Dr. Hugh Blanton ENTC 3331. Magnetostatics Dr. Blanton - ENTC 3331 - Magnetostatics 3 Magnetostatics Magnetism Chinese—100 BC Arabs—1200 AD Magnetite—Fe

Dr. Hugh Blanton

ENTC 3331

Magnetostatics

Dr. Blanton - ENTC 3331 - Magnetostatics 3

MagnetostaticsMagnetostatics

• Magnetism• Chinese—100 BC• Arabs—1200 AD

• Magnetite—Fe3O4

• Found near Magnesia (now Turkey)

• Permanent magnet• Not fundamental to

magnetostatics.• A permanent magnet

is equivalent to a polar material in electrostatics.

• • Equivalent to

electrostatics• The theoretical

structure of magnetostatics is very similar to electrostatics.

• But there is one important empirical fact that accounts for all the differences between the theory of magnetostatics and electrostatics.• There is no magnetic

monopole!

0

Hβ

tt


+

+

N

S

+

+

N

N

SS

A magnetic monopole does not exist—A magnetostatic field has no sources or sinks!

0 HHββ

divdiv


Elementary charge

is a source

Coulomb’s Law

(Elementary) DC current

is not a source

Ampere’s Law

+

+

Sdiv E

I

I

0H

div


Current DensityCurrent Density

• Moving chargescurrent.

• Charges move to the right with constant velocity, u.• Over a period of time, the charges move

distance, l.

u

l

v s

tul


• The amount of charge through an area, s, during t:

slq V volume

tsuq V


• Generalization:

u

u

s

projection of u

su ˆ

tq V sΔu ˆ

projection of onto the surface normal

u


Current DensityCurrent Density

• The definition of current density is:

• Therefore,

tq V sΔu ˆ

sJ ˆ

t

q

2mA

VuJ


Electrical CurrentElectrical CurrentsJ ˆdI

S

Without resistance

Convection

e.g. electron beam

Typically in vacuum or dielectric medium

With resistance

Conduction

e.g. copper wire

Typically in a conducting medium

Electrical Currents


Conducting MediaConducting Media

• Two types of charge carriers:• Negative charges• Positive charges +

-


Medium Negative Charges

Positive Charges

Conductors Free electrons

Semiconductors Electrons Holes

Ions Negative ions Positive ions


• Like mechanics, there is a resistance to motion.• Therefore, an external force is required

to maintain a current flow in a resistive conductor.

.. extextCoulomb qEF


• Since in most conductors, the resistance is proportional to the charge velocity.

.

.. 1ext

extextCoulomb

E

uuqEqF

constant of proportionality

(mobility)


• In semiconductors:• electron mobility

• electrons move against the direction

• hole mobility

• holes move in the same direction as

.exte

eE

u

E

.exth

hE

u

E


• SinceuJ

V

extextV EEJ

Ohm’s law

conductivity


• It follows that for• a perfect dielectric•

• and for a perfect conductor• • since current is finite.

• inside all conductors.

0J

0extE

0E


• Since

• for all conductors.

• All conductors are equipotential, but may have surface charge.

0ˆ VdVC

lE


Electrical ResistanceElectrical Resistance

• For a conductor

• Show that for a conductor of

cylindrical shape.

RI

V

A

lR

A1 A2


• Potential difference between A1 and A2.

• Current through A1 and A2.

lEdEVVV xx 2

112ˆˆˆ lxx

AEddI x

AA

sEsJ ˆˆ


I

VR

A

l

AE

lER

x

x

A

lR

The reciprocal of conductivity Resistivity (ohms/meter).

Do not confuse charge distribution!


• The electrical field can be expressed in terms of the charge density, .

• What is the equivalent expression for the magnetic field, .

E

dVrV

rE ˆ4

12

H


• Qualitatively,

•

• circular field lines

•

2

1

rH

IH


Jean-Baptiste Biot & Felix Savart developed the quantitative description for the magnetic field.


2

ˆˆ

4 r

dId

rlH

Ild

r

rPpoint of interest

differential section of conductor

contributes to field atH

d)(rH

)(rP

field comes out of plane due to

the cross product


• Total field through integration over .

• The line integration is not convenient• Wires are irregularly bent, but• Wires typically have constant cross-sections, s.

