The Theory of Electromagnetic Fieldusers.utcluj.ro/~lcret/Curs Electrotechnics/curs_4_2009_Laura.pdf · this non-zero curl equals the electric current density, unlike the electrostatic

Course no. 4

March 19 - 2009

Technical University of Cluj-Napoca

http://www.et.utcluj.ro/Curs_Electromagnetics2006_AC.htmhttp://www.et.utcluj.ro/~lcret

The Theory of Electromagnetic Field

Chapter 3

Magnetostatics fields


Introduction notes


Magnetic FieldsMagnetic Fields (B)(B)

II

CurrentsCurrents Permanent MagnetsPermanent Magnets


• The origin of magnetism lies in moving electric charges.Moving (or rotating) charges generate magnetic fields.

• An electric current generates a magnetic field.

• A magnetic field will exert a force on a moving charge.

• A magnetic field will exert a force on a conductor that carries an electric current.


•• Stationary charge:

• vq = 0

• E ≠ 0 B = 0

•Moving charge:

• vq ≠ 0 and vq = constant

• E ≠ 0 B ≠ 0

•Accelerating charge:

• vq ≠ 0 and aq ≠ 0

• E ≠ 0 B ≠ 0 Radiating field

A stationary charge produces an electric field only.

A uniformly moving charge produces an electric and magnetic field.

A accelerating charge produces an electric and magnetic field and a radiating electromagnetic field.


Units and definitions:Magnetic field vector

Magnetic induction

Magnetic flux densityBv

GT 4101 =

Tesla Gauss

SI unit211

mWbT =

Magnetic field strength

Hv

B Hµ= ⋅v v

Weber


Permeability

r oµ µ µ= ⋅Permeability of free space

Relative permeability for a medium

74 10oHm

µ π − ⎧ ⎫= ⋅ ⋅ ⎨ ⎬⎩ ⎭

Permeability of the medium

Exact constant

⎭⎬⎫

⎩⎨⎧⇔

⎭⎬⎫

⎩⎨⎧

mWb

mH

Magnetostatics fieldsRelative permeability µr and susceptibility χm

BismuthMercuryGoldSilverLeadCopperWater

Vacuum

AirAluminiumPalladium

CobaltNickelIron

0.999830.9999680.9999640.999980.9999830.9999910.999991

1.000

1.000000361.0000211.00082

2506006000

-1.66 E-4-3.2 E-5-3.6 E-5-2.60 E-5-1.7 E-5-0.98 E-5-0.88 E-5

0

3.6 E-72.5 E-58.2 E-4

---------

Diam

agneticParam

agnetic

Ferromagnetic

http://www.ndt-ed.org/EducationResources/CommunityCollege/MagParticle/Physics/MagneticMatls.htm


Current loop Coil or solenoid

Magnetostatics fieldsMagnetic fluxThe magnetic flux through an open surface S is given by:

[ ],Ψ = ⋅ =∫∫ur ur

S

B d A W eb er W b

In an electrostatic field, the flux passing thought a closed surface is the same as the charge enclosed. Thus it is possible to have isolated electric charges, which also reveals that electric flux lines are not necessarily closed.

Unlike electric flux lines, magnetic flux lines always close upon themselves. This is due to the fact that it is not possible to have isolated magnetic poles (or magnetic charges). Thus, the total flux through a closed surface in a magnetic field must be zero:

0Σ

⋅ =∫∫ur urB d A

This equation is referred to as the law of conservation of magnetic flux or Gauss’s law for magnetostatic fields.


Laws of magnetostaticsThe magnetic flux law

Definition:The total magnetic flux through a closed surface is always zero.

Integral form of the law0Σ

⋅ =∫∫ur urB d A

0ΣΣ

⋅ = ⋅ =∫∫ ∫∫∫ur ur ur

V

B d A d iv B d v

0=ur

divB Differential form of the law

( )12 2 1 2 1 0s n ndiv B n B B B B= ⋅ − = − =ur uur uur

Boundary conditions


Laws of magnetostaticsThe magnetization law

From experimental studies, it is found that the magnetization vector (the temporary component) is strongly related to the magnetic field strength. For most common magnetic materials, these two vectors are collinear and proportional for a wide range of values of H (linear materials and isotropic).

