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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
Dr. Blanton - ENTC 3331 - Magnetostatics 4
+
+
N
S
+
+
N
N
SS
A magnetic monopole does not exist—A magnetostatic field has no sources or sinks!
0 HHββ
divdiv
Dr. Blanton - ENTC 3331 - Magnetostatics 5
Elementary charge
is a source
Coulomb’s Law
(Elementary) DC current
is not a source
Ampere’s Law
+
+
Sdiv E
I
I
0H
div
Dr. Blanton - ENTC 3331 - Magnetostatics 6
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
Dr. Blanton - ENTC 3331 - Magnetostatics 7
• The amount of charge through an area, s, during t:
slq V volume
tsuq V
Dr. Blanton - ENTC 3331 - Magnetostatics 8
• Generalization:
u
u
s
projection of u
su ˆ
tq V sΔu ˆ
projection of onto the surface normal
u
Dr. Blanton - ENTC 3331 - Magnetostatics 9
Current DensityCurrent Density
• The definition of current density is:
• Therefore,
tq V sΔu ˆ
sJ ˆ
t
q
2mA
VuJ
Dr. Blanton - ENTC 3331 - Magnetostatics 10
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
Dr. Blanton - ENTC 3331 - Magnetostatics 11
Conducting MediaConducting Media
• Two types of charge carriers:• Negative charges• Positive charges +
-
Dr. Blanton - ENTC 3331 - Magnetostatics 12
Medium Negative Charges
Positive Charges
Conductors Free electrons
Semiconductors Electrons Holes
Ions Negative ions Positive ions
Dr. Blanton - ENTC 3331 - Magnetostatics 13
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 14
• Since in most conductors, the resistance is proportional to the charge velocity.
.
.. 1ext
extextCoulomb
E
uuqEqF
constant of proportionality
(mobility)
Dr. Blanton - ENTC 3331 - Magnetostatics 15
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 17
• It follows that for• a perfect dielectric•
• and for a perfect conductor• • since current is finite.
• inside all conductors.
0J
0extE
0E
Dr. Blanton - ENTC 3331 - Magnetostatics 18
• Since
• for all conductors.
• All conductors are equipotential, but may have surface charge.
0ˆ VdVC
lE
Dr. Blanton - ENTC 3331 - Magnetostatics 19
Electrical ResistanceElectrical Resistance
• For a conductor
• Show that for a conductor of
cylindrical shape.
RI
V
A
lR
A1 A2
Dr. Blanton - ENTC 3331 - Magnetostatics 20
• Potential difference between A1 and A2.
• Current through A1 and A2.
lEdEVVV xx 2
112ˆˆˆ lxx
AEddI x
AA
sEsJ ˆˆ
Dr. Blanton - ENTC 3331 - Magnetostatics 21
I
VR
A
l
AE
lER
x
x
A
lR
The reciprocal of conductivity Resistivity (ohms/meter).
Do not confuse charge distribution!
Dr. Blanton - ENTC 3331 - Magnetostatics 22
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 24
Jean-Baptiste Biot & Felix Savart developed the quantitative description for the magnetic field.
Dr. Blanton - ENTC 3331 - Magnetostatics 25
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
Dr. Blanton - ENTC 3331 - Magnetostatics 26
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 27
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 28
• What force does such a field exert onto a stationary current?• What is equivalent to:
H mF
EF
qe
Dr. Blanton - ENTC 3331 - Magnetostatics 29
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 30
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 31
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 32
• The experimental facts also show that:• and
• • Thus, the magnetic force for a straight
conductor is:
Im F
lm F
βlHlF
ˆˆ IdIC
m
Dr. Blanton - ENTC 3331 - Magnetostatics 33
Important ConsequencesImportant Consequences
• The force on a closed, current carrying loop is zero.
0ˆ HlF
C
m dI
closed loop = 0
Dr. Blanton - ENTC 3331 - Magnetostatics 34
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
Dr. Blanton - ENTC 3331 - Magnetostatics 35
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 36
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 37
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 38
• Note the following
221
2
2coslx
l
z
xl x
P
1θ
112 coscoscos
2θ
1θ
2222
242
ˆ
2
2coslxx
IlφHβ
lx
l
Dr. Blanton - ENTC 3331 - Magnetostatics 40
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 41
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.
Dr. Blanton - ENTC 3331 - Magnetostatics 42
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
Dr. Blanton - ENTC 3331 - Magnetostatics 43
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 44
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 45
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 46
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
Dr. Blanton - ENTC 3331 - Magnetostatics 47
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
Dr. Blanton - ENTC 3331 - Magnetostatics 48
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 49
• is always tangential on circles about the wire and its magnitude decreases with 1/r.H
Dr. Blanton - ENTC 3331 - Magnetostatics 50
• What is inside the wire?
• Again, use an Ampere’s circuital law.
H
Ir
a
Cz
rdHIdC
φφlH ˆˆˆ 2
0
Dr. Blanton - ENTC 3331 - Magnetostatics 51
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 52
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 53
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 54
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 55
• In a given region of space, the vector potential of the magnetostatic field is given by
• Determine)sin2(ˆcos5ˆ xy yxA
β
Dr. Blanton - ENTC 3331 - Magnetostatics 56
0sin2cos5
ˆˆˆ
0
sin2
cos5
xy
x
y
rotrot zyx
zyx
Aβ
yx sin5sin
0
0
β
Dr. Blanton - ENTC 3331 - Magnetostatics 57
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 58
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
Dr. Blanton - ENTC 3331 - Magnetostatics 59
dxxdxxx
25.0
25.0
25.0
25.0cos4
cos8
cos8
xu dxdu
8sin4
1
8sin
8sin
4
1sin4
1 81
81
x
Dr. Blanton - ENTC 3331 - Magnetostatics 60
• 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
Dr. Blanton - ENTC 3331 - Magnetostatics 61
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
Dr. Blanton - ENTC 3331 - Magnetostatics 62
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
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