1

EEE 498/598EEE 498/598Overview of Electrical Overview of Electrical

EngineeringEngineering

Lecture 8: Magnetostatics: Lecture 8: Magnetostatics: Mutual And Self-inductance; Mutual And Self-inductance; Magnetic Fields In Material Magnetic Fields In Material

Media; Magnetostatic Media; Magnetostatic Boundary Conditions; Boundary Conditions;

Magnetic Forces And TorquesMagnetic Forces And Torques

2Lecture 8

Lecture 8 ObjectivesLecture 8 Objectives

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

3Lecture 8

Flux LinkageFlux Linkage

Consider two magnetically coupled circuitsConsider two magnetically coupled circuits

C1

I1

S1 S2 C2

I2

4Lecture 8

Flux Linkage (Cont’d)Flux Linkage (Cont’d) The magnetic flux produced The magnetic flux produced II11 linking the surface linking the surface SS22 is given by is given by

If the circuit If the circuit CC22 comprises comprises NN22 turns and the circuit turns and the circuit CC11 comprises comprises NN11 turns, then the total turns, then the total flux linkage is given byflux linkage is given by

2

2112

S

sdB

2

2121122112

S

sdBNNNN

5Lecture 8

Mutual InductanceMutual Inductance

The The mutual inductancemutual inductance between two between two circuits is the magnetic flux linkage to circuits is the magnetic flux linkage to one circuit per unit current in the other one circuit per unit current in the other circuit: circuit:

1

1221

1

1212 I

NN

IL

6Lecture 8

Neumann Formula for Neumann Formula for Mutual InductanceMutual Inductance

2

2

211

21

211

21

1

1221

1

1212

C

S

ldAI

NN

sdBI

NN

I

NN

IL

7Lecture 8

Neumann Formula for Neumann Formula for Mutual Inductance Mutual Inductance

(Cont’d)(Cont’d)

1 12

1101 4 C R

ldIA

1 2

2

12

21210

211

2112

4 C C

C

R

ldldNN

ldAI

NNL

8Lecture 8

Neumann Formula for Neumann Formula for Mutual Inductance Mutual Inductance

(Cont’d)(Cont’d) The Neumann formula for The Neumann formula for

mutual inductance tells us thatmutual inductance tells us that LL1212 = L = L2121

the mutual inductance depends the mutual inductance depends only on the geometry of the only on the geometry of the conductors and not on the currentconductors and not on the current

9Lecture 8

Self InductanceSelf Inductance Self inductanceSelf inductance is a special case of mutual is a special case of mutual

inductance.inductance. The The self inductanceself inductance of a circuit is the ratio of the self of a circuit is the ratio of the self

magnetic flux linkage to the current producing it: magnetic flux linkage to the current producing it:

1

112

1

1

1111 I

N

IL

10Lecture 8

Self Inductance (Cont’d)Self Inductance (Cont’d)

For an isolated circuit, we call For an isolated circuit, we call the self inductance, the self inductance, inductanceinductance,, and evaluate it usingand evaluate it using

I

N

IL

2

11Lecture 8

Generation of Magnetic Generation of Magnetic FieldField

I

I

iron core

air gap with constant B field

N

S

12Lecture 8

Equivalent of a Magnetic Equivalent of a Magnetic DipoleDipole

I

N

S

• Magnetic dipole can be viewed as a pair of magnetic charges by analogy with electric dipole.

13Lecture 8

Forces Exerted on a Forces Exerted on a Magnetic Dipole in a Magnetic Dipole in a

Magnetic FieldMagnetic Field

N

S B

14Lecture 8

Current Loops (Magnetic Current Loops (Magnetic Dipoles) in AtomsDipoles) in Atoms

Electron orbiting nucleusElectron orbiting nucleus Electron spinElectron spin Nuclear spinNuclear spin negligible

A complete understanding of these atomic mechanisms requires application of quantum mechanics.