χ= ⋅uur uur

t mM H Valid just for linear materials

where: is the magnetic susceptibility of the material. χmWhen the magnetic susceptibility depends on the magnetic field strength H, it is said that the medium is nonlinear, because all the field relations become nonlinear equations. When the magnetic susceptibility depends on the position in the volume of the magnetic body, it is said that the problem is inhomogeneous, as opposed to the homogeneous case when the properties of the material are constant throughout the volume. Moreover, the magnetic properties may depend on the direction of theapplied field. This is called anisotropy of the magnetic material.


Laws of magnetostaticsThe relation between B, H and M vectors

( )0µ= ⋅ + +ur uur uuur uuur

t pB H M M

The vector sum, between the magnetization vectors (both components) and the magnetic field strength, multiplied with the permeability of the vacuum, is equal, at any moment and point, with the magnetic flux density:

For materials without permanent magnetization: ( )0µ= ⋅ +ur uur uuur

tB H M

For linear materials without permanent magnetization:

( ) ( ) ( )0 0 0 1µ µ χ µ χ µ= ⋅ + = ⋅ + ⋅ = ⋅ + ⋅ = ⋅ur uur uuur uur uur uur uur

t m mB H M H H H H

µ= ⋅ur uurB H

For materials with anisotropy and without permanent magnetization:


Laws of magnetostaticsThe relation between B, H and M vectors

Then, the relation between the magnetic flux vector and the magnetic field strength is a tensor one:

µ= ⋅ur uurB H

µ µ µµ µ µµ µ µ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

x xx xy xz x

y yx yy yz y

z zx zy zz z

B HB HB H

Fortunately, it often suffices to assume that the medium is homogeneous, linear and isotropic. This is the simplest possible case.Final note on the physical meaning of the relative magnetic permeability: it shows how many times the magnetic field strength is decreased or increased in the volume of the magnetic material due to the effect of the magnetization.

In general the magnetization vector consists in 2 components:- a temporary component (Mt) and a permanent one (Mp) Note:


i

Heθ

rs

Determine the “circulation of H -field” along an arbitrary circuit c

C2

Consequences:1. More currents through C2 add up;2. Currents outside C2 do not contribute;3. Position of current inside C2 is not

important.

rH

ds

=r. dθ

dθ1 2C C

H d s H d s i⋅ = ⋅ =∫ ∫uur r uur r

C1

Ampere’s law


Ampere’s law can be derived in its differential form as follows:

1 CC C S

H d s i H d s J d A⋅ = ⇒ ⋅ = ⋅∫ ∫ ∫∫uur r uur r ur ur

C CS S

ro t H d A J d A ro t H J⋅ = ⋅ ⇒ =∫∫ ∫∫uur ur ur ur uur ur

Ampere’s differential form (v = 0)

It is now clear that the electric currents are the curl-sources of the magnetic filed. Notice that the magnetic field has a non-zero curl, and this non-zero curl equals the electric current density, unlike the electrostatic field, which is a curl-free field.

ro t H J=uur ur

0ro t E =


Electric field Magnetic field

IrrotationalIrrotational FieldField Rotational FieldRotational Field

ro t H J=uur ur

0ro t E =


Note: until now we have considered that the surface S is immobile.

In the most general case of media in movement (surface has a relative speed towards the media), the Ampere's law in differential form must be completed as:

( ) 2, /ρ ⋅∂

= +∂

+ ×+ur

uu r rr r ur uv v rot D vDrotH J A m

t

Conduction current densityDisplacement current density Convection current density

Roentgen current density

From the physical point of view, the correction current appears due to the displacement, with the speed v (towards the surface SC), of bodies charged with charges (having the volume density of the charge ) and the Roentgen current appears due to the displacement with the speed v of polarized bodies (having volumetric density of the polarized charge ).

