9-Magnetostatics Part 1

8/7/2019 9-Magnetostatics Part 1

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ElectromagneticsElectromagnetics TheoryTheory

Magnetostatics: Part 1

Outline:

1. Biot -Savarts Law

2. Amperes Circuit Law

3. Magnetic Flux Density Maxwells Equations

4. Maxwells Equation for Static Fields

5. Magnetic Scalar and Vector Potentials

6. Electric Boundary Conditions

7. Capacitance


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OverviewOverview

Terms Electric Magnetic

Field Intensity E H

Flux Density D B

RelationD = IE

I = permittivity

B=QH

Q = permeability

Maxwells Equation

Basic Laws

Flux

!v

!

E

Dv

V

JH

B

!v

! 0

! SD d] ! SB d]

!

!

encl

r

Qd

r

QQ

SD

aF2

21

4TI

!

v!

encl

R

I

R

I

l

al2

0

4T

Q

Electrostatic static

electric fields

ag etostatic staticag etic fields


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BiotBiot--SavartsSavarts LawLaw

Biot-Savarts law states that the differential ma netic field intensity dH produced at apoint P, as shown in Fi ure P, by the differential current element I dl is proportional to

the product I dl and the sine of the an le E between the element and the line joinin P

to the element and is inversely proportional to the square of the distance R between P

and the element.

sin

R

ddH

E

w

2

sin

R

dlkIdH

E

!T4

1!k

24

sin

R

lI

TE!

3244 R

dI

R

dId

R

TT

RlalH

v!

v!


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244 R

d

R

dd

R

TTRlal

H v!v!

R= distance vector between dland point P

dl = alon the direction of the currentI

aR= points from the current elements to point P.

Determinin the direction ofdH

Conventional representation ofH (orI)

(a) out of the pa e and

(b) into the pa e.

aR


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!v

R

R

v24T

aJH

v!

s

S

24T

aKH

Volume currentLine current

v

!

L

R

R

dI

24T

al

Surface current


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To determine the field due to a straightline current carryin filamentaryconductor of finite len th AB as in the fi ure shown.

JEETVaH )cos(cos

412 !

I

Tri o Identities

UU

UU

UUU

UU

sin

1csc

csccot1

csccot

tan

1

cot

22

2

!

!

!

!

d

d

? A J

J

V

VT

V

V

V

T

aH

aRl

aaRal

RlH

2/322

3

4

,

,

4

z

dzI

dzd

zanddzdBut

Idd

zz

J

E

E

J

E

E

EETV

EVEEV

T

EEVEV

aH

aH

!

!

!!

2

1

2

1

sin4

cos

cos

4

,cos,cot

33

22

2

dI

ec

decI

decdzzLetting


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Special Case 1: For a semi-infinite conductor (with respect to P), A(0, 0, 0) and B(0, 0, g)and E1 = 90r while E2 = 0r

JTV

aH )90cos0(cos4

rrI

JTV

a4

I!

Special Case 2: For an infinite conductor (with respect to P), A(0, 0, -g) and B(0, 0, g)

and E1 = 90r while E2 = 0r

? A JV

arr 0cos0cos4

I

JTV

a2

I!

To find unit vectoraJ is not always easy. A simple approach it to obtain it from:

VJ aaa v! N where al= unit vector alon the line of currentaV = unit vector perpendicular to the line

of current


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ExampleExample

The conductin trian ular loop in the fi ure shown carries a current of 10 A. FindH at

(0, 0, 5).


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Amperes Circuit LawAmperes Circuit Law

Amperes circuit law states that the line inte ral ofH around a closedpath is the sameas the net current Ienc enclosed by the path.

! encldlHAmperes law is similar to Gausss law. Gausss law is applied when char e

distribution is symmetrical while Amperes law is applied when current distributionis symmetrical.

By applyin Stokess theorem to the left-hand side of the equation above:

v!!SL

enclddI SHlH

But,

S

enclI SJ Therefore, JH !v

Ma ells r

ati

Fr m here, it sh l be bserve thatvH =J { 0, that is a ma netostatic field is notconservative.


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Magnetic Flux DensityMagnetic Flux Density Maxwells EquationMaxwells Equation

The ma netic flux density B is similar to the electric flux density D. As D =I0E , thema netic flux density is related to the ma netic field intensity H accordin to:

HQ! where Q0 = 4Tx 10-7 H/m

Q0 is known as the permeability offree space.

The ma netic flux throu h a surface S is iven by

!S

dSB] where the ma netic flux ] is in webers (Wb) andthe ma netic flux density is in Webers per square

meter (WB/m2) or teslas (T).


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In an electrostatic field, the flux passin throu h a closed surface is the same as the char e

enclosed; ] = Qencl = . Unlike electric flux lines, ma netic flux lines always close

upon themselves as in (b). This is because it is notpossible to have isolatedmagnetic poles

(or magnetic charges).An isolated ma netic char e does not exist.

SD d

By cuttin a ma net bar into

smaller and smaller pieces, we

would still et a bar with both

north and south poles to ether.


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Thus, the total flux throu h a closed surface in a ma netic field can be written as:

This equation is referred to as the law ofconservation ofthe magnetic flux orGausss

law for magnetostatic fields.

By applyin the diver ence theorem,

! 0SB d

! vS dvd BB

0! BMa ells t

ati


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Maxwells Equation for Static FieldsMaxwells Equation for Static Fields

Differential Form Integral Form Remarks

Gausss law

Non-existence of ma netic

monopole

Conservative nature ofelectrostatic field

Amperes law

vV! D

0! B

0!v E

JH !v

!S v

vdvd VSD

!S d 0SB

!L S dd SJlH !S d 0lE

We can summarize all of the four Maxwells equation as shown in the table below. Thechoice between differential or inte ral forms depend on the problem which we want to

solve. A static field can only be electric or ma netic if it satisfies the correspondin

Maxwells equations.


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Magnetic Vector PotentialsMagnetic Vector Potentials

Electric Magnetic

Relations E = V(V/m) B = vA(Wb/m2)

Point char e

Line char e/current

Surface

char e/current

Volume

char e/current

-

Ma netic flux

throu h a surface

!

!

!

v

L

R

dv

R

d

R

Id

T

Q

T

Q

T

Q

4

4

4

JA

KA

lA

ddd!

d

dd!

d

dd!

d!

v

v

S

S

L

L

vdV

SdV

ldV

QV

rr

r

r

rr

r

r

rr

r

r

rr

r

V

TI

V

TI

V

TI

TI

0

0

0

0

4

1)(

4

1)(

4

1)(

4)(

! L dlA]

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9-Magnetostatics Part 1