Chapter 5 Steady Magnetic Fields Magnetic Flux Density, Field Equations Boundary Conditions 1....
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Transcript of Chapter 5 Steady Magnetic Fields Magnetic Flux Density, Field Equations Boundary Conditions 1....
Chapter 5 Steady Magnetic Fields
Magnetic Flux Density, Field Equations Boundary Conditions
1. Magnetic Flux Density, Flux, and Field Lines
2. Equations for Steady Magnetic Fields in Free Space
3. Vector & Scalar Magnetic Potentials
4. Magnetization of Media
5. Equations for Steady Magnetic Fields in A Medium
6. Boundary Conditions for Steady Magnetic Fields
1. Magnetic Flux Density, Flux, and Field Lines
A magnetic field exerts a force on a moving charge. Hence, the
force acting on the moving charges, the current element, or the
torque acting on a small current loop can be used to quantify the
magnetic fields.
Experiments show that the magnetic force acting on a moving
charge is related not only to the magnitude and the speed of the
charge, but also to the direction of motion.
F
B
v
In-line Direction
The magnetic force will be maximum when the charge is moving
along a certain direction, and will be zero when the motion is
perpendicular to it. We define the direction in which the force is zero
as the in-line direction, as shown in the following figure.
Assuming the maximum force is Fm, if
the angle between the direction of charge
motion and the in-line direction is , the
force will besinmF
The magnitude of the force F is proportional to the product of
the magnitude of the charge q and the magnitude of the velocity v.
This force is called Lorentz force.
We define a vector B whose magnitude is with the direction
being the in-line direction. The relationship between the vector B,
the charge q, the velocity v, and the force F is
qv
Fm
BvF q
Where vector B is called magnetic flux
density, and the unit is tesla ( T ).
Lorentz force is always perpendicular to the direction of charge
motion. Consequently, the Lorenz force can only change the
direction of the charge in motion and there is no work done in this
action.
In-line Direction
F
B
v
The current element is a segment of current-carrying wire.
The magnitude of the line element vector dl stands for the length of
the current element I , and the direction is that of the current I.
F
BIdl If the current flowing in the current element
is I,then
qqtt
qI dd
d
dd
d
dd v
lll
And the force F acting on the current element in a magnetic field
with magnetic flux density B is
BlF dI
if the current is parallel to the magnetic flux density B, the force will
be zero. If it is perpendicular to B, the force is maximum.
The direction of the magnetic force on a current is always perpen-
dicular to the direction of the current flow.
The torque on a small current loop.
cd
ba F
FB
S
When viewed from a large distance, the
current loop may be considered a magnetic
dipole.
ISBBIlIlBlFlT 2
where S is the area of the frame.
The magnetic field in the plane of the
frame current can be taken to be a uniform.
The small current loop is a plane square frame with four sides of
length l each, and the direction of flow current is shown in figure.
If the magnetic flux density B is parallel to the plane of the
frame, no force will act on the sides ab and cd, while the directions of
the forces on the sides ad and bc are opposite. The magnitude of the
torque T on the frame current is
F
d c
ba
F
F
FB
S
d c
ba F
F
BBn
Bt
F
F
S
If B is perpendicular to the plane of the
frame, the forces on the four sides are
directed outside and will cancel each other.
The torque acting on the frame current is
zero.
If the angle between the vector B and the
normal to the plane of the frame is , the
vector B may be resolved into two
components Bn and Bt . Then, the magnitude
of the torque T on the current loop is
sint ISBISBT
Requiring the direction of the directed surface S and the
direction of the current to obey the right hand rule, the above
equation can be written in the following vector form as
)( BST I
It is valid for any small current loop. In general, the product IS is
called the magnetic moment of the current loop, and it is denoted as
m, so thatSm I
The above equation can be written as
BmT
which states that if the magnetic moment m is parallel to the magnetic
flux density B, the torque acting on the frame is zero. If they are
perpendicular to each other, the torque is maximum.
The magnetic flux density can also be described using a set of curves.
The tangential direction at a point on the curve stands for the direction
of magnetic flux density, and these curves are called magnetic field lines.
The vector equation for the magnetic field line is
0d lB
The magnetic field lines cannot also be intersected.
