present. The presence of the induced magnetic dipoles modifies the magnetic field both inside and outside of the magnetized material. The presence of the induced magnetic dipoles modifies the magnetic field both inside and outside of the magnetized material.

Lecture 8 17 Permanent Magnets Most materials lose their magnetization when the external magnetic field is removed. Most materials lose their magnetization when the external magnetic field is removed. A material that remains magnetized in the absence of an applied magnetic field is called a permanent magnet. A material that remains magnetized in the absence of an applied magnetic field is called a permanent magnet.

Lecture 8 18 Magnetization Vector The magnetization or net magnetic dipole moment per unit volume is given by The magnetization or net magnetic dipole moment per unit volume is given by average magnetic dipole moment [Am 2 ] Number of dipoles per unit volume [m -3 ] [A/m]

Lecture 8 19 Magnetic Materials The effect of an applied electric field on a magnetic material is to create a net magnetic dipole moment per unit volume M. The effect of an applied electric field on a magnetic material is to create a net magnetic dipole moment per unit volume M. The dipole moment distribution sets up induced secondary fields: The dipole moment distribution sets up induced secondary fields: Total field Field in free space due to sources Field due to induced magnetic dipoles

Lecture 8 20 Volume and Surface Magnetization Currents A magnetized material may be represented as an equivalent volume ( J m ) and surface ( J sm ) magnetization currents. A magnetized material may be represented as an equivalent volume ( J m ) and surface ( J sm ) magnetization currents. These charge distributions are related to the magnetization vector by These charge distributions are related to the magnetization vector by

Lecture 8 21 Volume and Surface Magnetization Currents (Cont’d) Magnetization currents are equivalent currents that account for the effect of the magnetized material, and are analogous to equivalent volume and surface polarization charge densities in a polarized dielectric. Magnetization currents are equivalent currents that account for the effect of the magnetized material, and are analogous to equivalent volume and surface polarization charge densities in a polarized dielectric. If the magnetization vector is constant throughout a magnetized material, then the volume magnetization current density is zero, but the surface magnetization current is nonzero. If the magnetization vector is constant throughout a magnetized material, then the volume magnetization current density is zero, but the surface magnetization current is nonzero.

Lecture 8 22 Ampere’s Law in Magnetic Media Ampere’s law in differential form in free space: Ampere’s law in differential form in free space: Ampere’s law in differential form in a magnetized material: Ampere’s law in differential form in a magnetized material:

Lecture 8 23 Magnetic Field Intensity define the magnetic field intensity as

Lecture 8 24 General Forms of Ampere’s Law The general form of Ampere’s law in differential form becomes The general form of Ampere’s law in differential form becomes The general form of Ampere’s law in integral form becomes The general form of Ampere’s law in integral form becomes

Lecture 8 25 Permeability Concept For some materials, the net magnetic dipole moment per unit volume is proportional to the H field For some materials, the net magnetic dipole moment per unit volume is proportional to the H field magnetic susceptibility (dimensionless) the units of both M and H are A/m.

Lecture 8 26 Permeability Concept (Cont’d) Assuming that Assuming that we have we have The parameter  is the permeability of the material. The parameter  is the permeability of the material.

Lecture 8 27 Permeability Concept (Cont’d) The concepts of magnetization and magnetic dipole moment distribution are introduced to relate microscopic phenomena to the macroscopic fields. The concepts of magnetization and magnetic dipole moment distribution are introduced to relate microscopic phenomena to the macroscopic fields. The introduction of permeability eliminates the need for us to explicitly consider microscopic effects. The introduction of permeability eliminates the need for us to explicitly consider microscopic effects. Knowing the permeability of a magnetic material tells us all we need to know from the point of view of macroscopic electromagnetics. Knowing the permeability of a magnetic material tells us all we need to know from the point of view of macroscopic electromagnetics.