15Lecture 8

Current Loops (Magnetic Current Loops (Magnetic Dipoles) in Atoms Dipoles) in Atoms

(Cont’d)(Cont’d) In the absence of an applied In the absence of an applied

magnetic field, the infinitesimal magnetic field, the infinitesimal magnetic dipoles magnetic dipoles in most materialsin most materials are randomly oriented, giving a net are randomly oriented, giving a net macroscopic magnetization of zero.macroscopic magnetization of zero.

When an external magnetic field is When an external magnetic field is applied, the magnetic dipoles have a applied, the magnetic dipoles have a tendency to align themselves with tendency to align themselves with the applied magnetic field.the applied magnetic field.

16Lecture 8

Magnetized MaterialsMagnetized Materials

A material is said to be A material is said to be magnetized magnetized when induced magnetic dipoles when induced magnetic dipoles are present.are present.

The presence of the induced The presence of the induced magnetic dipoles modifies the magnetic dipoles modifies the magnetic field both inside and magnetic field both inside and outside of the magnetized outside of the magnetized material.material.

17Lecture 8

Permanent MagnetsPermanent Magnets

Most materials lose their Most materials lose their magnetization when the external magnetization when the external magnetic field is removed.magnetic field is removed.

A material that remains A material that remains magnetized in the absence of an magnetized in the absence of an applied magnetic field is called a applied magnetic field is called a permanent magnetpermanent magnet..

18Lecture 8

Magnetization VectorMagnetization Vector

The The magnetizationmagnetization oror net magnetic net magnetic dipole moment per unit volumedipole moment per unit volume is given is given byby

mNM average magnetic dipole moment [Am2]Number of

dipoles per unit volume [m-3]

[A/m]

19Lecture 8

Magnetic MaterialsMagnetic Materials

The effect of an applied electric field on a The effect of an applied electric field on a magneticmagnetic material is to create a net magnetic material is to create a net magnetic dipole moment per unit volume dipole moment per unit volume MM..

The dipole moment distribution sets up The dipole moment distribution sets up induced secondary fields:induced secondary fields:

indapp BBB

Total field Field in free space due to sources

Field due to induced magnetic dipoles

20Lecture 8

Volume and Surface Volume and Surface Magnetization CurrentsMagnetization Currents

A magnetized material may be A magnetized material may be represented as an equivalent volume represented as an equivalent volume ((JJmm) and surface () and surface (JJsmsm) magnetization ) magnetization currents.currents.

These charge distributions are related These charge distributions are related to the magnetization vector byto the magnetization vector by

nsm

m

aMJ

MJ

ˆ

21Lecture 8

Volume and Surface Volume and Surface Magnetization Currents Magnetization Currents

(Cont’d)(Cont’d) Magnetization currents are equivalent Magnetization currents are equivalent

currents that account for the effect of the currents that account for the effect of the magnetized material, and are analogous to magnetized material, and are analogous to equivalent volume and surface polarization equivalent volume and surface polarization charge densities in a polarized dielectric.charge densities in a polarized dielectric.

If the magnetization vector is constant If the magnetization vector is constant throughout a magnetized material, then throughout a magnetized material, then the volume magnetization current density the volume magnetization current density is zero, but the surface magnetization is zero, but the surface magnetization current is nonzero.current is nonzero.

22Lecture 8

Ampere’s Law in Ampere’s Law in Magnetic MediaMagnetic Media

Ampere’s law in differential form Ampere’s law in differential form in free in free spacespace::

Ampere’s law in differential form Ampere’s law in differential form in a in a magnetized materialmagnetized material::

JB 0

mJJB 0

23Lecture 8

Magnetic Field Magnetic Field IntensityIntensity

• define the magnetic field intensity as

JMB

JMB

MJJJB m

0

00

000

24Lecture 8

General Forms of General Forms of Ampere’s LawAmpere’s Law

The general form of Ampere’s law in differential form becomesThe general form of Ampere’s law in differential form becomes

The general form of Ampere’s law in integral form becomesThe general form of Ampere’s law in integral form becomes

JH

encl

SC

IsdJldH

25Lecture 8

Permeability ConceptPermeability Concept

For some materials, the For some materials, the net net magnetic dipole moment per unit volumemagnetic dipole moment per unit volume is proportional to the is proportional to the HH field field

HM mmagnetic

susceptibility(dimensionless)

• the units of both M and H are A/m.

