Lect07

(1)

EEE 498/598

Overview of Electrical

Engineering

Lecture 7: Magnetostatics:

Ampere’s Law Of Force; Magnetic

Flux Density; Lorentz Force;

Biot-savart Law; Applications Of

savart Law; Applications Of

Ampere’s Law In Integral Form;

(2)

Lecture 7 Objectives



To begin our study of

magnetostatics with Ampere’s law

of force; magnetic flux density;

Lorentz force; Biot-Savart law;

applications of Ampere’s law in

integral form; vector magnetic

potential; magnetic dipole; and

magnetic flux.

(3)

Overview of Electromagnetics

Maxwell’s equations Fundamental laws of

classical electromagnetics

Special

cases Electro-_statics Magneto-_statics _magnetic

Electro-waves

Kirchoff’s Laws Statics: 0

 

t





d

Geometric Optics

Transmission Line Theory Circuit

Theory Input from

other disciplines

(4)

Magnetostatics

 Magnetostatics_{Magnetostatics} is the branch of is the branch of

electromagnetics dealing with the

effects of electric charges in steady

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

 The fundamental law of The fundamental law of magnetostaticsmagnetostatics is is

Ampere’s law of force Ampere’s law of force._.

 Ampere’s law of force_{Ampere’s law of force} is analogous to is analogous to

Coulomb’s law

(5)

Magnetostatics (Cont’d)



In magnetostatics, the magnetic

field is produced by steady

currents. The magnetostatic

field does not allow for

 inductive coupling between inductive coupling between

circuits

(6)

Ampere’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.

 Ampere’s law of force_{Ampere’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.

 Ampere’s law of force involves Ampere’s law of force involves

complete circuits since current must

flow in closed loops.

(7)

Ampere’s Law of Force

(Cont’d)

 Experimental facts:Experimental 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

F₁₂

F₂₁

 

I₁ I₂

F₁₂

(8)

Ampere’s Law of Force

(Cont’d)

 Experimental Experimental

facts:

 A short current-A short

current-carrying wire

carrying wire

oriented

perpendicular to a

long

current-carrying wire

carrying wire

experiences no

force.



I₁

F₁₂= 0

(9)

Ampere’s Law of Force

(Cont’d)



Experimental facts:

 The magnitude of the force is The magnitude of the force is

inversely proportional to the

distance squared.

 The magnitude of the force is The magnitude of the force is

proportional to the product of the

currents carried by the two wires.

(10)

Ampere’s Law of Force

(Cont’d)

 The direction of the force established The direction of the force established

by the experimental facts can be

mathematically represented by



₁₂



12

ˆ

_F

_a

₂

_a

₁

_a

_R

a





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

(11)

Ampere’s Law of Force

(Cont’d)

 The force acting on a current The force acting on a current

element

element II2 ₂ddll22 by a current element by a current element II1 1

d

dll₁₁ is given by_{is given by}





2 12

1 1

2 2

0

12 12

ˆ

4 R

a

l

d

I

l

d

I

F





R





(12)

Ampere’s Law of Force

(Cont’d)

 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 2 12 1 2 2 1 0 12

ˆ

4

_{C C}

R

a

l

d

l

d

I

F





(13)

Ampere’s Law of Force

(Cont’d)



The force on

C

₁₁

due to

C

₂₂

is equal

in magnitude but opposite in

direction to the force on

C

22

due

to

C

11

.

12

21 F

(14)

Magnetic Flux Density

 Ampere’s force law describes an “action at a Ampere’s force law describes an “action at 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.

 In Ampere’s law, we can define an In Ampere’s law, we can define an

appropriate field that may be regarded as

the means by which currents exert force on

each other.

(15)

Magnetic Flux Density

(Cont’d)

 The The magnetic flux densitymagnetic flux density can be can be

introduced by writing















2 1 12 12 2 2 2 12 1 1 0 2 2 12

ˆ

4

C C C R

B

l

d

I

R

a

l

d

I

l

d

I

F





(16)

Magnetic Flux Density

(Cont’d)

 wherewhere







1

12

2 12 1 1

0 12

ˆ

4

_C

R

a

l

d

I

B





the magnetic flux density at the location of

(17)

Magnetic Flux Density

(Cont’d)

 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

force

force ddFF given by given by

B

l

Id

F

(18)

Magnetic Flux Density

(Cont’d)

 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 is_is given _given

by







C

B

l

d

I

(19)

Force on a Moving

Charge

 A moving point charge placed in a A moving point charge placed in a

magnetic field experiences a force

given by

B v

Q

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

B

v

Q

(20)

Lorentz 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

exist, then it experiences a total force given

by

 The Lorentz force equation is useful for The Lorentz force equation is useful for

determining the equation of motion for

electrons in electromagnetic deflection

systems such as CRTs.