Lecture 8 28 Relative Permeability The relative permeability of a magnetic material is the ratio of the permeability of the magnetic material to the permeability of free space The relative permeability of a magnetic material is the ratio of the permeability of the magnetic material to the permeability of free space

Lecture 8 29 Diamagnetic Materials In the absence of applied magnetic field, each atom has net zero magnetic dipole moment. In the absence of applied magnetic field, each atom has net zero magnetic dipole moment. In the presence of an applied magnetic field, the angular velocities of the electronic orbits are changed. In the presence of an applied magnetic field, the angular velocities of the electronic orbits are changed. These induced magnetic dipole moments align themselves opposite to the applied field. These induced magnetic dipole moments align themselves opposite to the applied field. Thus,  m < 0 and  r < 1. Thus,  m < 0 and  r < 1.

Lecture 8 30 Diamagnetic Materials (Cont’d) Usually, diamagnetism is a very miniscule effect in natural materials - that is  r  1. Usually, diamagnetism is a very miniscule effect in natural materials - that is  r  1. Diamagnetism can be a big effect in superconductors and in artificial materials. Diamagnetism can be a big effect in superconductors and in artificial materials. Diamagnetic materials are repelled from either pole of a magnet. Diamagnetic materials are repelled from either pole of a magnet.

Lecture 8 31 Paramagnetic Materials In the absence of applied magnetic field, each atom has net non-zero (but weak) magnetic dipole moment. These magnetic dipoles moments are randomly oriented so that the net macroscopic magnetization is zero. In the absence of applied magnetic field, each atom has net non-zero (but weak) magnetic dipole moment. These magnetic dipoles moments are randomly oriented so that the net macroscopic magnetization is zero. In the presence of an applied magnetic field, the magnetic dipoles align themselves with the applied field so that  m > 0 and  r > 1. In the presence of an applied magnetic field, the magnetic dipoles align themselves with the applied field so that  m > 0 and  r > 1.

Lecture 8 32 Paramagnetic Materials (Cont’d) Usually, paramagnetism is a very miniscule effect in natural materials - that is  r  1. Usually, paramagnetism is a very miniscule effect in natural materials - that is  r  1. Paramagnetic materials are (weakly) attracted to either pole of a magnet. Paramagnetic materials are (weakly) attracted to either pole of a magnet.

Lecture 8 33 Ferromagnetic Materials Ferromagnetic materials include iron, nickel and cobalt and compounds containing these elements. Ferromagnetic materials include iron, nickel and cobalt and compounds containing these elements. In the absence of applied magnetic field, each atom has very strong magnetic dipole moments due to uncompensated electron spins. In the absence of applied magnetic field, each atom has very strong magnetic dipole moments due to uncompensated electron spins. Regions of many atoms with aligned dipole moments called domains form. Regions of many atoms with aligned dipole moments called domains form. In the absence of applied magnetic field, the domains are randomly oriented so that the net macroscopic magnetization is zero. In the absence of applied magnetic field, the domains are randomly oriented so that the net macroscopic magnetization is zero.

Lecture 8 34 Ferromagnetic Materials (Cont’d) In the presence of an applied magnetic field, the domains align themselves with the applied field. In the presence of an applied magnetic field, the domains align themselves with the applied field. The effect is a very strong one with  m >> 0 and  r >> 1. The effect is a very strong one with  m >> 0 and  r >> 1. Ferromagnetic materials are strongly attracted to either pole of a magnet. Ferromagnetic materials are strongly attracted to either pole of a magnet.

Lecture 8 35 Ferromagnetic Materials (Cont’d) In ferromagnetic materials: In ferromagnetic materials: the permeability is much larger than the permeability of free space the permeability is much larger than the permeability of free space the permeability is very non-linear the permeability is very non-linear the permeability depends on the previous history of the material the permeability depends on the previous history of the material

Lecture 8 36 Ferromagnetic Materials (Cont’d) In ferromagnetic materials, the relationship B =  H can be illustrated by means of a magnetization curve (also called hysteresis loop ). In ferromagnetic materials, the relationship B =  H can be illustrated by means of a magnetization curve (also called hysteresis loop ). B H coercivity remanence (retentivity)

Lecture 8 37 Ferromagnetic Materials (Cont’d) Remanence (retentivity) is the value of B when H is zero. Remanence (retentivity) is the value of B when H is zero. Coercivity is the value of H when B is zero. Coercivity is the value of H when B is zero. The hysteresis phenomenon can be used to distinguish between two states. The hysteresis phenomenon can be used to distinguish between two states.