26Lecture 8

Permeability Concept Permeability Concept (Cont’d)(Cont’d)

Assuming thatAssuming that

we havewe have

The parameter The parameter is the is the permeabilitypermeability of the material. of the material.

HM m

HHMHB m 100

27Lecture 8

Permeability Concept Permeability Concept (Cont’d)(Cont’d)

The concepts of magnetization and magnetic The concepts of magnetization and magnetic dipole moment distribution are introduced to dipole moment distribution are introduced to relate microscopic phenomena to the relate microscopic phenomena to the macroscopic fields.macroscopic fields.

The introduction of The introduction of permeabilitypermeability eliminates the eliminates the need for us to explicitly consider microscopic need for us to explicitly consider microscopic effects.effects.

Knowing the Knowing the permeabilitypermeability of a magnetic material of a magnetic material tells us all we need to know from the point of tells us all we need to know from the point of view of macroscopic electromagnetics.view of macroscopic electromagnetics.

28Lecture 8

Relative PermeabilityRelative Permeability

The The relative permeabilityrelative permeability of a of a magnetic material is the ratio of magnetic material is the ratio of the permeability of the magnetic the permeability of the magnetic material to the permeability of material to the permeability of free spacefree space

0 r

29Lecture 8

Diamagnetic MaterialsDiamagnetic Materials

In the absence of applied magnetic field, In the absence of applied magnetic field, each atom has net zero magnetic dipole each atom has net zero magnetic dipole moment.moment.

In the presence of an applied magnetic In the presence of an applied magnetic field, the angular velocities of the field, the angular velocities of the electronic orbits are changed.electronic orbits are changed.

These induced magnetic dipole moments These induced magnetic dipole moments align themselves align themselves oppositeopposite to the applied to the applied field.field.

Thus, Thus, mm < 0 and < 0 and rr < 1. < 1.

30Lecture 8

Diamagnetic Materials Diamagnetic Materials (Cont’d)(Cont’d)

Usually, diamagnetism is a very Usually, diamagnetism is a very miniscule effect in natural miniscule effect in natural materials - that is materials - that is rr 1. 1.

Diamagnetism can be a big effect in Diamagnetism can be a big effect in superconductorssuperconductors and in and in artificial materialsartificial materials..

Diamagnetic materials are repelled Diamagnetic materials are repelled from either pole of a magnet.from either pole of a magnet.

31Lecture 8

Paramagnetic MaterialsParamagnetic Materials

In the absence of applied magnetic field, In the absence of applied magnetic field, each atom has net non-zero (but weak) each atom has net non-zero (but weak) magnetic dipole moment. These magnetic dipole moment. These magnetic dipoles moments are randomly magnetic dipoles moments are randomly oriented so that the net macroscopic oriented so that the net macroscopic magnetization is zero.magnetization is zero.

In the presence of an applied magnetic In the presence of an applied magnetic field, the magnetic dipoles align field, the magnetic dipoles align themselves with the applied field so that themselves with the applied field so that mm > 0 and > 0 and rr > 1. > 1.

32Lecture 8

Paramagnetic Materials Paramagnetic Materials (Cont’d)(Cont’d)

Usually, paramagnetism is a Usually, paramagnetism is a very miniscule effect in natural very miniscule effect in natural materials - that is materials - that is rr 1. 1.

Paramagnetic materials are Paramagnetic materials are (weakly) attracted to either pole (weakly) attracted to either pole of a magnet.of a magnet.

33Lecture 8

Ferromagnetic Ferromagnetic MaterialsMaterials

Ferromagnetic materials include iron, Ferromagnetic materials include iron, nickel and cobalt and compounds nickel and cobalt and compounds containing these elements.containing these elements.

