1 ene 325 electromagnetic fields and waves lecture 5 ampére’s law, scalar and vector magnetic...

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ENE 325ENE 325Electromagnetic Electromagnetic Fields and WavesFields and Waves

Lecture 5Lecture 5 Ampére’s law, Scalar and Ampére’s law, Scalar and Vector Magnetic Potentials, Vector Magnetic Potentials, Magnetic Force, Torque, and Magnetic Force, Torque, and Magnetic MaterialMagnetic Material

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Review (1)Review (1)

Ampere’s circuital law - the integration of around any closed path is equal to the net current enclosed by that path.

‘Curl’ is employed to find the point form of Ampère’s circuital law.

Curl of or is the maximum circulation of per unit area as the area shrinks to zero

encH d L I��

H��

H��

H��

limS 0

H d LH J

S

��

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Review (2)Review (2)

Magnetic flux density is related to the magnetic field intensity in the free space by

Weber/m2 or Tesla (T)where 0 is the free space permeability, given in units of henrys per meter, or

0 = 410-7 H/m. Magnetic flux (units of Webers) passing through a surface is found by

B��

H��

0B H��

B d S ��

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OutlineOutline Curl and point form of Ampére’s lawére’s law Magnetic flux density Scalar and vector magnetic potentials Magnetic force and torque Magnetic material and permeability

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Curl and the point form of Curl and the point form ofAmpAmpéé re’s circuital law re’s circuital law (1)(1) ‘Curl ’ is employed to find the point form A

mpère’s circuital law, analogous to ‘Diverg ence’ to find the point form of Gauss’s law.

Curl of or is the maximum circul ation of per unit area as the area shrink

s to zero.

limS 0

H d LH J

S

��

H��

H��

H��

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Curl and the point form of Curl and the point form ofAmpAmpéé re’s circuital law re’s circuital law (2)(2) ‘‘Curl´Curl´ operator perform a derivative of vec operator perform a derivative of vec

tor and returns a vector quantity tor and returns a vector quantity. . For Cart For Cart esian coordinates, can be esian coordinates, can be written aswritten as

. ��

x y z

x y z

a a a

H x y z

H H H

H��

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Physical view of curlPhysical view of curl

a) Field lines indicating divergence A simple way to see the b) Field lines indicating curl direction of curl using

right hand rule

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Stokes’s TheoremStokes’s Theorem Stokes’s Theorem Stokes’s Theorem relates a closed line inte relates a closed line inte

gral into a surface integral gral into a surface integral

H d L H d S ��

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Magnetic flux density, BMagnetic flux density, B

Magnetic flux density is related to the magnetic field intensity in the free space by

Magnetic flux (units of Webers) passing through a surface is found by

B��

H��

0B H��

Weber/m2 or Tesla (T)

1 Tesla = 10,000 Gauss. where 0 is the free space permeability, given in units of

henrys per meter, or 0 = 410-7 H/m.

B d S ��

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Gauss’s law for magnetic Gauss’s law for magnetic fieldsfields

B d S 0��

or . ��B 0

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EX1 A solid conductor of circular cross section is made of a homogeneous nonmagnetic material. If the radius a = 1 mm, the conductor axis lies on the z axis, and the total current in the direction is 20 A, finda) H at = 0.5 mm

b) B at = 0.8 mm

c) The total magnetic flux per unit length inside the conductor

za

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Maxwell’s equations for stat Maxwell’s equations for stat ic fields ic fields

Integral form Differential form

��

��

��

��

enc

enc

D d S Q

B d S 0

E d L 0

H d L I

vD

B 0

E 0

H J

��

��

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The scalar and vector The scalar and vector magnetic potentials (1)magnetic potentials (1) Scalar magnetic potential (Vm) is

the simple practical concept to determine the electric field. Similarly, the scalar magnetic potential, Vm, is defined to relate to the magnetic field but there is no physical interpretation.

E V ��

H��

Assume mH V ��

( )mH J V 0 ��

To make the above statement true, J = 0.

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The scalar and vector The scalar and vector magnetic potentials (2)magnetic potentials (2)

From 0B H 0 ��

( )0 mV 0

2mV 0

Laplace’s equationThis equation’s solution to determine the potential field requires that the potential on the boundaries is known.

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The difference between V (electric potential) and Vm

(scalar magnetic potential) is that the electric potential is a function of the positions while there can be many Vm values

for the same position. encH d L I

��

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While for the electrostatic case

0E ��

0E d L ��

a

abb

V E d L �� does not depend on path.

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Vector magnetic potential (A) is useful to find a magnetic filed for antenna and waveguide.

From

Let assume

so

and

B 0 ��

( )B A ��

( )A 0 ��

0

1H A

��

0

1H A J 0

��

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It is simpler to use the vector magnetic potential to determine the magnetic field. By transforming from Bio-savart law, we can write

0 .4

Id LA

R

��

The differential form 0 .4

Id Ld A

R

��

��

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Ex2Ex2 Determine the magnetic field Determine the magnetic field from the infinite length line of from the infinite length line of current using the vector magnetic current using the vector magnetic potentialpotential

x

y

z

z

22z

P

zd L dza

��

Find A��

at point P(, , z)

0

2 24

zIdzad A

z

��

then 0 0

1 1 zdAd H d A a

��

3/ 22 24

I dzd H a

z

��

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Vector magnetic potential Vector magnetic potential for other current for other current distributionsdistributions For current sheet

For current volume

0

4S

KdSA

R

��

0

4vol

JdvA

R

��

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Magnetic force Magnetic force

Force on a moving charge

Force on a differential current element

F qv B ��

d F dQv B ��

N

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Hall effectHall effect

Hall effect is the voltage exerted from the separation of electrons and positive ions influenced by the magnetic force in the conductor. This Hall voltage is perpendicular to both magnetic field and the charge velocity.

