1 EEE 498/598 Overview of Electrical Engineering Lecture 7: Magnetostatics: Amperes Law Of Force;...

1

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

EngineeringEngineeringLecture 7: Magnetostatics: Lecture 7: Magnetostatics:

Ampere’s Law Of Force; Magnetic Ampere’s Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-Flux Density; Lorentz Force; Biot-

savart Law; Applications Of savart Law; Applications Of Ampere’s Law In Integral Form; Ampere’s Law In Integral Form;

Vector Magnetic Potential; Vector Magnetic Potential; Magnetic Dipole; Magnetic FluxMagnetic Dipole; Magnetic Flux

2Lecture 7

Lecture 7 ObjectivesLecture 7 Objectives

To begin our study of To begin our study of magnetostatics with Ampere’s law magnetostatics with Ampere’s law of force; magnetic flux density; of force; magnetic flux density; Lorentz force; Biot-Savart law; Lorentz force; Biot-Savart law; applications of Ampere’s law in applications of Ampere’s law in integral form; vector magnetic integral form; vector magnetic potential; magnetic dipole; and potential; magnetic dipole; and magnetic flux. magnetic flux.

3Lecture 7

Overview of ElectromagneticsOverview of Electromagnetics

Maxwell’sequations

Fundamental laws of classical electromagnetics

Special cases

Electro-statics

Magneto-statics

Electro-magnetic

waves

Kirchoff’s Laws

Statics: 0t

d

Geometric Optics

TransmissionLine

Theory

CircuitTheory

Input from other

disciplines

4Lecture 7

MagnetostaticsMagnetostatics

MagnetostaticsMagnetostatics is the branch of is the branch of electromagnetics dealing with the electromagnetics dealing with the effects of electric charges in steady effects of electric charges in steady motion (i.e, steady current or DC).motion (i.e, steady current or DC).

The fundamental law of The fundamental law of magnetostaticsmagnetostatics is is Ampere’s law of forceAmpere’s law of force..

Ampere’s law of forceAmpere’s law of force is analogous to is analogous to Coulomb’s lawCoulomb’s law in electrostatics. in electrostatics.

5Lecture 7

Magnetostatics (Cont’d)Magnetostatics (Cont’d)

In magnetostatics, the magnetic In magnetostatics, the magnetic field is produced by steady field is produced by steady currents. The magnetostatic currents. The magnetostatic field does not allow forfield does not allow for inductive coupling between inductive coupling between

circuitscircuits coupling between electric and coupling between electric and

magnetic fieldsmagnetic fields

6Lecture 7

Ampere’s Law of ForceAmpere’s Law of Force

Ampere’s law of forceAmpere’s law of force is the “law of action” is the “law of action” between current carrying circuits.between current carrying circuits.

Ampere’s law of forceAmpere’s law of force gives the magnetic gives the magnetic force between two force between two current carrying circuitscurrent carrying circuits in an otherwise empty universe.in an otherwise empty universe.

Ampere’s law of force involves Ampere’s law of force involves complete circuits since current must complete circuits since current must flow in closed loops. flow in closed loops.

7Lecture 7

Ampere’s Law of Force Ampere’s Law of Force (Cont’d)(Cont’d)

Experimental Experimental facts:facts: Two parallel wires Two parallel wires

carrying current in carrying current in the same direction the same direction attract.attract.

Two parallel wires Two parallel wires carrying current in carrying current in the opposite the opposite directions repel.directions repel.

I1 I2

F12F21

I1 I2

F12F21

8Lecture 7


Experimental Experimental facts:facts: A short current-A short current-

carrying wire carrying wire oriented oriented perpendicular to a perpendicular to a long current-long current-carrying wire carrying wire experiences no experiences no force.force.

I1

F12 = 0

I2

9Lecture 7


Experimental facts:Experimental facts: The magnitude of the force is The magnitude of the force is

inversely proportional to the inversely proportional to the distance squared.distance squared.

The magnitude of the force is The magnitude of the force is proportional to the product of the proportional to the product of the currents carried by the two wires.currents carried by the two wires.