The flux of the magnetic flux density B through a directed surface
S is called magnetic flux, and it is denoted as , given by
SB d S
The unit of magnetic flux is weber (Wb).
As the electric field lines, the density of the magnetic field lines can
describe the intensity of the magnetic field. A larger density of
magnetic field lines stands for stronger magnetic field intensity.
2. Equations for Steady Magnetic Fields in Free Space
The magnetic flux density B of a steady magnetic field in
vacuum satisfies the following equations
Il 0 d lB
S 0dSB
Left equation is called Ampere’s circuital law, where 0 is the
permeability of vacuum, H/m , and I is the current
enclosed by the closed curve.
70 10π4
Ampere’s circuital law: The circulation of the magnetic flux
density in vacuum around a closed curve is equal to the current
enclosed by the curve multiplied by the permeability of vacuum.
The magnetic field lines are closed everywhere, with no beginning or
end. This may be called the principle of magnetic flux continuity.
Right equation shows that the total magnetic flux through a closed
surface is equal to zero.
From Stokes’ theorem we have Sl
SBlB d)(d
And considering S
I SJ d
0d)( 0 S SJB
Then, from Ampere’s circuital law we have
Since the above equation holds for any surface, the integrand
should be zero, leading to
JB 0
which states that the curl of the magnetic flux density of a steady
magnetic field at a point in vacuum is equal to the product of the
current density at the point and the permeability of vacuum.
From the divergence theorem we have
VS
V
d d BSB
V
V
0d B
0 B
Since the equation holds everywhere, the integrand should be zero,
i.e.
which states that the divergence of the magnetic flux density of a steady
magnetic field is equal to zero everywhere.
Consequently, we find the differential form of the equations for the
steady magnetic field in vacuum as
JB 0 0 B
The steady magnetic field in vacuum is a solenoidal field.
Considering , we obtain 0d
SSB
Based on Helmholtz’s theorem, the magnetic flux density B
should be )()()( rArrB
VV
d
)(
π4
1)(
rr
rBr V
V
d
)(
π4
1)(
rr
rBrA
Where
0)( r VV
d
)(
4π)(
0
rr
rJrA
which shows that the magnetic flux density of a steady magnetic field
at a point in vacuum is equal to the curl of the vector function A at the
point.
Considering , we haveJB 0 0 B
and )()( rArB
If the distribution of the current is known, the vector magnetic
potential A at a point can be found, and we can calculate the
magnetic flux density at the point.
VV
d
) ()(
4π)( 3
0
rr
rrrJrB
which is called the Biot-Savart’s law.
The current can be distributed in a volume, on a surface, or in a
line, and they are called volume current, surface current, and line
current, respectively.
lJJ ddd ISV S
The relationship between the magnetic flux density and the
current is
An equivalent relation among these currents is as
Where JS is the surface current density (A/m), and the direction of dl
is the flow direction of the line current.
The vector magnetic potentials and the magnetic flux densities
caused by a surface current and a line current are, respectively
SS
S
d
)(
π4)( 0
rr
rJrA
S
S
S
d)()(
π4)(
30
rr
rrrJrB
l rr
lrA
d
π4)( 0 I
l rr
rrlrB
30 )(d
π4)(
I
For some steady magnetic fields, it will be simple to calculate
the magnetic flux density based on Ampere’s circuital law.
For this, we need to find a closed curve along which the magnitude of
the magnetic flux density is constant everywhere, and the direction
coincides with the tangential direction of the curve. Then the vector
integral becomes a scalar integral, B can be taken out of the integral,
and it can be determined.
Il 0 d lB
Example 1. Calculate the magnetic flux density of an infinitely
long line current of I.
rO
z
y
x
dl
I
r ′ r – r ′
e
Solution: Select the cylindrical coordinate
system, and let the line current be along the z-
axis, then the direction of the vector
is that of B. And the direction of the cross
product vector is that of the unit vector e ,
and the direction of B is that of e , i.e.
)(d rrl
eB B
which states that the magnetic field lines are a set of the circles with
the center at the z-axis, and the magnetic flux densities is independent
of the variable .
Since the line current is infinitely long, the fields must have no
dependence on z.
