3 slides

Dr. Rakhesh Singh Kshetrimayum
3. Magnetostatic fields
Dr. Rakhesh Singh Kshetrimayum
2/16/20131 Electromagnetic FieldTheory by R. S. Kshetrimayum

3.1 Introduction to electric currents
Electric currents
Ohm’s law Kirchoff’s law Joule’s law
Boundary
conditions
2/16/2013Electromagnetic FieldTheory by R. S. Kshetrimayum2
Kirchoff’s current
law
Kirchoff’s voltage
law
Fig. 3.1 Electric currents

So far we have discussed electrostatic fields associated with
stationary charges
What happens when these charges started moving with
uniform velocity?
It creates electric currents and electric currents creates
magnetic fieldsmagnetic fields
In electric currents, we will study
Ohm’s law,
Kirchoff’s law
Joule’s law
Behavior of current density at a media interface

3.1.1 Current density
What is this?
For a particular surface S in a conductor, i is the flux of the
current density vector over that surface or
mathematically
i j ds= •∫
r r
j
r
3.1.2 Ohm’s law
It states that the current passing through a homogeneous
conductor is proportional to
the potential difference applied across it and
the constant of proportionality is 1/R which is dependent on
the material parameters of the conductor
S
i j ds= •∫

Mathematically,
From the relation between
current density (j) and current (i),
V
i V i
R
∝ ⇒ =
current density (j) and current (i),
electric potential (V) with electric field (E) and
resistance (R) with resistivity (ρ) in an isotropic material
we can obtain the Ohm’s law in point form as
V Edl Edlds E
jds i j E
dlR dl
ds
σ
ρ ρρ
= = = = ⇒ = =

where is the conductivity and
is the resistivity of the isotropic material
σ
1
ρ
σ
=
Material σ (S/m)
Rubber 10-15
Table 3.1 Conductivities of some common materials
Rubber 10-15
Water 2×10-14
Gold 4×107
Aluminum 3×107
Copper 5×107

Some points on perfect conductors and electric fields:
Perfect conductors or metals have infinite conductivity ideally
An infinite conductivity means for any non-zero electric field
one would get an infinite current density which is physically
impossibleimpossible
Perfect conductors do not have any electric fields inside it
Perfect conductors are always an equipotential surface
At the surface of the perfect conductor, the tangential
component of the electric field must be zero

3.2 Equation of continuity and KCL
j
r
ds
r
Fig. 3.3 Equation of continuity








−=•⇒







−=−= ∫∫∫ V
v
SV
v dv
dt
d
sdjdv
dt
d
dt
dq
i ρρ
rr

The above equation is integral form of equation of continuity
It states that any change of charge in a region must be
accompanied by a flow of charge across the surface bounding
the region
It is basically a principle of conservation of chargeIt is basically a principle of conservation of charge
By applying the divergence theorem
0V V
V V V
d d
jdv dv j dv
dt dt
ρ ρ 
∇• = − ⇒ ∇• + = 
 
∫ ∫ ∫
r r

Since the volume under consideration is arbitrary
Differential form of the equation of the continuity
0Vd
j
dt
ρ
∇• + =
r
At steady state, there can be no points of changing charge
density
Vd
j
dt
ρ
∇ • = −
r
∫ =•⇒=•∇
S
sdjj 00
rrr

The net steady current through any closed surface is zero
If we shrink the closed surface to a point, it becomes
Kirchoff’s current law (KCL)
0I =∑
KCL states that at any node (junction) in an electrical circuit,
the sum of currents flowing into that node is equal to the sum of
currents flowing out of that node
∑

3.3 Electromotive force and KVL
Fig. 3.4 Proof of KVL

When a resistor is connected between terminals 1 and 2 of
the battery,
The total electric field intensity’s (total electric field
comprise of electrostatic electric field as well as the
impressed electric fieldcaused by chemical action) relation toimpressed electric fieldcaused by chemical action) relation to
the current density is given as
where the superscript “c” is for conservative field and
the superscript “n” is for non-conservative field
( )c n
j E Eσ= +
r rr

Conservative electric field exists both inside the battery and
along the wire outside the battery,
While the impressed non-conservative electric field exists
inside the battery only
The line integral of the total electric field around the closedThe line integral of the total electric field around the closed
circuit gives
( ) ld
j
ldEE
C C
nc
r
r
rrr
•=•+∫ ∫σ

