Lect07 Emt

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1

EEE 498/598

Overview of Electrical Engineering

Lecture 7: Magnetostatics: Amperes LawOf Force; Magnetic Flux Density; Lorentz

Force; Biot-savart Law; Applications OfAmperes Law In Integral Form; Vector

Magnetic Potential; Magnetic Dipole;Magnetic Flux

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Lecture 72

Lecture 7 Objectives

To begin our study of magnetostatics with

Amperes law of force; magnetic flux

density; Lorentz force; Biot-Savart law;applications of Amperes law in integral

form; vector magnetic potential; magnetic

dipole; and magnetic flux.

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Lecture 73

Overview of Electromagnetics

Maxwells

equations

Fundamental laws of

classical electromagnetics

Special

cases

Electro-

statics

Magneto-

staticsElectro-

magnetic

waves

Kirchoffs

Laws

Statics: 0

t

d

Geometric

Optics

Transmission

Line

Theory

CircuitTheory

Input fromother

disciplines

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Lecture 74

Magnetostatics

Magnetostaticsis the branch of electromagnetics

dealing with the effects of electric charges in steady

motion (i.e, steady current or DC).

The fundamental law of magnetostaticsis

Amperes law of force.

Amperes law of forceis analogous to Coulombs

lawin electrostatics.

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Lecture 75

Magnetostatics (Contd)

In magnetostatics, the magnetic field is

produced by steady currents. The

magnetostatic field does not allow for inductive coupling between circuits

coupling between electric and magnetic fields

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Lecture 76

Amperes Law of Force

Amperes law of forceis the law of action

between current carrying circuits.

Amperes law of forcegives the magnetic force

between two current carrying circuitsin an

otherwise empty universe.

Amperes law of force involves complete circuits

since current must flow in closed loops.

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Lecture 77

Amperes Law of Force (Contd)

Experimental facts:

Two parallel wires

carrying current in thesame direction attract.

Two parallel wires

carrying current in the

opposite directionsrepel.

I1 I2

F12F21

I1 I2

F12F21

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Lecture 78


Experimental facts:

A short current-

carrying wire orientedperpendicular to a

long current-carrying

wire experiences no

force.

I1

F12 = 0

I2

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Lecture 79


Experimental facts:

The magnitude of the force is inversely

proportional to the distance squared.The magnitude of the force is proportional to

the product of the currents carried by the two

wires.

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Lecture 710


The direction of the force established by the

experimental facts can be mathematically

represented by

1212

12 RF aaaa

unit vector in

direction of force on

I2due toI1

unit vector in direction

ofI2fromI1

unit vector in directionof currentI1

unit vector in directionof currentI2

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Lecture 711


The force acting on a current elementI2 dl2by a

current elementI1 dl1is given by

2

12

1122012

12

4 R

aldIldIF R

Permeability of free space

0= 410-7 F/m

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Lecture 712


The total force acting on a circuit C2having a

current I2by a circuit C1having current I1is

given by

2 1

12

2

12

1221012

4C C

R

R

aldldIIF

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Lecture 713


The force on C1due to C2is equal in

magnitude but opposite in direction to the

force on C2due to C1.

1221 FF

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Lecture 714

Magnetic Flux Density

Amperes force law describes an action at a

distance analogous to Coulombs law.

In Coulombs law, it was useful to introduce the

concept of an electric fieldto describe the

interaction between the charges.

In Amperes law, we can define an appropriate

field that may be regarded as the means by

which currents exert force on each other.

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Lecture 715

Magnetic Flux Density (Contd)

The magnetic f lux densitycan be introduced

by writing

2

2 1

12

1222

2

12

1102212

4

C

C C

R

BldI

R

aldIldIF

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Lecture 716


where

1

1 22

12

11012

4C

R

R

aldIB

the magnetic flux density at the location ofdl2due to the currentI1in C1

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Lecture 717


Suppose that an infinitesimal current elementIdl

is immersed in a region of magnetic flux density

B. The current element experiences a force dF

given by

BlIdFd

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Lecture 718


The total force exerted on a circuit Ccarrying

current Ithat is immersed in a magnetic flux

density Bisgiven by

C

BldIF

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Lecture 719

Force on a Moving Charge

A moving point charge placed in a magnetic

field experiences a force given by

BvQ

The force experienced

by the point charge isin the direction into the

paper.