In this case, the circulation of the magnetic flux density around
the magnetic field line of radius r is
rB π2d lB
Based on Ampere’s circuital law, we have
r
IB
π2 0
We can prove that this equation is valid for
the magnetic fields outside the cylindrical
conducting wire of certain cross section and
carrying a current I.
I B
Example 2. Calculate the magnetic flux density produced by a
current loop of radius a and carrying the current I.
r
z
y
x
a
r'r - r'
e'
'
x
yO
a
r''
'e'–ex
ey
e'
Solution: Employing the cylindrical
coordinate system, let the center of the
loop to be at the origin, and the plane
of the loop in the xy-plane. Because of
the symmetry of the structure, the
fields must be independent of .
Finally, we obtain
20
π4
sin)(
r
IS erA
where is the area of the loop.2 π aS
For the convenience of calculation,
let the field points be in the xz-plane.
Consider the magnetic moment of the loop current is ,
the above equation becomes
ISzem
30
π4)(
r
rmrA
Using , we
have
AB
sincos2 π4
)(3
0 eerB rr
IS
The above results show that the vector magnetic potential A caused
by a current loop is inversely proportional to the square of the distance
r, the magnetic flux density B is inversely proportional to the cube of
the distance r, and both are related to the elevation angle.
which is correct for any magnetic
dipole with the magnetic moment m
and at the origin.
m
rA(r)
x
z
y
3. Vector & Scalar Magnetic Potentials
The relationship between the vector magnetic potential A and
the magnetic flux density B is
AB
The vector magnetic potential is different from the scalar electric
potential, but it is mainly an auxiliary function (辅助函数) for
deriving the field quantities.
Since , we have0 A AA 2 A
BA 2
In view of this, the vector magnetic potential A satisfies vector Poisson’s
equation.
when the distribution of the current is unknown, we have to solve the
equations for the steady magnetic fields based on the boundary conditions
(边界条件) . For this reason, we need to derive the differential
equations for the vector magnetic potential.
JA 0 2
In the source-free region, the above equation becomes Laplace’s
equation. 02 A
In rectangular coordinate system, Both Poisson’s and Laplace’s
equations can be decomposed into three scalar equations for the three
coordinate components, respectively. Therefore, both the method of
Green’s function and the method of separation of variables can be used
to solve the scalar Poisson’s and Laplace’s equations for all rectangular
components of the vector magnetic potential.
S
Φ SA d)(We know , then S
Φ
dSB
Using Stokes’ theorem, we obtain l
Φ
d lA
Obviously, it is very simple to calculate the magnetic flux by using
the vector magnetic potential.
In addition, the method of images can also be used .
In a source-free region, J = 0, and . Thus the magnetic
flux density B is irrotational (无旋) . In this case, the magnetic
flux density can be expressed in terms of the gradient of a scalar field
by letting
0 B
m0 B
where m is called the scalar magnetic potential.
In view of this, the scalar magnetic potential satisfies Laplace’s
equation. In this way, from the boundary conditions we can solve
Laplace’s equation to obtain the scalar magnetic potential, and the
magnetic flux density can be determined.
Nevertheless, the scalar magnetic potential is used only for source-
free regions.
0m2 0 BDue to , we have
4. Magnetization (磁化) of Media
In an atom the electrons are continuously orbited around the
nucleus, and closed loop currents are formed. Such a current loop
represents a magnetic dipole.
Under the influence of an external magnetic field, these dipoles
are rearranged, and a macroscopic( 宏 观 ) magnetic moment is
resulted in. This phenomenon is called magnetization.
In general, the arrangement of these magnetic dipoles is random,
due to thermal motion (热运动) , leading to zero overall magnetic
moment, or the lack of magnetization.
On the other hand, the electrons and the atomic nuclei themselves
are in spinning motion (自旋运动) , and they also form magnetic
dipoles with magnetic moments.
Medium
Composite field Ba+ Bs
Applied field Ba
Unlike the polarization of dielectric, the composite magnetic field
in the medium could be larger or smaller than the external
magnetic field, while the internal electric field is always smaller.
Magnetization Process
Secondary field Bs
Magnetization
Based on the magnetization process of the media, the behavior of
magnetization can be classified into three types: diamagnetic 抗磁性 ,
paramagnetic 顺词性 , ferromagnetic 铁磁性 and ferrimagnetic 亚铁磁性 media.
