3 slides

Dr. Rakhesh Singh Kshetrimayum

3. Magnetostatic fields

Dr. Rakhesh Singh Kshetrimayum

2/16/20131 Electromagnetic Field Theory by R. S. Kshetrimayum

3.1 Introduction to electric currents

Electric currents

Ohm’s law Kirchoff’s law Joule’s lawBoundary conditions

2/16/2013Electromagnetic Field Theory 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

jr

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),

Vi 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 Ejds i j E

dlR dl

ds

σρ ρρ

= = = = ⇒ = =


3.1 Introduction to electric currents� where is the conductivity and

� is the resistivity of the isotropic material

σ

1ρ

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 impossible impossible � � 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

jr

dsr

Fig. 3.3 Equation of continuity


−=•⇒

−=−= ∫∫∫

V

v

SV

v dvdt

dsdjdv

dt

d

dt

dqi 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 charge� It is basically a principle of conservation of charge

� By applying the divergence theorem

0V V

V V V

d djdv 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

jdt

ρ∇ • + =

r

� At steady state, there can be no points of changing charge density

Vd

jdt

ρ∇ • = −

r


∫ =•⇒=•∇S

sdjj 00rrr


� 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 field caused by chemical action) relation to impressed electric field caused 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 nj 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 closed � The line integral of the total electric field around the closed circuit gives


( ) ldj

ldEEC C

ncr

rrrr

•=•+∫ ∫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,


( ) ( ) ldj

ldEldEEC C

nncr

rrrrrr

•==•=•+∫ ∫∫ σξ

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's Voltage 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

dWdP 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 the contribution 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 11 2

1 2 1 2 2 2

0 0 0t t t

t

J J JJ Jn E E n

J

σ

σ σ σ σ σ

× − = ⇒ × − = ⇒ − = ⇒ =

r rr 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 the boundary condition on the normal components of the electric flux densities as follows:

1 2 1 21 2 1 1 2 2 1 2 1

1 2 1 2

n nn n S S n n S n

J JD D E E J

ε ερ 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’slaw

Gauss’s law for

Magnetic vector potential

Boundary

Self and mutual inductance


Gauss’s law for magnetic fields

MagnetizationAmpere’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 and rr

Idlr

� 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 rdB 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


∫ =•=ΨS

sdB 0rr

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

( )∫ ∫ =•∇=•=ΨS V

dvBsdB 0rrr

volume, the integrand itself must be identically zero which gives differential form of 3rdMaxwell’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 of current flow, then your fingers’ grip points in the direction of magnetic field

� Andre Ampere 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 4thMaxwell’s equation

� By application of Stoke’s theorem

∫ =•C

enclosedIldB 0µrr

� 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

sdJsdBldBrrrrrr

0µ


� Incomplete differential form of 4thMaxwell’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 RB

R

µ

π

×∫

r urur

$ $ = (x-x') +(y-y') +(z-z') R x y zur

$

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

QI 1

= ( ) d '4

OB lR

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 ' )

dll l

R R R∇ × ∇ × ∇ ×

rr r

primed variables of the source point, we have,

d ' = 0l∇×r

' = ( )

4

OI d l

BR

µ

π∴ ∇ ×∫

rur



� 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 dlB A

R R

µ µ

π π∇ × ⇒∫ ∫

r rur ur


� Generalizing line current density in terms of the volume current density,

0 = dv ' 4

VJA

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 nπur


� 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

AM m

∆∑

uur uur

imuur

� 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)

01

= lim [ ]i

vi

M mv 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

Mr

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 Mur uur

d Aur

d mur


� The magnetic vector potential and magnetic flux density could be calculated as

( ) '

2

0

2

0 ˆ4

ˆ

4dvrM

rr

rmdAd ×=

×=

rr

r

π

µ

π

µ

could be calculated as

'

3

= dv '

4

=

o

V

M rA

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'

4A M

rπ× ∇∫

1 1' ( ) ' ( ) + '

MM M

r r r∇ × = ∇ × ∇ ×

uuruur uur

Q

1 1 ' ( ) = ' - ' ( )

MM M

r r r∴ × ∇ ∇ × ∇ ×

uuruur uur


� The proof for the above equality, we will solve in example

'

0 1 A= ( ' - ' ) dv'

4v

MM

r r

µ

π∇ × ∇ ×∫

uuruurr

∫∫ ×−=×∇''

'''

SV

sdr

Mdv

r

M rrr

Q

� The proof for the above equality, we will solve in example 3.5


( ) ( )

( ) ( ) '0''0

'0''0

ˆ1

4

1

4

1

4

1

4

''

''

dsnMr

dvMr

sdMr

dvMr

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

''4ds

r

Jdv

r

JA

S

sb

V

vb ∫∫ +=

rrr

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∆

Jr

h∆

h∆

SJr

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 0rr


1 2

d + d =0S 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 2B =B

n n⇒



3.11.2 Tangential components of the magnetic field intensity

� Applying Ampere’s law to the closed path

where I is the total current enclosed by the closed path PQRS

∫ ∫∫∫∫ =•+•+•+•=•PQRSP SPRSQRPQ

IldHldHldHldHldHrrrrrrrrrr

� 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 = dl V

PQ

H H x J y h⇒ • • ∆∫ ∫uur uuur uur

h 0Lim

V SJ h J

∆ →∆ =

uur uurQ

� is the definition of surface current density

� From the property of vector scalar triple product, we have,

h 0Lim

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 ( - ) = S

z 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

1LIWm =

2

2 mWL

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 21NM

ψΛ= =

� The total energy in the magnetic field is due to the sum of energies

21 2 2121

1 1

NM

I I

ψΛ= =

2 2

1 2 12 1 1 2 2 12 1 2

1 1

2 2m

W W W W L I L I M I I= + + = + ±


3.13 SummaryElectric currents

Ohm’s law Kirchoff’s law Joule’s lawBoundary conditions

Ejprr

•=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= =

=∑ ∑

Ejprr

•=Ej σ= 21 nn JJ =

2

1

2

1

σ

σ=

t

t

J

J

−=

2

2

1

11

σ

ε

σ

ερ ns J

3.13 Summary

MagnetostaticsBiot Savart’s law

Gauss’s law for magnetic fields

Magnetic vector potential

Fig. 3.10 (b) Magnetostatics in a nutshell

Self and mutual inductance �

0

2

4

dl RdB I

R

µ

π

×=

rr

Jµ∫r

ur

21 2 2121

1 1

NM

I I

ψΛ= =

L=Λ/I=NΨ/I


magnetic fields

MagnetizationAmpere’s law

Boundary conditions

Magnetic vector potential in materials

∫ =•=ΨS

sdB 0rr

∫ =•C

enclosedIldB 0µrr

0 = dv ' 4

VJ

AR

µ

π ∫ur

01

1 = lim [ ]

N

iv

i

AM m

v m∆ →=∆∑

uur uur

''0

''4ds

r

Jdv

r

JA

S

sb

V

vb ∫∫ +=

rrr

π

µ

Bn1=Bn2 ( ) SJHHzrrr

=−× 21ˆ

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