BvQFm vQlId

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Lecture 720

Lorentz Force

If a point charge is moving in a region where both

electric and magnetic fields exist, then it experiences

a total force given by

The Lorentz force equation is useful fordetermining the equation of motion for electrons in

electromagnetic deflection systems such as CRTs.

BvEqFFF me

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Lecture 721

The Biot-Savart Law

The Biot-Savart lawgives us theB-field

arising at a specified pointPfrom a given

current distribution. It is a fundamental law of magnetostatics.

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Lecture 722

The Biot-Savart Law (Contd)

The contribution to theB-field at a pointP

from a differential current elementIdlis given

by

3

0

4

)(

R

RldIrBd

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Lecture 723


lId

PR

r r

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Lecture 724


The total magnetic flux at the pointPdue to the

entire circuit Cis given by

C

R

RldIrB

3

0

4)(

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Lecture 725

Types of Current Distributions

L ine current density(current) - occurs for

infinitesimally thin filamentary bodies (i.e.,

wires of negligible diameter).

Surface current density(current per unit

width) - occurs when body is perfectly

conducting.

Volume current density(current per unit

cross sectional area) - most general.

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Lecture 726

The Biot-Savart Law (Contd) For a surface distribution of current, the B-S law

becomes

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

becomes

S

s sdR

RrJrB

3

0

4)(

V

vdR

RrJrB3

0

4)(

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Lecture 727

Amperes Circuital Law in

Integral Form

Amperes Circuital Lawin integral formstates that the circulation of the magneticflux density in free space is proportional to

the total current through the surfacebounding the path over which the circulationis computed.

encl

C

IldB 0

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Lecture 728

Amperes Circuital Law in

Integral Form (Contd)

By convention, dSis

taken to be in the

direction defined by the

right-hand rule appliedto dl.

S

encl sdJI

Since volume current

density is the mostgeneral, we can write

Ienclin this way.

S

dl

dS

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Lecture 729

Amperes Law and Gausss Law

Just as Gausss law follows from Coulombs law,

so Amperes circuital law follows from Amperes

force law.

Just as Gausss law can be used to derive the

electrostatic field from symmetric charge

distributions, so Amperes law can be used to

derive the magnetostatic field from symmetriccurrent distributions.

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Lecture 730

Applications of Amperes Law

Amperes law in integral form is an integral

equationfor the unknown magnetic flux density

resulting from a given current distribution.

encl

C

IldB 0

known

unknown

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Lecture 731


(Contd)

In general, solutions to integral equations

must be obtained using numerical

techniques. However, for certain symmetric current

distributions closed form solutions to

Amperes law can be obtained.

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Lecture 732


(Contd)

Closed form solution to Amperes law

relies on our ability to construct a suitable

family ofAmperian paths

.AnAmperian pathis a closed contour to

which the magnetic flux density is

tangential and over which equal to aconstant value.

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Lecture 733

Magnetic Flux Density of an Infinite

Line Current Using Amperes Law

Consider an infinite line current along the z-axis

carrying current in the +z-direction:

I

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Lecture 734

Magnetic Flux Density of an Infinite Line

Current Using Amperes Law (Contd)

(1) Assume from symmetry and the right-hand rule

the form of the field

(2) Construct a family of Amperian paths

BaB

circles of radius where

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Lecture 735



(3) Evaluate the total current passing through the

surface bounded by the Amperian path

S

encl sdJI

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Lecture 736



Amperian path

IIencl

I

x

y

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Lecture 737



(4) For each Amperian path, evaluate the integral

BlldBC

2BldBC

magnitude ofB

on Amperian

path.

lengthof Amperian

path.

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Lecture 738



(5) Solve for Bon each Amperian path

lIB encl0

2

0IaB

A l i S k Th

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Lecture 739

Applying Stokess Theorem to

Amperes Law

S

encl

SC

sdJI

sdBldB

00

Because the above must hold for any

surface S, we must have

JB 0Differential form

of Amperes Law

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Lecture 740

Amperes Law in Differential

Form

Amperes law in differential form implies

that theB-field is conservativeoutside of

regions where current is flowing.

d l l f

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Lecture 741

Fundamental Postulates of

MagnetostaticsAmperes law in differential form

No isolated magnetic charges

JB 0

0

B

Bis solenoidal

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Lecture 742

Vector Magnetic Potential

Vector identity: the divergence of the curl of

any vector field is identically zero.

Corollary: If the divergence of a vector field is

identically zero, then that vector field can be

written as the curl of some vector potential

field.