ρv


( )ρ∂= + + ⋅ + ×

∂

uruur ur r ur r

vDrotH J v rot D vt

In the most general case, a magnetic field can be produced by:• conduction currents• displacement currents• convection currents• Roentgen currents

( )( )

ρ⎛ ⎞∂

⋅ = + + ⋅ + × ⋅⎜ ⎟∂⎝ ⎠∫ ∫∫ur

uur r ur r ur r urv

C S S

DH d s J v rot D v d At

( )

⋅ = + + +∫uur r

D C R

C S

H d s i i i iAmpere's integral form


Flux linkage


Flux linkage

The flux linkage may be self flux linkage and mutual flux linkage. A single coil has only its own self flux linkage, i.e. this is the flux created by its own current, which flows through its own turns.The mutual flux linkage is defined only if a pair of magnetically coupled coils exists.

The mutual flux linkage of coil 1 is due to the magnetic field of the coil 2, which induced emf in coil 1.



1

11 1 1 ,Ψ = ⋅∫∫uur ur

S

B d A W b

1

1 11 3 ,

4µπ⋅ ×

=⋅ ∫

ur ruur

C

I d s rB Tr

1 1

1 11 1 1 1 13 ,

4µ µ

π

⎛ ⎞⋅ ×⎜ ⎟Ψ = ⋅ Ψ ⋅⎜ ⎟⋅⎝ ⎠

∫∫ ∫ur r

ur

S C

I d s r d A W b Ir


[ ] [ ]µ⋅ ⋅

Ψ= = = ⋅

⋅ ⋅

∫∫ ∫∫

∫ ∫

ur ur uur ur

uur r uur rS C S C

C C

B d A H d A ,L

I H d s H d s


[ ] [ ]µ⋅

= ⋅⋅

∫∫

∫

uur ur

uur rS C

C

H d AL , H

H d s[ ]2

1

ε Σ

⋅= ⋅

⋅

∫∫

∫

ur ur

ur r

E d AC , F

E d s

CG

εσ

=Just as it was shown that for capacitors:

µ ε⋅ = ⋅extL CIt can be shown that:

Note: represents only the external inductance, related to the external flux of an inductor.

In an inductor such as a coaxial or parallel-wire transmission line, the inductance produced by the flux internal to the conductor is called the internal inductance while that produced by the flux external to it is called external inductance. The total inductance L is:

extL

intL

= +int extL L L


Magnetostatic field energy

1

12

n

m i i

i

W iψ=

= ⋅ ⋅∑The above relation gives us the expression of the energy stored in the magnetic field of several electric circuits due to currents

Magnetic field density21 1 12 2 2

em

Ww H H H B HV

µ µ= = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅

12mw B H= ⋅ ⋅ur uur

In, general:

12m e

V V

W w dv B H dv= ⋅ = ⋅ ⋅∫∫∫ ∫∫∫ur uur


AnalogyElectrostatics Magnetostatics Scalar Potential - V Vector Potential - A

E-field H-field

Permitivity Permeability

Volume Charge Density Current Density

Surface Charge Density Surface Current Density

Capacitance - C Inductance - L

Laplace’s Equation Laplace’s Equation

Poisson’s Equation Poisson’s Equation

Electromagnetic field

IntroductionSo far we have:

vdivE ρε

=ur

0rotE =ur

rotH J=uur ur

0divB =ur

These equations are OK for static fields, i.e. those fields independent of time. When fields vary as a function of time the curl equations acquire an additional term.

0rotE =ur

gets a tB∂∂

−v

gets a Dt

∂∂

ur

rotH J=uur ur

Electrostatic field Magnetostatic field


Faraday’s Experiments

N S

iv

• Michael Faraday discovered induction in 1831.• Moving the magnet induces a current i. • Reversing the direction reverses the current.• Moving the loop induces a current.• The induced current is set up by an induced EMF.



i

di/dtEMF

(right)(left)

v = 0

• Changing the current in the right-hand coil induces a current in the left-hand coil.