10Lecture 7


The direction of the force established The direction of the force established by the experimental facts can be by the experimental facts can be mathematically represented bymathematically represented by

1212

ˆˆˆˆ 12 RF aaaa

unit vector in direction of force on

I2 due to I1

unit vector in direction of I2 from I1

unit vector in direction of current I1

unit vector in direction of current I2

11Lecture 7


The force acting on a current The force acting on a current element element II2 2 ddll22 by a current element by a current element II1 1

ddll11 is given by is given by 2

12

1122012

12ˆ

4 R

aldIldIF R

Permeability of free space0 = 4 10-7 F/m

12Lecture 7


The total force acting on a circuit The total force acting on a circuit CC22 having a current having a current II22 by a circuit by a circuit CC11 having current having current II11 is given by is given by

2 1

12

212

1221012

ˆ

4 C C

R

R

aldldIIF

13Lecture 7


The force on The force on CC11 due to due to CC22 is equal is equal in magnitude but opposite in in magnitude but opposite in direction to the force on direction to the force on CC22 due due to to CC11..

1221 FF

14Lecture 7

Magnetic Flux DensityMagnetic Flux Density

Ampere’s force law describes an “action at Ampere’s force law describes an “action at a distance” analogous to Coulomb’s law.a distance” analogous to Coulomb’s law.

In Coulomb’s law, it was useful to introduce In Coulomb’s law, it was useful to introduce the concept of an the concept of an electric fieldelectric field to describe the to describe the interaction between the charges.interaction between the charges.

In Ampere’s law, we can define an In Ampere’s law, we can define an appropriate field that may be regarded as appropriate field that may be regarded as the means by which currents exert force on the means by which currents exert force on each other.each other.

15Lecture 7

Magnetic Flux Density Magnetic Flux Density (Cont’d)(Cont’d)

The The magnetic flux densitymagnetic flux density can be can be introduced by writingintroduced by writing

2

2 1

12

1222

212

1102212

ˆ

4

C

C C

R

BldI

R

aldIldIF

16Lecture 7


wherewhere

1

12

212

11012

ˆ

4 C

R

R

aldIB

the magnetic flux density at the location of dl2 due to the current I1 in C1

17Lecture 7


Suppose that an infinitesimal Suppose that an infinitesimal current element current element IdIdll is immersed in a is immersed in a region of magnetic flux density region of magnetic flux density BB. . The current element experiences a The current element experiences a force force ddFF given by given by

BlIdFd

18Lecture 7


The total force exerted on a circuit The total force exerted on a circuit CC carrying current carrying current II that is immersed that is immersed in a magnetic flux density in a magnetic flux density BB isis given given by by

C

BldIF

19Lecture 7

Force on a Moving Force on a Moving ChargeCharge

A moving point charge placed in a A moving point charge placed in a magnetic field experiences a force magnetic field experiences a force given bygiven by

BvQ

The force experienced by the point charge is in the direction into the paper.

BvQF m vQlId

20Lecture 7

Lorentz ForceLorentz Force If a point charge is moving in a region If a point charge is moving in a region

where both electric and magnetic fields where both electric and magnetic fields exist, then it experiences a total force exist, then it experiences a total force given bygiven by

The Lorentz force equation is useful for The Lorentz force equation is useful for determining the equation of motion for determining the equation of motion for electrons in electromagnetic deflection electrons in electromagnetic deflection systems such as CRTs. systems such as CRTs.

BvEqFFF me

21Lecture 7

The Biot-Savart LawThe Biot-Savart Law

The The Biot-Savart lawBiot-Savart law gives us the gives us the BB--field arising at a specified point field arising at a specified point PP from a given current from a given current distribution.distribution.

It is a fundamental law of It is a fundamental law of magnetostatics.magnetostatics.

22Lecture 7

The Biot-Savart Law The Biot-Savart Law (Cont’d)(Cont’d)

The contribution to the The contribution to the BB-field at a -field at a point point PP from a differential current from a differential current element element IdIdll’’ is given by is given by

30

4)(

R

RldIrBd

23Lecture 7


lId

PR

r r

24Lecture 7


The total magnetic flux at the point The total magnetic flux at the point PP due to the entire circuit due to the entire circuit CC is given by is given by

C R

RldIrB

30

4)(

25Lecture 7

Types of Current Types of Current DistributionsDistributions

Line current densityLine current density ((currentcurrent) - occurs for ) - occurs for infinitesimally thin filamentary bodies infinitesimally thin filamentary bodies (i.e., wires of negligible diameter).(i.e., wires of negligible diameter).

Surface current densitySurface current density ( (current per unit current per unit widthwidth) - occurs when body is perfectly ) - occurs when body is perfectly conductingconducting..