E

v

B



q

F

(21)

The Biot-Savart Law



The

Biot-Savart law

gives us the

B

-

-field arising at a specified point

field arising at a specified point

P

from a given current

_{from a given current}

distribution.



It is a fundamental law of

magnetostatics.

(22)

The Biot-Savart Law

(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}

3 0

4 )

(

R

l

d

I

r

B

d











(23)

The Biot-Savart Law

(Cont’d)

l

Id



P

R

(24)

The Biot-Savart Law

(Cont’d)

 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

l

d

I

r

B

0 ₃

4 )

(





(25)

Types of Current

Distributions

 Line current densityLine current density ((currentcurrent) - occurs for ) - occurs for

infinitesimally thin filamentary bodies

(i.e., wires of negligible diameter).

 Surface current densitySurface current density ( (current per unit current per unit

width

width) - occurs when body is perfectly ) - occurs when body is perfectly conducting

conducting..

 Volume current densityVolume current density ((current per unit current per unit

cross sectional area

(26)

The Biot-Savart Law

(Cont’d)

 For a surface distribution of current, For a surface distribution of current,

the B-S law becomes

 For a volume distribution of current, For a volume distribution of current,

the B-S law becomes

 



     S

s _d_s

R

R r

J r

B 0 ₃

4 ) (





 



     V v d R R r J r

B 0 ₃

4 )

(





(27)

Ampere’s Circuital Law

in Integral Form

 Ampere’s Circuital Law_{Ampere’s Circuital Law} in integral form states that in integral form states that

“the circulation of the magnetic flux density in

free space is proportional to the total current

through the surface bounding the path over

which the circulation is computed.”

encl

I

l

d

B







₀

(28)

Ampere’s Circuital Law

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 J d s

I

Since volume current density is the most general, we can write

I_encl in this way.

S

dl

(29)

Ampere’s Law and

Gauss’s Law

 Just as Gauss’s law follows from Just as Gauss’s law follows from

Coulomb’s law, so Ampere’s circuital

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

symmetric charge distributions, so

Ampere’s law can be used to derive

the magnetostatic field from

(30)

Applications of Ampere’s

Law

 Ampere’s law in integral form is an Ampere’s law in integral form is an integral integral

equation

equation for the unknown magnetic flux _{for the unknown magnetic flux}

density resulting from a given current

distribution.

encl C

I

l

d

B







₀



known

(31)

Applications of Ampere’s

Law (Cont’d)



In general, solutions to

integral

equations

must be obtained using

_{must be obtained using}

numerical techniques.



However, for certain symmetric

current distributions closed form

solutions to Ampere’s law can be

obtained.

(32)

Applications of Ampere’s

Law (Cont’d)



Closed form solution to Ampere’s

law relies on our ability to

construct a suitable family of

Amperian paths

.

_.



An

Amperian path

is a closed

contour to which the magnetic

flux density is tangential and over

which equal to a constant value.

(33)

Magnetic Flux Density of an

Infinite Line Current Using

Ampere’s Law

Consider an infinite line current along the z-axis carrying

current in the +z-direction:

(34)

Magnetic Flux Density of an

Infinite Line Current Using

Ampere’s Law (Cont’d)

(1) Assume from symmetry and the

right-hand rule the form of the field

(2) Construct a family of Amperian

paths

 



 

B

a

B



ˆ

circles of radius  where







(35)

(3) Evaluate the total current passing through

the surface bounded by the Amperian path



 

S

encl J d s

(36)

Amperian path

I I_encl 

I



x y

(37)

(4) For each Amperian path, evaluate

the integral

Bl l

d B

C

 



 

 2  B

l d

B  



magnitude of B

on Amperian path.

length of Amperian

(38)

(5) Solve for

(5) Solve for BB on each Amperian path_{on each Amperian path}

l I B  0 encl

  



2 ˆ 0I

a

(39)

Applying Stokes’s

Theorem to Ampere’s

Law

















S encl S C

s

d

J

I

s

d

B

l

d

B

0 0



 Because the above must hold for any

surface S, we must have

J

B







(40)

Ampere’s Law in

Differential Form



Ampere’s law in differential form

implies that the

B

-field is

conservative

outside of regions

_{outside of regions}

where current is flowing.

(41)

Fundamental Postulates

of Magnetostatics

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

 No isolated magnetic chargesNo isolated magnetic charges

J

B





₀





0 



(42)

Vector Magnetic

Potential

 Vector identity: “the divergence of the Vector identity: “the divergence of the

curl of any vector field is identically

zero.”