Diamagnetic: In these media, the composite magnetic moment is
zero under normal condition. When an external
magnetic field is applied, besides the spinning and
the orbital motion of the atom, the atomic orbit will
rotated about the applied magnetic field. This
motion is called precession 进动 .
The direction of the magnetic moment caused by
the atom is always opposite to the applied magnetic
field, and it results in a lower composite magnetic
field.
Bt
Silver, copper, bismuth, zinc, lead, mercury, and
other metals are diamagnetic.
Paramagnetic: The macroscopic composite magnetic moment is
zero due to thermal motion. Under the influence of applied magnetic
field, the direction of the magnetic moment of magnetic dipole is
turned toward the direction of the applied magnetic field. Hence, the
composite magnetic field will be increased. Aluminum, tin,
magnesium, wolfram, platinum, palladium, and others are
paramagnetic. Ferromagnetic: “Magnetic domains 磁 畴 ” exist in these
media. Under the influence of an applied magnetic field, a number
of magnetic domains will be turned. The directions of the magnetic
domains will give rise to a net magnetization that produces a
stronger total magnetic field. These media are called ferromagnetic.
Iron, cobalt, nickel, and others are ferromagnetic.
These ferromagnetic media often exhibit memory effect and
acquiring spontaneous magnetization 天然磁性 .
Ferrimagnetic: It is a metal oxide 金属氧化物 , and the magnetism
is weaker than that of ferromagnetic media, while the conductivity 导电 率 is very small. These media are called ferrimagnetic materials,
such as ferrites 铁氧体 .
The magnetization will generate the magnetic moments in the
medium. In order to measure the magnetization of the medium, we
define the vector sum of the magnetic moment per unit volume as the
magnetization intensity 磁化强度 , and it is denoted as M, given by
V
N
i
1
imM
where mi is the magnetic moment of the i-th magnetic dipole in .
is an infinitesimal volume.V
V
Ferrites are applied in microwave devices.
After magnetization, the resulted magnetic moment is produced by
the new currents in the media, and these currents are called magnetizing
currents 磁化电流 . The density of magnetizing current is denoted as J '.
SV
SV d)(
4πd
)(
4π)( n0 0
rr
erM
rr
rMrA
x
P
z
yr
dV'
O
V'r'
r - r'
S'
Where the first term stands for the vector
magnetic potential produced by the volume
magnetizing currents, and the second term is
that by the surface magnetizing currents.
The relationship between the density of magnetizing current and the
magnetization intensity M will be derived below.
After some derivations, we obtain
MJ neMJ SHence, we find
Example. A magnetic cylinder of radius a and length l is magnetized
uniformly along the axis of the cylinder. If the magnetization intensity is
M, find the magnetic flux density produced by the magnetizing current
at the point P of the axis and at a distance much larger than the radius of
the cylinder.
x
y
z
l
P(0,0, z)
O
a
SJ
Solution: Select cylindrical coordinate
system, and let the z-axis coincide with the
axis of the cylinder.
Since the magnetization is uniform, the
magnetization intensity is independent of the
coordinate variables. Hence,
0 MJ
We know the surface magnetizing current density is
neMJ S
where en is the unit vector of the outward normal to the surface.
eeeeMJ MM rzS n
x
y
z
l
P(0,0, z)
zdz'
O
a
SJ
These surface magnetizing currents form
loop currents on the lateral surface. The loop
current of width dz at z is ( dz) , and the
magnetic flux density dB caused by the current
loop at the point P(0, 0, z) (z >> a) is
SJ
zzz
Maz
d
)(2d
3
20
eB
The composite magnetic flux density produced by all of the
magnetizing currents at the point P is
zzz
Ma l
z
d)(
1
2
0 3
20 eB
22
20 1
)(
1
4 zlz
Maz
e
Since , the surface magnetizing currents exist only on the
lateral surface 侧面 of the cylinder, while the magnetizing currents
on the upper and the lower end faces are zero. Therefore, we have
MzeM
SJ
5. Equations for Steady Magnetic Fields 恒定磁场 in A Medium
In the magnetized medium, the magnetic field can be considered as
that produced by the conducting current I and the magnetizing
current I in vacuum. In this way, the circulation of the magnetic flux
density B around a closed curve is
)(d 0 IIl
lB
Il
lM
Bd
0 Considering , we have
lI
dlM
Let , thenHMB
0
Il
lH d
where H is called magnetic field intensity 磁场强度 , with a unit of
A/m. The above equation is called Ampere’s circuital law for a
medium, and it shows that in the medium the circulation of the
magnetic field intensity around a closed curve is equal to the
conducting current enclosed by the curve.
