module m2-1 electrical engineering -...

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Module M2-1Electrical Engineering

L E C T U R E 5 M A G N E T O S T A T I C S I I

S E P T E M B E R 1 6 , 2 0 1 3

Topics

Ampère’s circuital law

Magnetic flux density

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Magnetic flux

Gauss’s law for magnetism

Magnetization

Magnetic materials

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Ampère is a man of great discovery3

a French physicist and mathematician, born 1775 (Thai l d 8 d i th i f ็ ้ ิ )calendar 2318 during the reign of สมเดจ็พระเจา้ตากสิน)

Well known for work in electromagnetism such as Ampère’s circuital law

The unit of current (A) is named after him

Last time, we have seen thatcurrent produces the magnetic field

a wire carrying the current(arrow denotes the direction of current)

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(arrow denotes the direction of current)

The magnetic fieldexists around the wire

[VDO 27: magnetic field around a wire]http://www.youtube.com/watch?v=6bu84cSd3Zg

exists around the wire

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Ampère’s law relates the magnetic-field intensity to the current passing through a closed path

a wire carrying the current(arrow denotes the direction of current)

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a closed path(a circle)

a closed path

a closed path

The law is applicable to any closed path

But it is used often with symmetric closed paths such as circles (for ease of evaluation)

p

We may assign the direction to any closed path by using the right-hand rule

(current, flowingin the direction of arrow)

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in the direction of arrow)

closed path

Point the thumb of your right hand in the direction of

Direction of the path is along the other four fingers

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One form of Ampère’s law is given bya line integration

magnetic-fieldintensity

length alongthe closed path

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intensity

current enclosed inthe closed path

the closed path

the closed path

dot product

The line integral of H around a closed path is equal to the current traversing the surface bounded by that path

We can represent a path by a vector function

y-axis

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x-axis

Example in 2 dimensions: a circle of radius 1 is given by a vector function

for

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The derivative gives us the direction of the tangent line at position on the path

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y-axis

x-axis

is the component-wise derivative of

Example: the derivative of is

The symbol is called the line integral

Let a closed path be given by a vector function, for

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Note: In this course we focus on,

Definition: The line integral of H along C is

An integration of a function invariable just like

Note: In this course, we focus on2D closed paths. In a more advancedcourse, will have 3 components.

component-wise derivative:The magnetic field intensityat the position given by ,i.e., at the position

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Geometrically, is a sum ofinfinitesimal quantities along a closed path

...

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“...” is a mathematical symbolmeaning “and so forth.” In other words some elements

Divide the path into small arcs at points ’s

...

In other words, some elementsare not shown here.

Assign a vector to each arc, connecting the adjacent points Direction of approximates the tangent line to the arc Magnitude of approximates the arc length

Then, equals approximately the sum

Magnetic field intensity depends onthe position in the space

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For example consider the current I that flows out of the slide For example, consider the current I that flows out of the slide in an infinitely-long wire

Recall from last lecture: Magnitudes of H on any circle around the wire are constant Direction of H is tangent to the circle

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We can draw H at any points include points on a closed path

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under an influenceof the magnetic field

In both pictures, the line integrals of H equal I14

Equal I because of Ampère’s law, although the paths are very different

Magnitude of H is constanton the given closed path C

Magnitude of H is not constanton the given closed path C

Source: (p. 225, F. T. Ulaby, E. Michielssen, U. Ravaioli, 2010)

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To apply Ampère’s law,the current must flow through a closed path

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The above path does not enclose the current I

Hence, the line integral of H along this path is zero, even though H is not zero along the path

Adapted from (p. 225, F. T. Ulaby, E. Michielssen, U. Ravaioli, 2010)

Example: The magnetic-field intensityof a conducting-current wire

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The current I passes through an infinitely-long wire

Question: what is the magnetic field intensity at distant r from the wire?