Note that the line integral of the conservative field over a
closed loop is zero
The line integral of the non-conservative field in non-zero
and is equal to the emf of the battery source
Since non-conservative field outside the battery is zero,Since non-conservative field outside the battery is zero,
( ) ( ) ld
j
ldEldEE
C C
nnc
r
r
rrrrr
•==•=•+∫ ∫∫ σ
ξ
2
1

Note that i = jA or j=i/A
Therefore, the voltage drop across the resistor is
V=jl/σ=il/σA=iρl/A=iR
If there are more than one source of emf and more than one
resistor in the closed path, we get Kirchhoff'sVoltage Law
(KVL)(KVL)
KVL states that around a closed path in an electric circuit,
the algebraic sum of the emfs is equal to the algebraic sum of the voltage
drops across the resistances
1 1
M N
m n n
m n
i Rξ
= =
=∑ ∑

3.4 Joule’s law and power dissipation
Consider a medium in which charges are moving with an
average velocity v under the influence of an electric field
If ρv is the volume charge density, then the force
experienced by the charge in the volume dv is
ρ= =
r r r
If the charge moves a distance dl in a time dt, the work done
by the electric field is
VdF dqE dvEρ= =
r r r
( )V VdW dF dl dvE vdt E v dvdt E jdvdt j Edvdtρ ρ= • = • = • = • = •
rr r r r rr rr r

3.4 Joule’s law and power dissipation
Then, the elemental work done per unit time is
If we define the power density p as the power per unit
volume, then, point form of Joule’s law is
dW
dP j Edv
dt
= = − •
rr
volume, then, point form of Joule’s law is
The power associated with the volume (integral form of
Joule’s law) is given by
p j E= •
rr
V V
P pdv j Edv= = •∫ ∫
rr

3.5 Boundary conditions for current density
S∆
1σ
Fig. 3.5 Boundary conditions for current density
2σ

How does the current density vector changes when passing
through an interface of two media of different conductivities
σ1 and σ2?
Let us construct a pillbox whose height is so small that the
contribution from the curved surface of the cylinder to thecontribution from the curved surface of the cylinder to the
current can be neglected
Applying equation of continuity and computing the surface
integrals, we have,
( ) 211121 0ˆ0ˆˆ0 nn
S
JJJJnsJnsJnsdji =⇒=−•⇒=∆•−∆•⇒=•= ∫
rrrrrr

It states that the normal component of electric current
density is continuous across the boundary
Since, we have another boundary condition that the
tangential component of the electric field is continuous
across the boundary, that is,across the boundary, that is,
( ) 1 2 11 2 1
1 2
1 2 1 2 2 2
0 0 0t t t
t
J J JJ J
n E E n
J
σ
σ σ σ σ σ
 
× − = ⇒ × − = ⇒ − = ⇒ = 
 
r r
r r) )

The ratio of the tangential components of the current
densities at the interface is equal to the ratio of the
conductivities of the two media
We can also calculate the free charge density from the
boundary condition on the normal components of theboundary condition on the normal components of the
electric flux densities as follows:
1 2 1 2
1 2 1 1 2 2 1 2 1
1 2 1 2
n n
n n S S n n S n
J J
D D E E J
ε ε
ρ ρ ε ε ρ ε ε
σ σ σ σ
 
− = ⇒ = − ⇒ = − = − 
 

3.6 Introduction to magnetostatics
In static magnetic fields, the three fundamental laws are
Biot Savart’s Law,
Gauss’s law for magnetic fields and
Ampere’s circuital law
Biot Savart law gives the magnetic field due to a current
carrying elementcarrying element
From Gauss’s law for magnetic fields, we can understand that
the magnetic field lines are always continuous
In other words, magnetic monopole does not exist in nature
Ampere’s circuital law states that a current carrying loop
produces a magnetic field

Magnetostatics
Biot Savart’s
law
Gauss’s law for
Magnetic vector
potential
Boundary
Self and mutual
inductance
Gauss’s law for
magnetic fields
Magnetization
Ampere’s
law
Boundary
conditions
Fig. 3.6 Magnetostatics
Magnetic vector
potential in materials

3.6 Introduction to magnetostatics
It is easier to find magnetic fields from the curl of magnetic
vector potential whose direction is along the direction of
electric current density
Another topic we will study here is that how do magnetic
fields behave in a mediumfields behave in a medium
We will also try to find the
self and mutual inductance and
magnetic energy