By using Stokes’ theorem, from the above equation, we obtain
JH
which is the differential form of Ampere’s circuital law in a medium ,
and it states that in a medium the curl of the magnetic field intensity at
a point is equal to the density of the conduction current at the point.
The magnetizing currents do not affect that the magnetic field lines
are closed everywhere, hence, in medium the flux of the magnetic flux
density through a closed surface is still zero, and the divergence of the
magnetic flux density is zero everywhere, i.e.
S
0dSB 0 B
Since the magnetic field intensity is only related to the conducting
current 传导电流 , the magnetic field intensity simplifies the calculation
of the magnetic fields in medium just likes the electric flux density
simplified that of the electrostatic fields in dielectric.
For most of the materials, the magnetization intensity M is
proportional 成比例的 to the magnetic field intensity H, i.e.
HM m
where m is called magnetic susceptibility 磁化率 , and it could be a
positive or negative real number.
Considering , we haveHMB
0
HB )1( m0
Let , then )1( m0 HB
where is called permeability 磁导率 , and it is usually expressed by
the relative value as
m0
r 1
Nevertheless, whether it is the diamagnetic or the paramagnetic
material, magnetization is very weak. Hence, usually their relative
permeability can be considered to be equal to 1.
After a diamagnetic 抗磁性 material is magnetized the composite
magnetic field will be decreased, thus
1 , ,0 r 0 m
A paramagnetic 顺磁性 material is magnetized, the composite
magnetic field will be increased, thus1 , ,0 r 0 m
Recently, a macromolecule 高分子 magnetic material is developed,
and the relative permeability could be on the same order as the
permittivity 介电常数 .
The magnetization in ferromagnetic materials is very strong, so that
the value of the relative permeability could be very large.
The relative permabilities of
three kinds of magnetic materials
Materials r
Gold 0.9996
Silver 0.9998
Copper 0.9999
Materials r
Aluminum 1.000021
Magnesium 1.000012
Titanium 1.000180
Materials r
Nickel 250
Iron 4000
Mu-metal 磁性合金 105
Diamagnetic FerromagneticParamagnetic
Similar to the electrical properties of the dielectrics, the
magnetic properties of the media can also be homogeneous or
inhomogeneous, linear or non-linear, and isotropic or anisotropic.
HB
33 23 31
32 22 21
31 12 11
If the permeability does not vary with space, and the medium is
called homogeneous for the magnetic property, otherwise, it is
inhomogeneous. If the permeability is independent of the magnitude
and the direction of applied magnetic field, and the magnetic flux
density is proportional to the magnetic field intensity, it is called a
linear isotropic medium for its magnetic property.
The permeability of an anisotropic medium has 9 components, and
the relationship between B and H is
For homogeneous, linear, and isotropic magnetic media, we have
l
I
d lB JB
S
0d SH 0 H
Because of , from Helmholtz’s theorem we obtain JB
VV
d
)(
π4)(
rr
rJrA
which satisfies the following differential equation
JA 2
The upper equation is the special solution 特解 of the lower equation.
The above results show that for homogeneous, linear, and isotropic
magnetic media, the equations in free space can be used, provided the
permeability of vacuum 0 is replaced by that of the medium 。
6. Boundary Conditions for Steady Magnetic Fields
The derivation of the boundary conditions is similar to that of
electrostatic fields, and here we only list the results as follows:
1
2
B2
H1B1
H2en
(a) If there is no surface current at boundary,
then we find2t1t HH
For linear isotropic media, the above equation
can be rewritten as
2
2t
1
1t
BB
(b) The normal components of the magnetic flux density are
continuous. 2n1n BB
For linear isotropic media, we have
n22 n11 HH
Il
lH d S
0dSB
Both the magnitude and the direction of the magnetic field intensity
and magnetic flux density will be discontinuous. In fact, this
discontinuity is resulted by the surface magnetizing current at the
boundary.