• The induced current does not depend on the size of the current in the right-hand coil.

• The induced current depends on di/dt.



• Moving the magnet changes the flux Ψ (1) – motional EMF.• Changing the current changes the flux Ψ (2) - transformer EMF.Faraday: changing the flux induces an EMF (e).

Faraday’s law

1) 2)N S

iv i

di/dtEMF

(right)(left)

v = 0

Ψ= −

dedt

The emf induced around a loop

equals the rate of changeof the flux through that loop


Faraday formulated the law named after his nameThe induced electromagnetic force (EMF) - eemf or simply (e), in any closed conducting loop (circuit) is equal to the time rate of change of the magnetic flux linkage of the loop.

Γ

⎛ ⎞Ψ ⎜ ⎟= − = − ⋅⎜ ⎟⎝ ⎠∫∫

ur ur

S

d de Bdt dt

d A Integral form of Faraday’s law

The negative sign shows that the induced emf (and currents) would act in such a way that they would oppose the change of the flux creating it. This law is also known as Lentz’ law of EMF induction.If the circuit consists of N loops all of the same area and if Ψ is the flux through one loop, then the total induced emf is:

Λ Ψ=− = ⋅

d de Ndt dt

Electromagnetic fieldLenz’s Law: The direction of any magnetic induction effect (induced current) is such as to oppose the cause producing it. (Opposing change = inertia!)

A

E

B(increasing)

A

E

B(decreasing)

A

E

B(increasing)

A

E

B(decreasing)


Differential form of Faraday’s law

Ψ= −

dedt

S

dE d s B d Adt

ΓΓ

⎛ ⎞⎜ ⎟⋅ = − ⋅⎜ ⎟⎝ ⎠

∫ ∫∫ur r ur ur

Note: is an open surface.ΓS

e E d sΓ

= ⋅∫ur r

Γ

Ψ = ⋅∫∫ur ur

S

B d A


S

dE d s B d Adt

ΓΓ

⎛ ⎞⎜ ⎟⋅ = − ⋅⎜ ⎟⎝ ⎠

∫ ∫∫ur r ur ur

Applying Stokes’s theorem:

S S

dE d s rotE d A B d Adt

Γ ΓΓ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⋅ = ⋅ = − ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ ∫∫ ∫∫ur r ur ur ur ur

Suppose that the surface SГ is mobile with the velocity v, then the derivative with respect the time of the surface integral will be:

The surface is immobile The surface is mobile with velocity v

( )Γ Γ Γ

⎛ ⎞ ⎛ ⎞∂⎜ ⎟ ⎜ ⎟⋅ = ⋅ + × ⋅⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠∫∫ ∫∫ ∫∫

urur ur ur ur r ur

S S S

d BB d A d A rot B v d Adt t


S

dE d s B d Adt

ΓΓ

⎛ ⎞⎜ ⎟⋅ = − ⋅⎜ ⎟⎝ ⎠

∫ ∫∫ur r ur ur ( )

Γ Γ Γ

⎛ ⎞ ⎛ ⎞∂⎜ ⎟ ⎜ ⎟⋅ = ⋅ + × ⋅⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠∫∫ ∫∫ ∫∫

urur ur ur ur r ur

S S S

d BB d A d A rot B v d Adt t

( )Γ Γ Γ

⎛ ⎞ ⎛ ⎞∂⎜ ⎟ ⎜ ⎟⋅ = − ⋅ − × ⋅⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠∫∫ ∫∫ ∫∫

urur ur ur ur r ur

S S S

BrotE d A d A rot B v d At

( )BrotE rot v Bt

∂= − + ×

∂

urur r ur

Differential form of Faraday’s law

Electromagnetic fieldThe induced electromagnetic force (emf) around a circuit can be separated into two terms:

• Transformer emf component, due to the time-rate of change of the B-field :

• Motional emf component, due to the motion of the circuit:

BrotEt

∂= −

∂

urur

transformer

S S

Be E d s rotE d A d At

Γ ΓΓ

∂= ⋅ = ⋅ = − ⋅

∂∫ ∫∫ ∫∫ur

ur r ur ur ur

Note: the induced electric field is non-conservative !!!! 0rotE ≠ur

The electric field is conservative only in electrostatics regime !!!!