3.7 Biot Savart’s law
The magnetic field due to a current carrying segment is
proportional to
its length and
the current it is carrying and
the sine of the angle between andr
r
Idl
r
the sine of the angle between and
inversely proportional to the square of distance r of the point
of observation P from the source current element
Mathematically,
0
2 2 2
4
dl r dl r dl r
dB I dB kI dB I
r r r
µ
π
× × ×
∝ ⇒ = ⇒ =
r r r
$ $ $r r r
r Idl

3.8 Gauss’s law for magnetic fields
In studying electric fields, we found that electric charges
could be separated from each other such that a positive
charge existed independently from a negative charge
Would the same separation of magnetic poles exist?
A magnetic monopole has not been observed or found in
naturenature
We find that magnetic field lines are continuous and do not
originate or terminate at a point
Enclosing an arbitrary point with a closed surface, we can
express this fact mathematically integral form of 3rd
Maxwell’s equations
∫ =•=Ψ
S
sdB 0
rr

3.8 Gauss’s law for magnetic fields
Using the divergence theorem,
In order this integral to be equal to zero for any arbitrary
volume, the integrand itself must be identically zero which
( )∫ ∫ =•∇=•=Ψ
S V
dvBsdB 0
rrr
volume, the integrand itself must be identically zero which
gives differential form of 3rd Maxwell’s equations
=0B∇•
ur

3.9 Ampere’s circuital law
In 1820, Christian Oersted observed that compass needles
were deflected when an electrical current flowed through a
nearby wire
Right hand grip rule: if your thumb points in the direction of
current flow, then your fingers’ grip points in the direction ofcurrent flow, then your fingers’ grip points in the direction of
magnetic field
AndreAmpere formulated that the line integral of magnetic
field around any closed path equals µ0 times the current
enclosed by the surface bounded by the closed path

Incomplete integral form of 4th Maxwell’s equation
By application of Stoke’s theorem
∫ =•
C
enclosedIldB 0µ
rr
In order the integral to be equal on both sides of the above
equation for any arbitrary surface, the two integrands must
be equal
( )∫ ∫∫ •=•×∇=•
C SS
sdJsdBldB
rrrrrr
0µ

Incomplete differential form of 4th Maxwell’s equation
Note that there is a fundamental flaw in this Ampere’s
circuital law
0=B Jµ∇ ×
ur uur
circuital law
Maxwell in fact corrected this Ampere’s circuital law by
adding displacement current in the RHS
Lorentz force equation: for a charge q moving in the uniform
field of both electric and magnetic fields, the total force on
the charge is
E MF F F qE qv B= + = + ×
r r r r rr

3.10 Magnetic vector potential
Some cases, it is expedient to work with magnetic vector
potential and then obtain magnetic flux density
Since magnetic flux density is solenoidal, its divergence is
zero
( ) =0B∇ •
ur
A vector whose divergence is zero can be expressed in term
of the curl of another vector quantity
=B A∇×
ur ur

From Biot Savart’s law,
It is a standard notation to choose primed coordinates for the
source and unprimed coordinates for the field or observation
3
'
=
4
O I dl R
B
R
µ
π
×
∫
r ur
ur
$ $= (x-x') +(y-y') +(z-z')R x y z
ur
$
source and unprimed coordinates for the field or observation
point
where the negative sign has been eliminated by reversing the
terms of the vector product
3
1
( ) = -
R
R R
∇
ur
Q
I 1
= ( ) d '
4
O
B l
R
µ
π
∴ ∇ ×∫
ur r

Since
Since the curl in unprimed variables is taken w.r.t. the
primed variables of the source point, we have,
1 ' 1
( ) d ' = ( ) - ( d ' )
dl
l l
R R R
∇ × ∇ × ∇ ×
r
r r
primed variables of the source point, we have,
d ' = 0l∇×
r
'
= ( )
4
O I d l
B
R
µ
π
∴ ∇ ×∫
r
ur

The integration and curl are w.r.t. to two different sets of
variables, so we can interchange the order and write the
preceding equation as
0 0' '
= [ ] =
4 4
I Idl dl
B A
R R
µ µ
π π
∇ × ⇒∫ ∫
r r
ur ur
Generalizing line current density in terms of the volume
current density,
0
= dv '
4
VJ
A
R
µ
π ∫
r
ur
4 4R Rπ π∫ ∫

v∆v∆
Fig. 3.8 (a) Electron orbit around nucleus creating
magnetic dipole moment; Magnetization in (b) non-
magnetic and (c) magnetic materials