0 A

V M i P i l

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Lecture 743


(Contd)

Since the magnetic flux density is

solenoidal, it can be written as the curl of

a vector field called the vector magnetic

potential.

ABB 0

V M i P i l

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Lecture 744


(Contd)The general form of the B-S law is

Note that

V

vd

R

RrJrB

3

0

4

)(

31

RR

R


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Lecture 745


(Contd) Furthermore, note that the deloperator operates

only on the unprimed coordinates so that

R

rJ

rJR

RrJ

RRrJ

1

13

V M i P i l

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Lecture 746


(Contd)

Hence, we have

vdR

rJrBV

4

0

rA

V M i P i l

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Lecture 747


(Contd) For a surface distribution of current, the vector

magnetic potential is given by

For a line current, the vector magnetic potential is

given by

sdR

rJ

rAS

s

4)( 0

L R

ldIrA

4)( 0

V M i P i l

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Lecture 748


(Contd)

In some cases, it is easier to evaluate the

vector magnetic potential and then use

B= A, rather than to use the B-S law

to directly findB.

In some ways, the vector magnetic

potential Ais analogous to the scalarelectric potential V.

V M i P i l

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Lecture 749


(Contd)

In classical physics, the vector magnetic

potential is viewed as an auxiliary function

with no physical meaning.

However, there are phenomena in

quantum mechanics that suggest that the

vector magnetic potential is a real (i.e.,measurable) field.

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Lecture 750

Magnetic Dipole

A magnetic dipolecomprises a small current

carrying loop.

The point charge (charge monopole) is the

simplest source of electrostatic field. The

magnetic dipole is the simplest source of

magnetostatic field. There is no such thing as

a magnetic monopole (at least as far asclassical physics is concerned).

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Lecture 751

Magnetic Dipole (Contd)

The magnetic dipole is analogous to the

electric dipole.

Just as the electric dipole is useful inhelping us to understand the behavior of

dielectric materials, so the magnetic dipole

is useful in helping us to understand thebehavior of magnetic materials.

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Lecture 752


Consider a small circular loop of radius bcarrying a steady currentI. Assume that the wireradius has a negligible cross-section.

x

y

b

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Lecture 753

Magnetic Dipole (Contd)The vector magnetic potential is evaluated

forR >> bas

sin4

sin

cos

sin

4

cossin1cossin

4

4)(

2

2

0

2

0

2

0 2

0

2

0

0

r

bIa

r

b

aa

Ib

dr

b

raa

Ib

R

bdaIrA

yx

yx

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Lecture 754


The magnetic flux density is evaluated for

R >> bas

sincos2

4

2

3

0 aabIr

AB r

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Lecture 755


Recall electric dipole

The electric field due to the electric charge

dipole and the magnetic field due to the

magnetic dipole are dualquantities.

sincos24 30

aar

pE r

Qdp momentdipoleelectric

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Lecture 756

Magnetic Dipole Moment

The magnetic dipole moment can be defined

as2

bIam z

Direction of the dipole moment

is determined by the directionof current using the right-hand

rule.

Magnitude of

the dipole

moment is the

product of the

current and

the area of the

loop.

M i Di l M

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Lecture 757

Magnetic Dipole Moment

(Contd)We can write the vector magnetic potential in

terms of the magnetic dipole moment as

We can write the B field in terms of the

magnetic dipole moment as

2020 4

4

sin

r

am

r

maA r

rmaam

rB r

1

4sincos2

4

0

3

0

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Lecture 758

Divergence of B-Field

The B-field is solenoidal, i.e. thedivergence of the B-field is identically equalto zero:

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

A magnetic charge can be viewed as anisolated magnetic pole.

0 B

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Lecture 759

Divergence of B-Field (Contd)

N

S

N

S

NS

No matter how small

the magnetic is

divided, it always has

a north pole and asouth pole.

The elementary

source of magneticfield is a magnetic

dipole.

I

N

S

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Lecture 760

Magnetic Flux

The magnetic flux

crossing an open

surface Sis given by

S

sdBS

B

C

Wb/m2Wb

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Lecture 761

Magnetic Flux (Contd)

From the divergence theorem, we have

Hence, the net magnetic flux leaving anyclosed surface is zero. This is another

manifestation of the fact that there are nomagnetic charges.

000

SV

sdBdvBB

M i Fl d V

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Magnetic Flux and Vector

Magnetic Potential

The magnetic flux across an open surface

may be evaluated in terms of the vector

magnetic potential using Stokess theorem:

C

SS

ldA

sdAsdB

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Lect07 Emt