H

ld

l r

Id2

ˆˆ

4

1 rlH

magnetic field strength


• Take advantage of:

ls

Jll ˆˆ

ˆˆ dt

qdV

td

qdVIdd

t

q

useful relationship

m

AdV

rr

Id

Vl22

ˆ

4

1ˆˆ

4

1 rJrlH

Biot-Savart Law


• What force does such a field exert onto a stationary current?• What is equivalent to:

H mF

EF

qe


• Experimental facts:• Flexible wire in a magnetic field, .• No current

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

H

H

l

0I


• Experimental facts:• Flexible wire in a magnetic field, .• Current up.

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

H

H

l

0I

mF

H

mF

I

right-handed rule


• Experimental facts:• Flexible wire in a magnetic field, .• Current down.

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

H

H

l

0I

mF

H

mF

I

right-handed rule


• The experimental facts also show that:• and

• • Thus, the magnetic force for a straight

conductor is:

Im F

lm F

βlHlF

ˆˆ IdIC

m


Important ConsequencesImportant Consequences

• The force on a closed, current carrying loop is zero.

0ˆ HlF

C

m dI

closed loop = 0


ExampleExample

• Linear conductor• Determine magnetic field .• Determine the force, , on another

conductor.mF H

Biot-Savart Law

l R

Id2

ˆˆ

4

1 RlH

z

xXl

ld dθ R 0, zxP

H

d

θ

x


• Substituting•

z

xXl

ld dθ

x

R 0, zxPθ

zdzld ˆˆ

dzdzd sinˆˆˆˆˆ φRzRl at P(x,z), points into the

plane

φ

R

ld

R

Note that for a small d, R is approximately unchanged when separated by dwhich implies:

cscsin

sin xx

RRx

l R

Id2

ˆˆ

4

1 RlH


• Note: z

xXl

ld dθ

x

R 0, zxPθ

R

ld

R cotcot xzz

x

dxdxdz

2

22

2 sin

cossin

sin

coscossinsin

sin

coscot xxz

dxxdz 22

cscsin

1

l R

Id2

ˆˆ

4

1 RlH


• Using the previous transformations:

2

122

2

2 csc

cscsin

4ˆ

ˆˆ

4

1

d

x

xI

R

Id

l

φRl

H

1

2

2

1

2

1

cos4ˆcos

4ˆsin

4ˆ

x

I

x

Id

x

IφφφH

21 coscos4ˆ

x

IφH

z

xXl

ld dθ

x

R 0, zxPθ

R

ld

R


• Note the following

221

2

2coslx

l

z

xl x

P

1θ

112 coscoscos

2θ

1θ

2222

242

ˆ

2

2coslxx

IlφHβ

lx

l


• For an infinitely long wire where

x

I

2

φHβ

xl


• Now, what is the force on a parallel conductor wire carrying the current, I? z

xd

I

2ˆ1 yH

2d

2d

y

field by I1 at location of I2

d

IlIm

2ˆˆ 1

2 yzF


z

x2

d2

d

yd

IlIm

2ˆˆ 1

2 yzF

x

d

lIIm

2

ˆ 21xF

• I1 attracts I2

• Similarly I2 attracts I1 with the same force.

• Attraction is proportional to 1/distance.


Maxwell’s Magnetostatic EquationsMaxwell’s Magnetostatic Equations

• Experimental fact: An equivalent to the electrostatic monopole field does not exist for magnetostatics.

Charge is the source of the electrostatic field

Vdiv D

No equivalent in magnetostatics

0β

div


• Let’s apply Gauss’s theorem to an arbitrary field:

• Gauss’s law of Magnetostatics• Mathematical expression of the experimental fact that a source of the magnetostatic field does not exist.

dVdivdVS

AsA

ˆ

0ˆ sββ ddVdivSV


• Experimental fact: The magnetostatic field is generally a rotational field.

• Apply Stoke’s theorem to any arbitrary field:

• Ampere’s Circuital Law

0 JHH

rot

lAsA ˆˆ ddrotCs

IdddrotsCs

sJlHsH ˆˆˆ


• Mathematical expression of the experimental fact that the line integral of the magnetostatic field around a closed path is equal to the current flowing through the surface bounded by this path.