3.10.1 Magnetization
The magnetic moment of an electron is defined as
where I is the bound current (bound to the atom and it is
$ $2
= I d = I Sm n nπ
ur
where I is the bound current (bound to the atom and it is
caused by orbiting electrons around the nucleus of the atom)
is the direction normal to the plane in which the electron
orbits and
d is the radius of orbit (see Fig. 3.8 (a))
$n

Magnetization is magnetic moment per unit volume
The magnetization for N atoms in a volume ∆v in which the
ith atom has the magnetic moment is defined as
1
= lim [ ]
N
i
A
M m
∆
∑
uur uur
im
uur
Materials like free space, air are nonmagnetic (µr is
approximately 1)
For non-magnetic materials: (see for example Fig. 3.8 (b), in
a volume , the vector sum of all the magnetic moments is
zero)
0
1
= lim [ ]i
v
i
M m
v m∆ →
=∆
∑

For magnetic materials: (see for instance Fig. 3.8 (c), in a
volume , the vector sum of all the magnetic moments is non-
zero)
Given a magnetization which is non-zero for a
magnetic material in a volume, the magnetic dipole moment
M
r
magnetic material in a volume, the magnetic dipole moment
due to an element of volume dv can be written as
The contribution of due to is
= dvdm M
ur uur
d A
ur
dm
ur

The magnetic vector potential and magnetic flux density
could be calculated as
( ) '
2
0
2
0
ˆ
4
ˆ
4
dvrM
rr
rmd
Ad ×=
×
=
rrr
π
µ
π
µ
could be calculated as
'
3
= dv'
4
=
o
V
M r
A
r
B A
µ
π
×
∴
∇ ×
∫
uur r
ur
ur ur

3.10.2 Magnetic vector potential in materials
Let us try to express this magnetic vector potential in terms
of bound surface and volume current density
1
' ( ) =
r
∇
$
Q 0 1
= ' ( ) dv'A M
µ
× ∇∫
ur uur
We also have,
2
' ( ) =
r r
∇Q 0
= ' ( ) dv'
4
A M
rπ
× ∇∫
1 1
' ( ) ' ( ) + '
M
M M
r r r
∇ × = ∇ × ∇ ×
uur
uur uur
Q
1 1
' ( ) = ' - ' ( )
M
M M
r r r
∴ × ∇ ∇ × ∇ ×
uur
uur uur

The proof for the above equality, we will solve in example
'
0 1
A= ( ' - ' ) dv'
4 v
M
M
r r
µ
π
∇ × ∇ ×∫
uur
uurr
∫∫ ×−=×∇
''
'''
SV
sd
r
M
dv
r
M r
rr
Q
The proof for the above equality, we will solve in example
3.5
( ) ( )
( ) ( ) '0''0
'0''0
ˆ
1
4
1
4
1
4
1
4
''
''
dsnM
r
dvM
r
sdM
r
dvM
r
A
SV
SV
×+×∇=
×+×∇=⇒
∫∫
∫∫
rr
rrrr
π
µ
π
µ
π
µ
π
µ

The above equation can be written in the form below
Where
bound volume current density is given by
''0
''4
ds
r
J
dv
r
J
A
S
sb
V
vb
∫∫ +=
rr
r
π
µ
bound volume current density is given by
bound surface current density is expressed as
=vbJ M∇ ×
uuur uur
ˆ= nsbJ M ×
uuur uur

Magnetized material can always be modeled in terms of
bound surface and volume current density
But they are fictitious elements and can not be measured
Only the magnetization is considered to be real and
measurablemeasurable

3.11 Magnetostatic boundary conditions
S∆
S∆
J
r
h∆
h∆
SJ
r
h∆
Fig. 3.9 Magnetostatic boundary conditions

3.11.1 Normal components of the magnetic flux density
Consider a Gaussian pill-box at the interface between two
different media, arranged as in the figure above
The integral form of Gauss’s law tells us that
∫ =•
rr
As the height of the pill-box ∆h tends to zero at the interface,
there will be no contribution from the curved surfaces in the
total magnetic flux, hence, we have
∫ =•
pillbox
sdB 0
rr

1 2
d + d =0
S S
B s B s⇒ • •∫ ∫
ur r ur r
1 2
1 2
1 2B ds - B ds =0n n
S S
⇒ ∫ ∫
(B - B )ds=0⇒ ∫
The normal components of the magnetic flux density are
continuous at the boundary
1 2
(B - B )ds=0n n
S
⇒ ∫
1 2
B =Bn n⇒