Volume current densityVolume current density ((current per unit current per unit cross sectional areacross sectional area) - most general.) - most general.

26Lecture 7

The Biot-Savart Law The Biot-Savart Law (Cont’d)(Cont’d) For a surface distribution of current, For a surface distribution of current,

the B-S law becomesthe B-S law becomes

For a volume distribution of current, For a volume distribution of current, the B-S law becomesthe B-S law becomes

S

s sdR

RrJrB

30

4)(

V

vdR

RrJrB

30

4)(

27Lecture 7

Ampere’s Circuital Law Ampere’s Circuital Law in Integral Formin Integral Form

Ampere’s Circuital LawAmpere’s Circuital Law in integral form states in integral form states that “the circulation of the magnetic flux that “the circulation of the magnetic flux density in free space is proportional to the density in free space is proportional to the total current through the surface bounding total current through the surface bounding the path over which the circulation is the path over which the circulation is computed.”computed.”

encl

C

IldB 0

28Lecture 7

Ampere’s Circuital Law Ampere’s Circuital Law in Integral Form (Cont’d)in Integral Form (Cont’d)

By convention, dS is taken to be in the

direction defined by the right-hand rule applied

to dl.

S

encl sdJI

Since volume currentdensity is the most

general, we can write Iencl in this way.

S

dl

dS

29Lecture 7

Ampere’s Law and Ampere’s Law and Gauss’s LawGauss’s Law

Just as Gauss’s law follows from Just as Gauss’s law follows from Coulomb’s law, so Ampere’s circuital Coulomb’s law, so Ampere’s circuital law follows from Ampere’s force law.law follows from Ampere’s force law.

Just as Gauss’s law can be used to Just as Gauss’s law can be used to derive the electrostatic field from derive the electrostatic field from symmetric charge distributions, so symmetric charge distributions, so Ampere’s law can be used to derive Ampere’s law can be used to derive the magnetostatic field from the magnetostatic field from symmetric current distributions.symmetric current distributions.

30Lecture 7

Applications of Ampere’s Applications of Ampere’s LawLaw

Ampere’s law in integral form is an Ampere’s law in integral form is an integral equationintegral equation for the unknown magnetic for the unknown magnetic flux density resulting from a given flux density resulting from a given current distribution.current distribution.

encl

C

IldB 0known

unknown

31Lecture 7

Applications of Ampere’s Applications of Ampere’s Law (Cont’d)Law (Cont’d)

In general, solutions to In general, solutions to integral integral equationsequations must be obtained using must be obtained using numerical techniques.numerical techniques.

However, for certain symmetric However, for certain symmetric current distributions closed form current distributions closed form solutions to Ampere’s law can be solutions to Ampere’s law can be obtained.obtained.

32Lecture 7

Applications of Ampere’s Applications of Ampere’s Law (Cont’d)Law (Cont’d)

Closed form solution to Ampere’s Closed form solution to Ampere’s law relies on our ability to law relies on our ability to construct a suitable family of construct a suitable family of Amperian pathsAmperian paths..

An An Amperian pathAmperian path is a closed is a closed contour to which the magnetic contour to which the magnetic flux density is tangential and over flux density is tangential and over which equal to a constant value.which equal to a constant value.

33Lecture 7

Magnetic Flux Density of an Magnetic Flux Density of an Infinite Line Current Using Infinite Line Current Using

Ampere’s LawAmpere’s LawConsider an infinite line current along the z-axis Consider an infinite line current along the z-axis

carrying current in the +z-direction:carrying current in the +z-direction:

I

34Lecture 7


Ampere’s Law (Cont’d)Ampere’s Law (Cont’d)

(1) Assume from symmetry and the (1) Assume from symmetry and the right-hand rule the form of the fieldright-hand rule the form of the field

(2) Construct a family of Amperian (2) Construct a family of Amperian pathspaths

BaB ˆ

circles of radius where

35Lecture 7



(3) Evaluate the total current passing through (3) Evaluate the total current passing through the surface bounded by the Amperian paththe surface bounded by the Amperian path

S

encl sdJI

36Lecture 7



Amperian path

IIencl

I

x

y

37Lecture 7



(4) For each Amperian path, evaluate (4) For each Amperian path, evaluate the integralthe integral

BlldBC

2BldBC

magnitude of Bon Amperian

path.

lengthof Amperian

path.