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Solution: The magnetic-field intensityof a conducting-current wire

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Answer:

`

side view(H points intothe slide)

top view

Answer: Direction of H is according to the right-hand rule (see the

above figures)

The magnitude of H is We will show this shortlyusing Ampère’s law

Solution: Magnitude of magnetic-field intensity18

Consider a closed path C being the circle of radius r:

top viewside view

Consider a closed path C being the circle of radius r:

Think of as points on the circle

for

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Solution (continued)19

Ampère’s law tells us thatp

The line integral evaluates intoRecall: the dot product


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By symmetry, the magnetic field intensities at all points on path C have the same magnitude, denoted by a scalar h


top viewside view

Hence, the magnitude of the magnetic field intensity is

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Topics



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Magnetic flux


Magnetization

Magnetic materials

Magnetic flux density B is related tothe magnetic field intensity H

In free space, the relationship in SI unit is

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Magnetic flux density(Wb/m2 , or equivalently,T)

Magnetic field intensity(A/m)

free space permeabilityH/m

Note:A – ampereC – coulombF – faradH – henry

Analogy in electrostatic, also in free space:

Electric displacement vector(C/m2)

Electric field (V/m)

free space permittivityF/m

T – teslaWb – weber

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Magnetic flux is an integral of the magnetic flux density B over a surface

Magnetic flux:

t ( 2)

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Analogy in electrostatic:

Magnetic flux density(Wb/m2)

an area vector (m2)

Magnetic flux(Wb)

an area vector (m2)displacement flux (C)

Source: http://en.wikipedia.org/wiki/Magnetic_flux

Electric displacement vector (C/m2)

p

flux of E (V-m)

Electric field (V/m)

an area vector (m2)

The symbol is called a surface integration26

Divide the surface into small rectangles, centered at and of the area

Associate each rectangle a unit vector normal (i e perpendicular) to the rectangle

Source: http://en.wikipedia.org/wiki/Magnetic_flux

(i.e., perpendicular) to the rectangle Compute the magnetic flux density for each

rectangle Then, equals approximately

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Here is an illustration of a surface integration

[VDO 26/ Magnetic flux: an integration of B over a surface]

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surface]http://www.youtube.com/watch?v=pB7oZNBIqqc

A surface could be closed or open28

Closed surfaces Open surfaces

Source: http://en.wikipedia.org/wiki/File:SurfacesWithAndWithoutBoundary.svg

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Topics



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Magnetic flux


Magnetization

Magnetic materials

Mathematically, Gauss’s law for magnetism states that...

The integration of B over a closed surface is zero:

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Analogy to electrostatics:

This circle means thatthe surface is closed

Note: We have seen this formof Guass’s law for electrostatics:

By multiplying both sides by , we may write the above equationi f d h

.

Total charge enclosedin the closed surface

in terms of and as shown.

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Intuitively, Gauss’s law for magnetismstates that ....

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Inside a closed surface the amount of north poles equals the

For simplicity, a closed surface is shown here in 2D

Source: http://ned.ipac.caltech.edu/level5/March03/Vallee2/Vallee2_2.html

Inside a closed surface, the amount of north poles equals the amount of south poles

This intuitive explanation is supported by experiments: magnetic poles are found to occur in pairs of north and south So the net amount of poles in a closed surface is zero

An Application of the magnetic flux

[VDO 22: magnetic flux and a superconducting plate]

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plate]http://www.youtube.com/watch?v=VyOtIsnG71U

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Review: we have seen these key equations33

Gauss’s law:

Gauss’s law for magnetism:

Ampère’s circuital law:

Topics



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Magnetic flux


Magnetization

Magnetic materials

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There are several ways to create and destroy the magnetic properties

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Here are some note on vocabularies: “to magnetize a material” = “to turn a material into a magnet” to magnetize a material to turn a material into a magnet

“to demagnetize a material” = “to destroy the magnetic properties of a material”

This VDO shows you methods of magnetization and demagnetization: [VDO 23/started from 1:02/

methods to magnetize and demagnetize]methods to magnetize and demagnetize]http://www.youtube.com/watch?v=Dka-cROHxBY

QUnder an influence of an external magnetic field

A

QA material becomes magnet because theirdipole moments align approximately in one direction.

Under an influence of an external magnetic field,how does a material such as iron become magnet?

A

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In the following slides, we will explainthis answer in detail.

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In our living world (macroscopic world), magnetismarises from magnets (hard and soft magneticarises from magnets (hard and soft magneticmaterials), electromagnets, and current flow.