In the absence of applied magnetic field, In the absence of applied magnetic field, each atom has very strong magnetic dipole each atom has very strong magnetic dipole moments due to uncompensated electron moments due to uncompensated electron spins.spins.

Regions of many atoms with aligned dipole Regions of many atoms with aligned dipole moments called moments called domainsdomains form. form.

In the absence of applied magnetic field, the In the absence of applied magnetic field, the domainsdomains are randomly oriented so that the are randomly oriented so that the net macroscopic magnetization is zero.net macroscopic magnetization is zero.

34Lecture 8

Ferromagnetic Materials Ferromagnetic Materials (Cont’d)(Cont’d)

In the presence of an applied In the presence of an applied magnetic field, the domains align magnetic field, the domains align themselves with the applied field.themselves with the applied field.

The effect is a very strong one The effect is a very strong one with with mm >> 0 and >> 0 and rr >> 1. >> 1.

Ferromagnetic materials are Ferromagnetic materials are strongly attracted to either pole of strongly attracted to either pole of a magnet.a magnet.

35Lecture 8

Ferromagnetic Materials Ferromagnetic Materials (Cont’d)(Cont’d)

In ferromagnetic materials:In ferromagnetic materials: the permeability is much larger the permeability is much larger

than the permeability of free spacethan the permeability of free space the permeability is very non-linearthe permeability is very non-linear the permeability depends on the the permeability depends on the

previous history of the materialprevious history of the material

36Lecture 8

Ferromagnetic Ferromagnetic Materials (Cont’d)Materials (Cont’d) In ferromagnetic materials, the relationship In ferromagnetic materials, the relationship

BB = = HH can be illustrated by means of a can be illustrated by means of a magnetization curvemagnetization curve (also called (also called hysteresis loophysteresis loop).).

B

H

coercivity

remanence(retentivity)

37Lecture 8

Ferromagnetic Materials Ferromagnetic Materials (Cont’d)(Cont’d)

Remanence (retentivity)Remanence (retentivity) is the value is the value of of BB when when HH is zero. is zero.

CoercivityCoercivity is the value of is the value of HH when when BB is zero.is zero.

The The hysteresishysteresis phenomenon can be phenomenon can be used to distinguish between two used to distinguish between two states.states.

38Lecture 8

Antiferromagnetic Antiferromagnetic MaterialsMaterials

Antiferromagnetic materials include Antiferromagnetic materials include chromium and manganese.chromium and manganese.

In antiferromagnetic materials, the In antiferromagnetic materials, the magnetic moments of individual magnetic moments of individual atoms are strong, but adjacent atoms are strong, but adjacent atoms align in opposite directions.atoms align in opposite directions.

The macroscopic magnetization of The macroscopic magnetization of the material is negligible even in the the material is negligible even in the presence of an applied field.presence of an applied field.

39Lecture 8

Ferrimagnetic MaterialsFerrimagnetic Materials

Ferrimagnetic materials include Ferrimagnetic materials include oxides of iron, nickel, or cobalt.oxides of iron, nickel, or cobalt.

The magnetic moments of adjacent The magnetic moments of adjacent atoms are aligned opposite to each atoms are aligned opposite to each other, but there is incomplete other, but there is incomplete cancellation of the moments because cancellation of the moments because they are not equal.they are not equal.

Thus, there is a net magnetic Thus, there is a net magnetic moment within a domain.moment within a domain.

40Lecture 8

Ferrimagnetic Materials Ferrimagnetic Materials (Cont’d)(Cont’d)

In the absence of applied magnetic field, In the absence of applied magnetic field, the the domainsdomains are randomly oriented so that are randomly oriented so that the net macroscopic magnetization is the net macroscopic magnetization is zero.zero.

In the presence of an applied magnetic In the presence of an applied magnetic field, the domains align themselves with field, the domains align themselves with the applied field.the applied field.

