4.5 Magnetic field intensity and relative permeability
Orbiting electrons around nucleus cause circulating currents and
consequently magnetic dipoles.
Electrons and nucleus of an atom spin on their own axes
create certain magnetic dipole moments.
In the absence of an external magnetic field the magnetic dipoles of the atoms of most
materials (except permanent magnets) have random orientations, resulting in no net
magnetic moment. The application of an external magnetic field causes both an
alignment of the magnetic moments of the spinning electrons and an induced magnetic
moment due to a change in the orbital motion of electrons.
The strength of this magnetizing effect is described by
the magnetization vector M, which is related to
magnetization current densities by
J m = " ! M ( A / m2 )
volume current density
surface current density J ms = M ! a n
(A/ m )
Therefore, the second postulate becomes
1
"!B = J +"!M
µ0
or
#
&B
$
M
) ( $ ' !! = J
0
%1µ4
24
3"
H
A cross section of a magnetized
material.
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.5 Magnetic field intensity and relative permeability
We define the fourth new fundamental field quantity, the magnetic field
intensity H, such that
H=
B
!M
µ0
The use of the H enables us to write a curl equation relating the magnetic field and
the distribution of free currents in any medium.
" ! H = J ( A / m2 )
where J ( A / m 2 ) is the volume density of free current.
When the magnetic properties of the medium are linear and isotropic, the magnetization
is directly proportional to the magnetic field intensity
M = ! mH
µr
or
B = µ0 (1 + ! m )H = µ0 µ r H
: Relative permeability.
µ r µ0 : Permeability
The permeability of most materials is very close to that of free space
µ0 . For ferromagnetic materials such as iron, and nickel, µ could
r
be very large (50-5000 and up to 106); the permeability depends not
only on the magnitude of H but also on the history of the material.
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.6 Boundary conditions for magnetostatic fields
Because " ! B = 0 , in light of " ! D = 0
associated with boundary condition
D1n = D2 n
we have
The normal component boundary condition
B1n = B2 n
or
µ1 H1n = µ 2 H 2 n
The boundary condition for the tangential components of magnetostatic field is
obtained from the integral form of the curl equation for H:
r
" H ! dl = I
C
We now choose the closed path abcda as the contour
C and letting bc = da approach zero, we have
r
$ H # dl = H1 # !w + H 2 # ("!w) = J sn !w
abcda
or
H1t ! H 2t = J sn ( A / m)
More concisely,
a n 2 " ( H1 ! H 2 ) = J s ( A / m )
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.6 Boundary conditions for magnetostatic fields
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.7 Inductances and inductors
Consider two neighboring closed loops, C1 and C2 bounding surfaces S1 and S2,
respectively. If a current I1 flows in C1, a magnetic field B1 will be created. Some of the
magnetic flux due to B1 will link with C2, that it will pass through the surfaces S2:
"12 = # B1 ! ds 2
(Wb)
From the Biot-Savart law, we see that B1 is directly proportional to I1; hence
!12 = L12 I1
In case C2 has N2 turns, the flux linkage !12 due to !12 is
The mutual inductance between two circuits is L12 =
"12 = N 2 !12
!12
I1
Its physical meaning is the magnetic flux linkage with
One circuit per unit current in the other.
The self inductance of loop C1 is defined as L11 =
It measures the magnetic flux linkage per unit
current in the loop itself.
!11
I1
Two magnetically
coupled loops.
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.7 Inductances and inductors
µ 0 rI
2!a 2
µI
B 2 = a" B" 2 = a" 0
2!r
a) Inside the inner conductor, 0 ! r ! a we can easily find that B1 = a" B" 1 = a"
b) Between the inner and outer conductors, a ! r ! b
we have
Assume the current I is uniformly distributed over the cross section of the inner conductor.
To calculate the external inductance per
unit length of the coaxial line, we
calculate the flux between a ! r ! b
# ext = $
b
a
µI
B" 1dr = 0
2!
$
b
a
dr µ 0 I b
=
ln( )
2!
r
a
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
4.7 Inductances and inductors
Lext =
The external inductance
" ext µ 0
b
=
ln( )
2!
I
a
To calculate the internal inductance per unit length of the coaxial line, we calculate
the flux between r and a generated by the current in the region of 0 to r.
d" int = #
a
r
µI
B% 2dr = 0 2
2$a
#
a
rdr =
r
µ0 I 2 2
(a ! r )
4$a 2
We now consider an annular ring in the inner conductor between radii r and r+dr. But the
Current in the annular ring is only a fraction 2!rdr / !a 2 = 2rdr / a 2 of the total current I.
d" int =
Lint
"
=
r =a
r =0
d$ int
I
=
2rdr
d! int
a2
µ0 1
!a 2 2a 2
"
a
0
(a 2 # r 2 )rdr =
µ0
8!
The total inductance is
L = Lint + Lext
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
Chapter 5
Steady Electric Currents
There are several types of electric currents caused by
the motion of free charges:
Conduction currents
In conductors and semiconductors are caused by drift motion of
conduction electrons and/or holes;
Electrolytic currents
are the result of migration of positive and negative ions
Convection currents
Result from motion of electrons and/or ions in a vacuum.
Our attention is the conduction currents
that are governed by Ohmʼs law
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
5.1 Equation of continuity and KCL
The current leaving the region is the total outward flux of the current density vector
through the surface S. We have
I = ! J # ds = "
S
dQ
d
= " ! $dv
dt
dt V
! $ # Jdv = " !
V
V
d%
dv
dt
$#J = "
d!
dt
The last equation is called equation of continuity.
For steady currents, charge density does not vary with time, we have
which is
!I
j
=0
"!J = 0
Kirchhoffʼs current law.
j
Combining
&$ # E = "
!%
and
!t
# = # 0 e $ (" / ! ) t
$ # E = " / ! we have
(C / m 3 )
It says that the charge density at a given location will decrease with time exponentially.
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
5.2 Boundary conditions for current density
The boundary condition for current density can be directly derived by
comparing the governing equations of current density and that of electric
field:
Equations for E
Boundary Condition
of E
"!D = 0
Dn1 = Dn 2
"!E = 0
Et 1 = Et 2
Equations for J
Boundary Condition
of J
"!J = 0
&J#
('$ ! = 0
%) "
J n1 = J n 2
J t1 / ! 1 = J t 2 / ! 2
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
5.3 Resistance calculations
We have discussed the procedure for finding the capacitance between two conductors
separated by a dielectric medium. These conductors may be of arbitrary shapes. In
terms of electric field quantities the basic formula for capacitance can be written as
$E " ds
D " ds
Q
!
!
S
S
C= =
=
V # ! E " dl # ! E " dl
L
L
where the surface integral is carried out over a surface
enclosing the positive conductor and the line integral in the
denominator is from the negative conductor to the positive
conductor.
Ohmʼs law, J = !E, ensures that the streamlines for J and E will be the same in an
isotropic medium. The resistance between the conductors is
V # !LE " dl # !LE " dl
R= =
=
I
! J " ds ! $E " ds
S
Therefore,
RC =
S
C "
=
G !
The equation holds if ! and ! of the medium have
the same space dependence or if the medium is
homogeneous. In these cases, if the capacitance
between two conductors is known, the resistance can
be obtained directly from the " / !
ratio without
recomputation.
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16
5.3 Resistance calculations (cont.)
Basic Electromagnetics, Dept. of Elec. Eng., The Chinese University of Hong Kong, Prof. K.-L. Wu / Prof. Th. Blu
Lesson 15&16