Magnet causes magnetism

Flow of current in a coil causes magnetismmagnetism causes magnetism

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Technically speaking, we say that the source of magnetism is a magnetic dipole moment. g g p

N

S

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• At atomic and molecular levels (microscopic world),electrons

: move in orbits around a nucleus similar to theearth moving in an orbit around the sun,

: rotate (spin) around their own axes similar to theearth rotating around its own axis.• Movement of electrons in orbits and electron spin

q i l t t h t d i th fare equivalent to charge movement and is the source ofmagnetism or dipole moments. We can call them orbitaldipole moment and spin dipole moment, respectively.

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• In fact, the origin of all magnetism in magneticmaterials are due to movement of electrons in orbits andelectron spin.

Magnetic dipole momentdue to electron moving in orbit around nucleus

Magnetic dipole momentdue to electron spinning around its own axis

• Net dipole moment of an atom is a vector sum of allorbital and spin dipole moments through theirinteraction. Imagine complexity of interactions of 26orbital moments and 26 spin moment in one iron atom.

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• When Fe atoms (and other magnetic atoms such asNi Co) form a solid net moments of atoms further

• Each domain will have anet dipole moment, call

Ni, Co) form a solid, net moments of atoms furtherinteract. This will cause net moments to point, to align,in the same direction within a small region called amagnetic domain.

pmagnetization, in one directiononly. Sizes of magnetic domainsare few tens of microns.• Different domains havedifferent magnetization.

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[VDO 25: Magnetic domain]http://www.youtube.com/watch?v=QgwReDkpq6Ehttp://www.youtube.com/watch?v QgwReDkpq6E

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• A piece of magnetic materials(such as magnets) have millions

f d i L ft i t thof domains. Left in nature, theyshow no magnetism as netmoments of different domainspoint in different directions. Wecan see domains with KerrMicroscopy technique.

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• We can force moments in different domains to align in the same direction by applying external magnetic fieldsfrom permanent magnets or electromagnets. • External magnetic fields will exert force on moments in domains so that they are parallel to the applied fields.Under such condition, we say that materials

- show net dipole moment,- exhibit magnetism- exhibit magnetism,- become magnetized.

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[VDO 24: Magnetized domains/ trim to 0:08-2:11]http://www.youtube.com/watch?v=85dIRfKMlwMhttp://www.youtube.com/watch?v 85dIRfKMlwM Note: The VDO does not consider an effect known as the

hysteresis effect. The purpose of this VDO is to illustrate a basic concept of magnetized domains.

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Topics



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Magnetic flux


Magnetization

Magnetic materials

In a material, magnetization (a vector) appears in a relationship of B and H

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Recall: in free space (vacuum), we have a relationship

free space(vacuum)

dipole momentin a domain

p

In a material (such iron), the l i hi i

(free space)

Magnetic fluxdensity

Magnetic fieldintensity

a material relationship is

(generic material)

Magnetization (the vector sum of allthe magnetic dipole moments)

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The magnetization M is related tothe magnetic field intensity H

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Relationship: material’s magnetic susceptibility(no unit dimensionless)

Values of depend on the materials:

(no unit, dimensionless)

vacuumwatergold air aluminum iron (99.96% pure)cobalt

By substitution, we have a relationship(generic material)

Source: (p. 137, S. M. Wenthworth, 2005)

for engineering purposes, these values are approximately zero

Materials are divided into 3 groups according to the basis of their magnetic susceptibilities

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watergold air aluminum iron (99.96% pure)cobalt

diamagnetic materials:Their ’s are negative

paramagnetic materials:Their ’s are positive

ferromagnetic materials:Their ’s are positiveTheir s are negative Their s are positive

and smallTheir s are positiveand large

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Summary

Ampère’s circuital law states the relationship between magnetic field intensity and the current:

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between magnetic field intensity and the current:

In free space, the magnetic flux density (B) is a scalar multiplication of magnetic field intensity:

The magnetic flux is an integral of the magnetic flux density over a surface:

Summary

Gauss’s law for magnetism states that the magnetic flux is zero for a closed surface:

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magnetic flux is zero for a closed surface:

Under an external magnetic field, a ferromagnetic material becomes a magnet because its domains align in one direction

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