38Lecture 7



(5) Solve for (5) Solve for BB on each Amperian pathon each Amperian path

l

IB encl0

2ˆ 0IaB

39Lecture 7

Applying Stokes’s Applying Stokes’s Theorem to Ampere’s Theorem to Ampere’s

LawLaw

S

encl

SC

sdJI

sdBldB

00

Because the above must hold for any surface S, we must have

JB 0 Differential formof Ampere’s Law

40Lecture 7

Ampere’s Law in Ampere’s Law in Differential FormDifferential Form

Ampere’s law in differential form Ampere’s law in differential form implies that the implies that the BB-field is -field is conservativeconservative outside of regions outside of regions where current is flowing.where current is flowing.

41Lecture 7

Fundamental Postulates Fundamental Postulates of Magnetostaticsof Magnetostatics

Ampere’s law in differential formAmpere’s law in differential form

No isolated magnetic chargesNo isolated magnetic charges

JB 0

0 B B is solenoidal

42Lecture 7

Vector Magnetic Vector Magnetic PotentialPotential

Vector identity: “the divergence of the Vector identity: “the divergence of the curl of any vector field is identically curl of any vector field is identically zero.”zero.”

Corollary: “If the divergence of a Corollary: “If the divergence of a vector field is identically zero, then vector field is identically zero, then that vector field can be written as the that vector field can be written as the curl of some vector potential field.”curl of some vector potential field.”

0 A

43Lecture 7

Vector Magnetic Vector Magnetic Potential (Cont’d)Potential (Cont’d)

Since the magnetic flux density Since the magnetic flux density is is solenoidalsolenoidal, it can be written as , it can be written as the curl of a vector field called the curl of a vector field called the the vector magnetic potentialvector magnetic potential..

ABB 0

44Lecture 7


The general form of the B-S law isThe general form of the B-S law is

Note thatNote that

V

vdR

RrJrB

30

4)(

3

1

R

R

R

45Lecture 7


Furthermore, note that the Furthermore, note that the deldel operator operator operates only on the unprimed operates only on the unprimed coordinates so thatcoordinates so that

R

rJ

rJR

RrJ

R

RrJ

1

13

46Lecture 7


Hence, we haveHence, we have

vd

R

rJrB

V

4

0

rA

47Lecture 7


For a surface distribution of current, For a surface distribution of current, the vector magnetic potential is given the vector magnetic potential is given byby

For a line current, the vector magnetic For a line current, the vector magnetic potential is given bypotential is given by

sd

R

rJrA

S

s

4

)( 0

L R

ldIrA

4

)( 0

48Lecture 7


In some cases, it is easier to In some cases, it is easier to evaluate the vector magnetic evaluate the vector magnetic potential and then use potential and then use BB = = AA, , rather than to use the B-S rather than to use the B-S law to directly findlaw to directly find BB..

In some ways, the vector In some ways, the vector magnetic potential magnetic potential AA is analogous is analogous to the scalar electric potential to the scalar electric potential VV..

49Lecture 7


In classical physics, the vector In classical physics, the vector magnetic potential is viewed as magnetic potential is viewed as an auxiliary function with no an auxiliary function with no physical meaning.physical meaning.

However, there are phenomena in However, there are phenomena in quantum mechanics that suggest quantum mechanics that suggest that the vector magnetic potential that the vector magnetic potential is a real (i.e., measurable) field. is a real (i.e., measurable) field.

50Lecture 7

Magnetic DipoleMagnetic Dipole

A A magnetic dipolemagnetic dipole comprises a small comprises a small current carrying loop.current carrying loop.

The point charge (The point charge (charge monopolecharge monopole) is the ) is the simplest source of electrostatic field. simplest source of electrostatic field. The magnetic dipole is the simplest The magnetic dipole is the simplest source of magnetostatic field. There is source of magnetostatic field. There is no such thing as a magnetic monopole no such thing as a magnetic monopole (at least as far as classical physics is (at least as far as classical physics is concerned).concerned).

51Lecture 7

Magnetic Dipole (Cont’d)Magnetic Dipole (Cont’d)

The magnetic dipole is analogous The magnetic dipole is analogous to the electric dipole.to the electric dipole.

Just as the electric dipole is useful Just as the electric dipole is useful in helping us to understand the in helping us to understand the behavior of dielectric materials, behavior of dielectric materials, so the magnetic dipole is useful in so the magnetic dipole is useful in helping us to understand the helping us to understand the behavior of magnetic materials.behavior of magnetic materials.