3.11.2Tangential components of the magnetic field intensity
ApplyingAmpere’s law to the closed path
where I is the total current enclosed by the closed path PQRS
∫ ∫∫∫∫ =•+•+•+•=•
PQRSP SPRSQRPQ
IldHldHldHldHldH
rrrrrrrrrr
where I is the total current enclosed by the closed path PQRS
which lies in the xy plane
Assume that x is along the direction of PQ in Fig. 3.9
At the interface, ∆h 0, the line integral along paths QR and
SP are negligible, hence,

d + d =I
PQ RS
H l H l• •∫ ∫
uur r uur r
$ $
1 2( - ) d l = dlV
PQ
H H x J y h⇒ • • ∆∫ ∫
uur uuur uur
h 0
Lim V SJ h J
∆ →
∆ =
uur uur
Q
is the definition of surface current density
From the property of vector scalar triple product, we have,
h 0
Lim V SJ h J
∆ →
∆ =Q
$
( ) $
1 2( - ) d l = d lS
PQ
H H y z J y⇒ • × •∫ ∫
uur uuur uur
$

The tangential component of the magnetic field intensity at
$
{ } { } $ $
1 2 1 2( - ) d l = ( - ) d l = d lS
PQ PQ
y z H H z H H y J y⇒ • × × • •∫ ∫ ∫
uur uuur uur uuur uur
$ $
1 2( - ) = Sz H H J⇒ ×
uur uuur uur
$
The tangential component of the magnetic field intensity at
the interface is continuous unless there is a surface current
density present at the interface

3.12 Self and mutual inductance
A circuit carrying current I produces a magnetic field which
causes a flux to pass through each turn of the circuit
If the circuit has N turns, we define the magnetic flux linkage
as
Also, the magnetic flux linkage enclosed by the current
.NψΛ = B dsψ = •∫
r r
Also, the magnetic flux linkage enclosed by the current
carrying conductor is proportional to the current carried by
the conductors
L= Λ/I=
where L is the constant of proportionality called the
inductance of the circuit (unit: Henry)
I LIΛ ∝ ⇒ Λ = ⇒
N
I
ψ

The magnetic energy stored in an inductor is expressed from
circuit theory as:
If instead of having a single circuit, we have two circuits
2
2
1
LIWm = 2
2 mW
L
I
⇒ =
If instead of having a single circuit, we have two circuits
carrying currents I1 and I2, a magnetic induction exists
between two circuits
Four components of fluxes are produced
The flux for example, is the flux passing through the
circuit 1 due to current in circuit 2
12 ,ψ

Define M12 =
Similarly,
1
212
S
B dsψ = •∫
ur uur
12 1 12
2 2
N
I I
ψΛ
=
21 2 21N
M
ψΛ
= =
The total energy in the magnetic field is due to the sum of
energies
21 2 21
21
1 1
N
M
I I
ψΛ
= =
2 2
1 2 12 1 1 2 2 12 1 2
1 1
2 2
mW W W W L I L I M I I= + + = + ±

3.13 Summary
Electric currents
Ohm’s law Kirchoff’s law Joule’s law
Boundary
conditions
Ejp
rr
•=Ej σ= JJ =
Fig. 3.10 (a) Electric currents in a nutshell
Kirchoff’s current
law
Kirchoff’s voltage
law
0I =∑
1 1
M N
m n n
m n
i Rξ
= =
=∑ ∑
Ejp
rr
•=Ej σ= 21 nn JJ =
2
1
2
1
σ
σ
=
t
t
J
J






−=
2
2
1
1
1
σ
ε
σ
ε
ρ ns J

3.13 Summary
Magnetostatics
Biot Savart’s law
Gauss’s law for
magnetic fields
Magnetic vector potential
Fig. 3.10 (b) Magnetostatics in a nutshell
Self and mutual
inductance0
2
4
dl R
dB I
R
µ
π
×
=
r
r
Jµ
∫
r
ur
21 2 21
21
1 1
N
M
I I
ψΛ
= =
L=Λ/I=NΨ/I
magnetic fields
MagnetizationAmpere’s law
Boundary conditions
Magnetic vector potential in materials
∫ =•=Ψ
S
sdB 0
rr
∫ =•
C
enclosedIldB 0µ
rr
0
= dv '
4
VJ
A
R
µ
π ∫
ur
0
1
1
= lim [ ]
N
i
v
i
A
M m
v m∆ →
=∆
∑
uur uur
''0
''4
ds
r
J
dv
r
J
A
S
sb
V
vb
∫∫ +=
rr
r
π
µ
Bn1=Bn2 ( ) SJHHz
rrr
=−× 21ˆ

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