The magnetic effects are weaker than in The magnetic effects are weaker than in ferromagnetic materials, but are still ferromagnetic materials, but are still substantial.substantial.

41Lecture 8

FerritesFerrites Ferrites are the most useful Ferrites are the most useful

ferrimagnetic materials.ferrimagnetic materials. Ferrites are ceramic material containing Ferrites are ceramic material containing

compounds of iron. compounds of iron. Ferrites are non-conducting magnetic Ferrites are non-conducting magnetic

media so eddy current and ohmic losses media so eddy current and ohmic losses are less than for ferromagnetic materials.are less than for ferromagnetic materials.

Ferrites are often used as transformer Ferrites are often used as transformer cores at radio frequencies (RF).cores at radio frequencies (RF).

42Lecture 8

Fundamental Laws of Fundamental Laws of Magnetostatics in Magnetostatics in

Integral FormIntegral Form

0

S

SC

sdB

sdJldH

HB

Gauss’s law for magnetic field

Ampere’s law

Constitutive relation

43Lecture 8

Fundamental Laws of Fundamental Laws of Magnetostatics in Magnetostatics in Differential FormDifferential Form

0

B

JH

HB

Ampere’s law

Gauss’s law for magnetic field

Constitutive relation

44Lecture 8

Fundamental Laws of Fundamental Laws of MagnetostaticsMagnetostatics

The integral forms of the fundamental The integral forms of the fundamental laws are more general because they apply laws are more general because they apply over regions of space. The differential over regions of space. The differential forms are only valid at a point.forms are only valid at a point.

From the integral forms of the From the integral forms of the fundamental laws both the differential fundamental laws both the differential equations governing the field within a equations governing the field within a medium and the boundary conditions at medium and the boundary conditions at the interface between two media can be the interface between two media can be derived.derived.

45Lecture 8

Boundary ConditionsBoundary Conditions

1

2na

Within a Within a homogeneous homogeneous medium, there are no medium, there are no abrupt changes in abrupt changes in HH or or BB. However, at the . However, at the interface between two interface between two different media different media (having two different (having two different values of values of , it is , it is obvious that one or obvious that one or both of these must both of these must change abruptly.change abruptly.

46Lecture 8

Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

The normal component of a The normal component of a solenoidalsolenoidal vector field is continuous across a vector field is continuous across a material interface:material interface:

The tangential component of a The tangential component of a conservativeconservative vector field is continuous vector field is continuous across a material interface:across a material interface:

nn BB 21

0,21 stt JHH

47Lecture 8

Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

The tangential component of The tangential component of HH is is continuous across a material continuous across a material interface, unless a surface interface, unless a surface current exists at the interface.current exists at the interface.

When a surface current exists at When a surface current exists at the interface, the BC becomesthe interface, the BC becomes

sn JHHa 21ˆ

48Lecture 8

Boundary Conditions Boundary Conditions (Cont’d)(Cont’d)

In a perfect conductor, both the In a perfect conductor, both the electric and magnetic fields electric and magnetic fields must vanish in its interior. Thus, must vanish in its interior. Thus,

sn

n

JHa

B

ˆ

0• a surface current must exist• the magnetic field just outside the perfect conductor must be tangential to it.

49Lecture 8

Overview of Magnetic Overview of Magnetic Forces and TorquesForces and Torques

The experimental basis of The experimental basis of magnetostatics is the fact that current magnetostatics is the fact that current carrying wires exert forces on one carrying wires exert forces on one another as described by Ampere’s law of another as described by Ampere’s law of force.force.

A number of devices are based on the A number of devices are based on the forces and torques produced by static forces and torques produced by static magnetic fields including DC electric magnetic fields including DC electric motors and electrical instruments such motors and electrical instruments such as voltmeters and ammeters.as voltmeters and ammeters.