Lecture 8 38 Antiferromagnetic Materials Antiferromagnetic materials include chromium and manganese. Antiferromagnetic materials include chromium and manganese. In antiferromagnetic materials, the magnetic moments of individual atoms are strong, but adjacent atoms align in opposite directions. In antiferromagnetic materials, the magnetic moments of individual atoms are strong, but adjacent atoms align in opposite directions. The macroscopic magnetization of the material is negligible even in the presence of an applied field. The macroscopic magnetization of the material is negligible even in the presence of an applied field.

Lecture 8 39 Ferrimagnetic Materials Ferrimagnetic materials include oxides of iron, nickel, or cobalt. Ferrimagnetic materials include oxides of iron, nickel, or cobalt. The magnetic moments of adjacent atoms are aligned opposite to each other, but there is incomplete cancellation of the moments because they are not equal. The magnetic moments of adjacent atoms are aligned opposite to each other, but there is incomplete cancellation of the moments because they are not equal. Thus, there is a net magnetic moment within a domain. Thus, there is a net magnetic moment within a domain.

Lecture 8 40 Ferrimagnetic Materials (Cont’d) In the absence of applied magnetic field, the domains are randomly oriented so that the net macroscopic magnetization is zero. In the absence of applied magnetic field, the domains are randomly oriented so that the net macroscopic magnetization is zero. In the presence of an applied magnetic field, the domains align themselves with the applied field. In the presence of an applied magnetic field, the domains align themselves with the applied field. The magnetic effects are weaker than in ferromagnetic materials, but are still substantial. The magnetic effects are weaker than in ferromagnetic materials, but are still substantial.

Lecture 8 41 Ferrites Ferrites are the most useful ferrimagnetic materials. Ferrites are the most useful ferrimagnetic materials. Ferrites are ceramic material containing compounds of iron. Ferrites are ceramic material containing compounds of iron. Ferrites are non-conducting magnetic media so eddy current and ohmic losses are less than for ferromagnetic materials. Ferrites are non-conducting magnetic media so eddy current and ohmic losses are less than for ferromagnetic materials. Ferrites are often used as transformer cores at radio frequencies (RF). Ferrites are often used as transformer cores at radio frequencies (RF).

Lecture 8 42 Fundamental Laws of Magnetostatics in Integral Form Gauss’s law for magnetic field Ampere’s law Constitutive relation

Lecture 8 43 Fundamental Laws of Magnetostatics in Differential Form Ampere’s law Gauss’s law for magnetic field Constitutive relation

Lecture 8 44 Fundamental Laws of Magnetostatics The integral forms of the fundamental laws are more general because they apply over regions of space. The differential forms are only valid at a point. The integral forms of the fundamental laws are more general because they apply over regions of space. The differential forms are only valid at a point. From the integral forms of the fundamental laws both the differential equations governing the field within a medium and the boundary conditions at the interface between two media can be derived. From the integral forms of the fundamental laws both the differential equations governing the field within a medium and the boundary conditions at the interface between two media can be derived.

Lecture 8 45 Boundary Conditions Within a homogeneous medium, there are no abrupt changes in H or B. However, at the interface between two different media (having two different values of , it is obvious that one or both of these must change abruptly. Within a homogeneous medium, there are no abrupt changes in H or B. However, at the interface between two different media (having two different values of , it is obvious that one or both of these must change abruptly.

Lecture 8 46 Boundary Conditions (Cont’d) The normal component of a solenoidal vector field is continuous across a material interface: The normal component of a solenoidal vector field is continuous across a material interface: The tangential component of a conservative vector field is continuous across a material interface: The tangential component of a conservative vector field is continuous across a material interface:

Lecture 8 47 Boundary Conditions (Cont’d) The tangential component of H is continuous across a material interface, unless a surface current exists at the interface. The tangential component of H is continuous across a material interface, unless a surface current exists at the interface. When a surface current exists at the interface, the BC becomes When a surface current exists at the interface, the BC becomes

Lecture 8 48 Boundary Conditions (Cont’d) In a perfect conductor, both the electric and magnetic fields must vanish in its interior. Thus, In a perfect conductor, both the electric and magnetic fields must vanish in its interior. Thus, a surface current must exist the magnetic field just outside the perfect conductor must be tangential to it.