SJBB 0 2t1t
Applying the right hand rule on the direction of the path and the
directed surface bound by the path, the above equation can be
rewritten in the following vector form
SJeMM n21 )(
1
2
en et
1M
2M
SJ
At the boundary, the relationship between the tangential components
of magnetic flux densities and the magnetizing current is
SJMM 2t1t
Considering , we have Il
lM d
Il 0 d lB
A medium with infinite permeability is called a perfect magnetic
conductor 理想导磁体 . The magnetic field intensity 磁场强度 should
be zero in a perfect magnetic conductor.
Otherwise, from we can see that infinite magnetic flux
density , an infinite electric current, and infinite energy are required.
HB
The tangential component of the magnetic field
intensity is continuous at the boundary. Hence, it
cannot exist on the surface of a perfect magnetic
conductor.
H
In other words, the magnetic field intensity must be perpendicular
to the surface of a perfect magnetic conductor.
Example 1. A loop magnetic core 环型磁芯 with a gap is closely
wound by a coil with N turns, as shown in the figure. When the coil
carries a current I, and the leakage 漏 magnetic flux outside the coil
is neglected, find the magnetic flux density and the magnetic field
intensity in the core and the gap.
Solution: Since the leakage magnetic flux
is neglected, the direction of the magnetic
flux density is around the circle, and it is
perpendicular to the end faces of the gap.
From the boundary condition, we know that
the magnetic flux density Bg in the gap is
equal to Bf in the core, i.e.
fg0fg HHBB
Since r0 >> a , the magnetic field in the core can be considered
to be uniform. Using Ampere’s circuital law in media, and taking
the circle of radius r0 as the integral path, then we have NI lH d
NIdrB
dB
) π2( 0f
0
g
Considering , we have fg BB
) π2(
00
0 fg drd
NI
eBB
Then) π2(
00 0
gg drd
NI
eB
H In the gap
) π2(
00
0 ff drd
NI
eB
H In the core
Example 2. An infinitely long wire carrying a direct electric
current 直流电 I is placed parallel to an infinite perfect magnetic
conducting plane. The distance between the wire and the plane is h;
find the magnetic field intensity in the upper half-space.
X
h
y
x =
0
I
O
Solution: The method of images 镜像法 is used. An infinitely long
electric current I is placed at the image position. The resultant
magnetic field intensity in the upper half-space is found as
r
I
r
I
π2 π221 eeHHH
r'
h
h
Py
x 0
I
H1H2
H1H2
H
O
r
I'
'
0
Since the tangential component 切向分量 of the magnetic field
intensity must be zero on the surface of the perfect magnetic conductor,
we haveII
Hence, the resultant 合成的 magnetic field intensity is
xy hyx
hy
hyx
hy
hyx
x
hyx
xIeeH
22222222 )()()()(π2
For any point on the boundary, y = 0 , we
obtainyhx
xIeH
)( π
22
we can see that the magnetic field intensity to be perpendicular
垂直的 to the boundary.
Example 3. An infinite line current I is nearby an infinite interface
formed between two media, and the permeabilities of the media are 1
and 2, respectively. Find the magnetic field intensities in the two
media.
I
2
1 =
Solution: Assume the current I is placed in medium 2, as shown in
the following figure.
I
H2
I '
H'
e '
e
+
1
I " e
H"
I
2
1 = +
I
H2
I '
H'
e '
e
1
I " e
H"
Uniqueness theorem 唯一性定理 shows that the field depends on the
source and the boundary. After making the above postulates, the upper
half-space is still a region with source, and the lower half-space is still a
source-free region.
II
II
2 1
2
2 1
2 1
2
In order to maintain the original boundary conditions, the
resultant fields should obey the boundary conditions, .2n1n2t1t , BBHH
III
III
1 2 2 We find
eeH)( π2
)(
π2 2 1
2 1 2 r
I
r
I22 2 HB
eH)( π 2 1
2 1
r
I11 1 HB Then
01 H eHB
r
I
π2
11 1
In this case, the image currents are . These results are
the same as before.
0, III
Obviously, if medium 1 is a perfect magnetic conductor, i.e. ,
then
1