 Corollary: “If the divergence of a vector 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 

(43)

Vector Magnetic

Potential (Cont’d)



Since the magnetic flux density

is

solenoidal

, it can be written as

the curl of a vector field called

the

vector magnetic potential

.

_.

A

B













(44)

Vector Magnetic

Potential (Cont’d)

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

 Note thatNote that

 



     V v d R R r J r

B 0 ₃

4 ) (





3 1 R R R _  

 

  

(45)

Vector Magnetic

Potential (Cont’d)

 Furthermore, note that the Furthermore, note that the del_del operator operator

operates only on the unprimed

coordinates so that

 

_{ }

 

                                 R r J r J R R r J R R r J 1 1 3

(46)

Vector Magnetic

Potential (Cont’d)



Hence, we have

 

d

v

R

r

J

r

B

V

















4

0

 

r A

(47)

Vector Magnetic

Potential (Cont’d)

 For a surface distribution of current, For a surface distribution of current,

the vector magnetic potential is given

by

 For a line current, the vector magnetic For a line current, the vector magnetic

potential is given by

 

_d_s

R r J

r A

S

s  











4 )

( 0





 I dl r

A





)

(48)

Vector Magnetic

Potential (Cont’d)



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 find

B

.

_.



In some ways, the vector

magnetic potential

A

is analogous

to the scalar electric potential

(49)

Vector Magnetic

Potential (Cont’d)



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

(50)

Magnetic Dipole

 A A magnetic dipolemagnetic dipole comprises a small comprises a small

current carrying loop.

 The point charge (The point charge (charge monopolecharge monopole) is the ) 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 as classical physics is

concerned).

(51)

Magnetic Dipole (Cont’d)



The magnetic dipole is analogous

to the electric dipole.



Just as the electric dipole is useful

in helping us to understand the

behavior of dielectric materials,

so the magnetic dipole is useful in

helping us to understand the

(52)

Magnetic Dipole (Cont’d)

 Consider a small circular loop of radius Consider a small circular loop of radius b_b

carrying a steady current

carrying a steady current II. Assume that the _{. Assume that the}

wire radius has a negligible cross-section.

x y

(53)

Magnetic Dipole

(Cont’d)

 The vector magnetic potential is The vector magnetic potential is

evaluated for

evaluated for R >> bR >> b as as













                     _ sin cos ˆ sin ˆ 4 cos sin 1 cos ˆ sin ˆ 4 ˆ 4 ) ( 2 2 0 2 0 2 0 2 0 0 b I r b a a Ib d r b r a a Ib R bd a I r A y x y x         _         



(54)

Magnetic Dipole (Cont’d)



The magnetic flux density is

evaluated for

R >> b

as

 













 sin

ˆ cos

2 ˆ 4

2 3

0 _I _b _a _a

r A

(55)

Magnetic Dipole

(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 field

due to the magnetic dipole are

due to the magnetic dipole are dualdual

quantities.









ˆ 2 cos ˆ sin

4 ₀ r3 a a p

E  _r 

Qd

p



electric

dipole

moment



(56)

Magnetic Dipole

Moment

 The magnetic dipole moment can The magnetic dipole moment can

be defined as

be defined as ₂

ˆ

_I

_b

a

m



_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.

(57)

Magnetic Dipole

Moment (Cont’d)

 We can write the vector magnetic We can write the vector magnetic

potential in terms of the magnetic

dipole moment as

 We can write the B field in terms We can write the B field in terms

of the magnetic dipole moment as

2 0 2 0 4 ˆ 4 sin ˆ r a m r m a A r











   







   

(58)

Divergence of

B

-Field

 The B-field is The B-field is solenoidalsolenoidal, i.e. the , i.e. the

divergence of the B-field is

identically equal to zero:

 Physically, this means that magnetic Physically, this means that magnetic

charges (monopoles) do not exist.

 A magnetic charge can be viewed as A magnetic charge can be viewed as

an isolated magnetic pole.

0 



(59)

Divergence of

B

-Field

(Cont’d)

N S

N S N S

 No matter how No matter how

small the magnetic

is divided, it

always has a north

pole and a south

pole.

 The elementary The elementary

source of magnetic

field is a magnetic

field is a magnetic I

(60)

Magnetic Flux

 The magnetic The magnetic

flux crossing an

open surface

open surface SS is _is

given by









S

s

d

B

S

B

C

Wb/m2

(61)

Magnetic Flux (Cont’d)

_{From the divergence theorem, we have}_{From the divergence theorem, we have}

_Hence,_Hence,_{the net magnetic flux leaving any closed surface is zero}_{the 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.}