Lecture 8 49 Overview of Magnetic Forces and Torques The experimental basis of magnetostatics is the fact that current carrying wires exert forces on one another as described by Ampere’s law of force. The experimental basis of magnetostatics is the fact that current carrying wires exert forces on one another as described by Ampere’s law of force. A number of devices are based on the forces and torques produced by static magnetic fields including DC electric motors and electrical instruments such as voltmeters and ammeters. A number of devices are based on the forces and torques produced by static magnetic fields including DC electric motors and electrical instruments such as voltmeters and ammeters.

Lecture 8 50 Magnetic Forces on Moving Charges The force on a charged particle moving with velocity v in a magnetostatic field characteristic by magnetic flux density B is given by The force on a charged particle moving with velocity v in a magnetostatic field characteristic by magnetic flux density B is given by

Lecture 8 51 Lorentz Force Equation The force on a charged particle moving with velocity v in a region where there exists both a magnetostatic field B and an electrostatic field E is given by The force on a charged particle moving with velocity v in a region where there exists both a magnetostatic field B and an electrostatic field E is given by

Lecture 8 52 Lorentz Force Equation (Cont’d) The Lorentz force equation can be used to obtain the equations of motion for charged particles in various devices including cathode ray tubes (CRTs), microwave klystrons and magnetrons, and cyclotrons. The Lorentz force equation can be used to obtain the equations of motion for charged particles in various devices including cathode ray tubes (CRTs), microwave klystrons and magnetrons, and cyclotrons. The Lorentz force equation also explains the Hall effect in conductors and semiconductors. The Lorentz force equation also explains the Hall effect in conductors and semiconductors.

Lecture 8 53 Magnetic Force on Current- Carrying Conductors When a current carrying wire is placed in a region permeated by a magnetic field, it experiences a net magnetic force given by When a current carrying wire is placed in a region permeated by a magnetic field, it experiences a net magnetic force given by

Lecture 8 54 Torque on a Current Carrying Loop Consider a small rectangular current carrying loop in a region permeated by a magnetic field. Consider a small rectangular current carrying loop in a region permeated by a magnetic field. x y I B F m1 F m2 L W

Lecture 8 55 Torque on a Current Carrying Loop (Cont’d) Assuming a uniform magnetic field, the force on the upper wire is Assuming a uniform magnetic field, the force on the upper wire is The force on the lower wire is The force on the lower wire is

Lecture 8 56 Torque on a Current Carrying Loop (Cont’d) The forces acting on the loop have a tendency to cause the loop to rotate about the x-axis. The forces acting on the loop have a tendency to cause the loop to rotate about the x-axis. The quantitative measure of the tendency of a force to cause or change rotational motion is torque. The quantitative measure of the tendency of a force to cause or change rotational motion is torque.

Lecture 8 57 Torque on a Current Carrying Loop (Cont’d) The torque acting on a body with respect to a reference axis is given by The torque acting on a body with respect to a reference axis is given by distance vector from the reference axis

Lecture 8 58 magnetic dipole moment of loop Torque on a Current Carrying Loop (Cont’d) The torque acting on the loop is The torque acting on the loop is

Lecture 8 59 Torque on a Current Carrying Loop (Cont’d) The torque acting on the loop tries to align the magnetic dipole moment of the loop with the B field The torque acting on the loop tries to align the magnetic dipole moment of the loop with the B field holds in general regardless of loop shape

Lecture 8 60 Energy Stored in Magnetic Field The magnetic energy stored in a region permeated by a magnetic field is given by The magnetic energy stored in a region permeated by a magnetic field is given by

Lecture 8 61 Energy Stored in an Inductor The magnetic energy stored in an inductor is given by The magnetic energy stored in an inductor is given by