50Lecture 8

Magnetic Forces on Magnetic Forces on Moving ChargesMoving Charges

The force on a charged particle The force on a charged particle moving with velocity moving with velocity vv in a in a magnetostatic field magnetostatic field characteristic by magnetic flux characteristic by magnetic flux density density BB is given by is given by

BvqF m

51Lecture 8

Lorentz Force EquationLorentz Force Equation

The force on a charged particle The force on a charged particle moving with velocity moving with velocity vv in a in a region where there exists both a region where there exists both a magnetostatic field magnetostatic field BB and an and an electrostatic field electrostatic field EE is given by is given by

BvEqF

52Lecture 8

Lorentz Force Equation Lorentz Force Equation (Cont’d)(Cont’d)

The Lorentz force equation can be The Lorentz force equation can be used to obtain the equations of used to obtain the equations of motion for charged particles in motion for charged particles in various devices including cathode ray various devices including cathode ray tubes (CRTs), microwave klystrons tubes (CRTs), microwave klystrons and magnetrons, and cyclotrons.and magnetrons, and cyclotrons.

The Lorentz force equation also The Lorentz force equation also explains the explains the Hall effectHall effect in conductors in conductors and semiconductors.and semiconductors.

53Lecture 8

Magnetic Force on Magnetic Force on Current-Carrying Current-Carrying

Conductors Conductors When a current carrying wire is placed When a current carrying wire is placed

in a region permeated by a magnetic in a region permeated by a magnetic field, it experiences a net magnetic field, it experiences a net magnetic force given byforce given by

BlIdFC

m

54Lecture 8

Torque on a Current Torque on a Current Carrying LoopCarrying Loop

Consider a small Consider a small rectangular rectangular current carrying current carrying loop in a region loop in a region permeated by a permeated by a magnetic field.magnetic field.

x

y

I

B Fm1

Fm2

L

W

55Lecture 8

Torque on a Current Torque on a Current Carrying Loop (Cont’d)Carrying Loop (Cont’d)

Assuming a uniform magnetic field, the force on the upper wire isAssuming a uniform magnetic field, the force on the upper wire is

The force on the lower wire isThe force on the lower wire is

ILBaF zm ˆ1

ILBaF zm ˆ2

56Lecture 8

Torque on a Current Torque on a Current Carrying Loop (Cont’d)Carrying Loop (Cont’d)

The forces acting on the loop The forces acting on the loop have a tendency to cause the have a tendency to cause the loop to rotate about the x-axis.loop to rotate about the x-axis.

The quantitative measure of the The quantitative measure of the tendency of a force to cause or tendency of a force to cause or change rotational motion is change rotational motion is torquetorque..

57Lecture 8

FrT

Torque on a Current Torque on a Current Carrying Loop (Cont’d)Carrying Loop (Cont’d)

The The torquetorque acting on a body with acting on a body with respect to a reference axis is respect to a reference axis is given bygiven by

distance vector from the reference axis

58Lecture 8

BILWaILWBa

FW

aFW

aT

zx

mymy

ˆˆ2

ˆ2

ˆ 21

magnetic dipole moment of loop

Torque on a Current Torque on a Current Carrying Loop (Cont’d)Carrying Loop (Cont’d)

The torque acting on the loop isThe torque acting on the loop is

59Lecture 8

BmT

Torque on a Current Torque on a Current Carrying Loop (Cont’d)Carrying Loop (Cont’d)

The torque acting on the loop The torque acting on the loop tries to align the magnetic tries to align the magnetic dipole moment of the loop with dipole moment of the loop with the B fieldthe B field

holds in general regardless of loop shape

60Lecture 8

Energy Stored in Energy Stored in Magnetic FieldMagnetic Field

The magnetic energy stored in a The magnetic energy stored in a region permeated by a magnetic region permeated by a magnetic field is given byfield is given by

dvHdvHBWVV

m 2

2

1

2

1

61Lecture 8

Energy Stored in an Energy Stored in an InductorInductor

The magnetic energy stored in The magnetic energy stored in an inductor is given byan inductor is given by

2

2

1LIWm