52Lecture 7

Magnetic Dipole (Cont’d)Magnetic Dipole (Cont’d) Consider a small circular loop of radius Consider a small circular loop of radius bb

carrying a steady current carrying a steady current II. Assume that the . Assume that the wire radius has a negligible cross-section.wire radius has a negligible cross-section.

x

y

b

53Lecture 7

Magnetic Dipole Magnetic Dipole (Cont’d)(Cont’d)

The vector magnetic potential is The vector magnetic potential is evaluated for evaluated for R >> bR >> b as as

sin4

ˆ

sincosˆsinˆ

4

cossin1cosˆsinˆ

4

ˆ

4)(

2

20

20

2

0 20

2

0

0

r

bIa

r

baa

Ib

dr

b

raa

Ib

R

bdaIrA

yx

yx

54Lecture 7

Magnetic Dipole (Cont’d)Magnetic Dipole (Cont’d)

The magnetic flux density is The magnetic flux density is evaluated for evaluated for R >> bR >> b as as

sinˆcos2ˆ4

23

0 aabIr

AB r

55Lecture 7

Magnetic Dipole Magnetic Dipole (Cont’d)(Cont’d)

Recall electric dipoleRecall electric dipole

The electric field due to the electric The electric field due to the electric charge dipole and the magnetic charge dipole and the magnetic field due to the magnetic dipole are field due to the magnetic dipole are dualdual quantities. quantities.

sinˆcos2ˆ

4 30

aar

pE r

Qdp moment dipole electric

56Lecture 7

Magnetic Dipole Magnetic Dipole MomentMoment

The magnetic dipole moment can The magnetic dipole moment can be defined asbe defined as 2ˆ bIam z

Direction of the dipole moment is determined by the direction of current using the right-hand rule.

Magnitude of the dipole moment is the product of the current and the area of the loop.

57Lecture 7

Magnetic Dipole Magnetic Dipole Moment (Cont’d)Moment (Cont’d)

We can write the vector magnetic We can write the vector magnetic potential in terms of the magnetic potential in terms of the magnetic dipole moment asdipole moment as

We can write the B field in terms We can write the B field in terms of the magnetic dipole moment asof the magnetic dipole moment as

20

20

4

ˆ

4

sinˆ

r

am

r

maA r

rmaam

rB r

1

4sinˆcos2ˆ

40

30

58Lecture 7

Divergence of Divergence of BB-Field-Field

The B-field is The B-field is solenoidalsolenoidal, i.e. the , i.e. the divergence of the B-field is divergence of the B-field is identically equal to zero:identically equal to zero:

Physically, this means that Physically, this means that magnetic charges (monopoles) do magnetic charges (monopoles) do not exist.not exist.

A magnetic charge can be viewed A magnetic charge can be viewed as an isolated magnetic pole.as an isolated magnetic pole.

0 B

59Lecture 7

Divergence of Divergence of BB-Field -Field (Cont’d)(Cont’d)

N

S

NS

NS

No matter how No matter how small the small the magnetic is magnetic is divided, it always divided, it always has a north pole has a north pole and a south pole.and a south pole.

The elementary The elementary source of source of magnetic field is a magnetic field is a magnetic dipole.magnetic dipole.

I

N

S

60Lecture 7

Magnetic FluxMagnetic Flux

The magnetic The magnetic flux crossing an flux crossing an open surface open surface SS is is given bygiven by

S

sdBS

B

C

Wb/m2Wb

61Lecture 7

Magnetic Flux (Cont’d)Magnetic Flux (Cont’d) From the divergence theorem, we haveFrom the divergence theorem, we have

Hence, Hence, the net magnetic flux leaving any closed surface is zerothe net magnetic flux leaving any closed surface is zero . This is another manifestation of the fact that there are no magnetic charges.. This is another manifestation of the fact that there are no magnetic charges.

000 SV

sdBdvBB

62Lecture 7

Magnetic Flux and Magnetic Flux and Vector Magnetic Vector Magnetic

PotentialPotential The magnetic flux across an open The magnetic flux across an open

surface may be evaluated in terms surface may be evaluated in terms of the vector magnetic potential of the vector magnetic potential using Stokes’s theorem:using Stokes’s theorem:

C

SS

ldA

sdAsdB

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1 EEE 498/598 Overview of Electrical Engineering Lecture 7: Magnetostatics: Amperes Law Of Force;...

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