XS

ld

H

IC

field vector of the

magnetostatic field

line differential

surface

contour

current flowing through the

surface


Long lineLong line

• Suppose we have an infinitely long line of charge:

• Recall that charge is the fundamental quantity for electrostatics

ro

lr

2

E


Long lineLong line

• Suppose we have an infinitely long line carrying current,I:• What is .• Orient wire along the z-axis

• Choose a circular Amperian contour about the wire.• Ampere circuital law

H H

r

ld

I

z


• Symmetry implies that is constant on the contour and is always tangential to the contour.• This implies that

IdC

lH ˆ

H

H

IrdHdHCC

φφlHφH ˆˆˆˆ

r

IHrHdrHI

22

2

0


• is always tangential on circles about the wire and its magnitude decreases with 1/r.H


• What is inside the wire?

• Again, use an Ampere’s circuital law.

H

Ir

a

Cz

rdHIdC

φφlH ˆˆˆ 2

0


• is current through the Amperian surface

• The magnitude of increases linearly inside the conductor.

rHrdHa

rI 2ˆˆ

2

02

2

φφ

2

2

a

rI

22 a

IrH

H


• It is interesting to note that the comparison of part (a) and (b) of this problem shows that for a convective current, I, the electrostatic and magnetostatic fields are perpendicular to each other.

• This is generally true in electrodynamics!

φH ˆH

rE ˆE


• The magnetostatic field is rotational without sources• •

• In electrostatics•

• • A scaler potential, V, exists, so that

0β

div

0 JHH

rot

0 EE

rot

0ˆ lE d

VE


• Can any potential be defined in magnetostatics?• Let’s take advantage of the general vector

identity•

• Define a vector potential, ,so that•

• It follows that in agreement with Maxwell equations

0 A

0)( A

rotdiv

A

HβA

rot

0β

div


• In a given region of space, the vector potential of the magnetostatic field is given by

• Determine)sin2(ˆcos5ˆ xy yxA

β


0sin2cos5

ˆˆˆ

0

sin2

cos5

xy

x

y

rotrot zyx

zyx

Aβ

yx sin5sin

0

0

β


• Magnetic flux, ,through an area S is given by the surface integral

• Use this equation and the solution to previous problem to calculate the magnetic flux, , for the field through a square loop.

sβ ˆds

x

y

0.25m

0.25m


dxdydS

zβsβ ˆˆ25.0

25.0

25.0

25.0

dxdyyx

25.0

25.0

25.0

25.0sin5cos

yu dydu

dxyxy81

81

25.0

25.0cos5cos

dxxx

25.0

25.0 8cos5cos

88cos5cos

8

)cos(cos xx


dxxdxxx

25.0

25.0

25.0

25.0cos4

cos8

cos8

xu dxdu

8sin4

1

8sin

8sin

4

1sin4

1 81

81

x


• Note that since , it follows from Stoke’s theorem that

• Calculate again using

Aβ

rot

CSS

ddrotd lAsAsβ ˆˆˆ

C

dlA ˆ

x0.25m

0.25m


81

81

81

81

81

81

81

81

81

81

81

81

ˆˆ

ˆˆ

ˆ

lAlA

lAlA

lA

dd

dd

d

xx

xx

yx

yx

C

81

81

81

81

81

81

81

81

8sin2

8sin2

8cos5

8cos5

dydy

dxdx


81

81

81

81

81

81

81

81

8sin2

8sin2

8cos5

8cos5

yyyy

xx

8sin8

1

4

1

8sin8

1

4

1

8sin8

1

4

1

8sin8

1

4

1

8cos8

5

8cos8

5

8cos8

5

8cos8

5


8sin8

1

4

1

8sin8

1

4

1

8sin8

1

4

1

8sin8

1

4

1

8cos8

5

8cos8

5

8cos8

5

8cos8

5

8sin2

1

8sin8

1

8sin8

1

8sin8

1

8sin8

1

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Dr. Hugh Blanton ENTC 3331. Magnetostatics Dr. Blanton - ENTC 3331 - Magnetostatics 3 Magnetostatics Magnetism Chinese—100 BC Arabs—1200 AD Magnetite—Fe