+

+

+

+

+

+

-

-

-

-

-

-

B

+ + + +

Fq

I

qF qv B ��

( )x zqv a Ba

yqvBa N

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Magnetic force on the Magnetic force on the current carrying conductor current carrying conductor (1)(1) For the current carrying conductor, consider the

magnetic force on the whole conductor not on the charges.

From

and

then

vJ v��

dQ = vdv

( )vd F dv v B ��

.d F J Bdv ��

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Magnetic force on the Magnetic force on the current carrying conductor current carrying conductor (2)(2)From

we can write

then

For a straight conductor in a uniform magnetic field (still maintains the closed circuit),

Force between differential current elements determine the force on the conductor influenced by the other nearby.

Jdv KdS Id L ��

d F Id L B ��

.F Id L B ��

F I L B ��

or F = ILBsin

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Ex3Ex3 Determine the force action Determine the force action on circuit 2 by circuit 1. on circuit 2 by circuit 1.

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Force and torque on a Force and torque on a closed circuit (1)closed circuit (1)

x

y

z

FR

O

T

dT R d F ��

Nm

where = torque (Nm)

= distance from the origin (m)

= Force (N)

If the current is uniform, .T R F ��

T��

R��

d F��

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Force and torque on a Force and torque on a closed circuit (2)closed circuit (2)

For a current loop, we can express torque as

If is constant or uniform, we can express torque as

Define magnetic dipole moment

.dT Id S B ��

B��

.T I S B ��

Id S dm��

where m = magnetic dipole moment (Am2).

Therefore, torque can be shown as .T m B ��

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Ex4Ex4 To illustrate some force and torque To illustrate some force and torque calculations, consider the rectangular calculations, consider the rectangular loop shown. Calculate the total force loop shown. Calculate the total force and torque contribution on each side. and torque contribution on each side. Let the current Let the current II flow in the loop lied in flow in the loop lied in the uniform magnetic field the uniform magnetic field tesla. tesla.

x y zx y zB B a B a B a

��

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Ex5Ex5 A 2.5 m length conductor is A 2.5 m length conductor is located at z = 0, x =4m and has a located at z = 0, x =4m and has a uniform current of 12 A in the uniform current of 12 A in the direction . Determine in this direction . Determine in this area if the force acting on the area if the force acting on the conductor is 1.2conductor is 1.21010-2-2 N in the N in the direction .direction .

ya B

��

( ) / 2x za a

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The nature of magnetic The nature of magnetic materialsmaterials

Combine our knowledge of the action of a magnetic field on a current loop with a simple model of an atom and obtain some appreciation of the difference in behavior of various types of materials in magnetic fields.

The magnetic properties of the materials depend on ‘magnetic moment’. Three types of magnetic moment are1. The circular orbiting of electrons around the positive nucleus results in the current and then the magnetic field m = IdS.2. Electron spinning around its own axis and thus generates a magnetic dipole moment. 3. Nuclear spin, this factor provides a negligible effect on the overall magnetic properties of materials.

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Types of magnetic material Types of magnetic material (1)(1)

diamagnetic The small magnetic filed produced by the motion of the electrons in their orbits and those produced by the electron spin combine to produce a net field of zero or we can say the permanent magnetic moment m0 = 0.

The external field would produce an internal magnetic field.

Some examples of materials that has diamagnetic effect are Metallic bismuth, hydrogen, helium, the other ‘inert’ gases, sodium chloride, copper, gold silicon, germanium, graphite, and sulfur.

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paramagnetic The net magnetic moment of each atom is not zero but the average over the volume is, due to random orientation of the atoms. The material shows no magnetic effects in the absence of the external field.

Whenever there is an external field and the alignment of magnetic moments acts to increase the value of , the material is called ‘paramagnetic’ but if it acts to decrease the value of , it is still called diamagnetic.

For example, Potassium, Oxygen, Tungsten, and

some rare earth elements.

B��

B��

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Ferromagnetic each atom has a relatively large dipole moment due to uncompensated electron spin moments. These moments are forced to line up in parallel fashion over region containing a large number of atoms, these regions are called ‘domains’. The domain moments vary in direction from domain to domain. The overall effect is therefore one of cancellation, and the material as a whole has no magnetic moment.

When the external field is applied, those domains which are in the direction of the applied field increase their size at the expense of their neighbors, and the internal field increases greatly over that of the external field alone. When the external field is removed, a completely random domain alignment is not usually attained, and a residual dipole field remains in the macroscopic structure.

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The magnetic state of material is a function of its magnetic history or ‘hysteresis’. For example, Iron, Nickel, and Cobalt.

Antiferromagnetic The forces between adjacent atoms cause the atomic moments to line up in anti parallel fashion. The net magnetic moment is zero. The antiferromagnetic materials are affected slightly by the presence of and external magnetic field.

For example, nickel oxide (NiO), ferrous sulfide (FeS), and cobalt chloride (CoCl2).

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Ferrimagnetic Substances show an antiparallel alignment of adjacent atomic moments, but the moments are not equal. A large response to an external magnetic field therefore occurs.

For example, the ferrites, the iron oxide magnetite (Fe3O4), a nickel-zinc ferrite, and a nickel ferrite.

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Superparamagnetic materials are composed of an assembly of ferromagnetic particles in a nonferromagnetic matrix. The domain walls cannot penetrate the intervening matrix material to the adjacent particles.

For example, the magnetic tape.

1 ene 325 electromagnetic fields and waves lecture 5 ampére’s law, scalar and vector magnetic...

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