0rotE =ur

E v B= ×ur r ur ( )motionale E d s v B d s

Γ Γ

= ⋅ = × ⋅∫ ∫ur r r ur r


Particular cases:∂

= − −∂

urur AE gradV

t1) The surface is immobile (v = 0):

The motional emf is zero: ( ) 0Γ Γ

= ⋅ = × ⋅ =∫ ∫ur r r ur r

motionale E d s v B d s

Γ Γ Γ

∂ ∂= − ⋅ = ⋅ = − ⋅

∂ ∂∫∫ ∫ ∫ur ur

ur ur r rtransformer

S

B Ae d A E d s d st t

The transformer emf is non-zero:

∂= −

∂

urur BrotE

tDifferential form of Faraday’s law:

2) The surface is mobile but the magnetic field B is constant in time:= × −

ur r urE v B gradV

The motional emf is non-zero: ( )Γ Γ

= ⋅ = × ⋅∫ ∫ur r r ur r

motionale E d s v B d s

Electromagnetic fieldMaxwell’s equations (v = 0)

Integral form Differential form Significance

Γ

∂Ψ⋅ = −

∂∫ur rE d s

t∂

= −∂

urur BrotE

t

Faraday’s lawMaxwell 1

Γ

∂⋅ = + ⋅

∂∫ ∫∫ur

uur r ur

S

DH d s i d At

∂= +

∂

uruur ur DrotH J

t

Ampere’s lawMaxwell 2

Σ

⋅ =∫∫ur urD d A Q ρ=

urvdivD Electric flux’s law

Maxwell 3

0Σ

⋅ =∫∫ur urB d A 0=

urdivB Magnetic flux’s law

Maxwell 4

Electromagnetic fieldMaxwell’s equations

∂= −

∂

urur BrotE

t

∂= +

∂

uruur DrotH

t

µ ∂= − ⋅

∂

uurur HrotE

t

ε ∂= + ⋅

∂

uruur ErotH

t


( )µ µ∂ ∂= − ⋅ ⇒ = − ⋅

∂ ∂

uurur ur uurHrotE rot rotE rotH

t tε ∂

= + ⋅∂

uruur ErotH

t

( )2 2

2 2( )µ ε µ ε∂ ∂= − ⋅ ⋅ ⇒ −∆ = − ⋅ ⋅

∂ ∂

ur urur ur urE Erot rotE grad divE E

t t

In free space: 0ρε

= =ur

VdivE

2

2µ ε ∂∆ = ⋅ ⋅

∂

urur EE

tor 2

2µ ε ∂∆ = ⋅ ⋅

∂

uuruur HH

t2

1µ ε µ ε⋅ = ⋅ ⋅r rcwith:

In vacuum:

2

2 2

2

2 2

1

1

∂∆ = ⋅

∂∂

∆ = ⋅∂

urur

uuruur

EEc t

HHc t

Wave’s equations


µ ∂= − ⋅

∂

uurur HrotE

tε ∂

= + ⋅∂

uruur ErotH

t

∂∂

urEt

uurrotH

urrotE

∂∂

uurHt

Special case: plane waves

satisfy the wave equation

A sin( t )ψ ω φ= +With the known solution of the form

2 2

2 2 2

1x c t

∂ ψ ∂ ψ∂ ∂

=( )( )

,

,

= ⋅

= ⋅

ur r

uur ry

z

E E x t j

H H x t k

Electromagnetic fieldPlane Electromagnetic Waves

x

Ey

Bz

c

Documents

The Theory of Electromagnetic Fieldusers.utcluj.ro/~lcret/Curs Electrotechnics/curs_4_2009_Laura.pdf · this non-zero curl equals the electric current density, unlike the electrostatic