FI 2201 Electromagnetism Magnetic Fields in Matter - Multisite ITB

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FI 2201 Electromagnetism

Alexander A. Iskandar, Ph.D.

Physics of Magnetism and Photonics Research Group

Magnetic Fields in Matter

MAGNETIZATION, AUXILARY FIELD AND LINEAR MEDIA

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Spin

• Spin is an intrinsic properties of a particle (an electron or atoms).

• Spin is the source of magnetic response of a material toSpin is the source of magnetic response of a material to external magnetic field.

Electromagnetism

Alexander A. Iskandar 3

Magnetism

• There are three forms of magnetism in matter, all having to do with spins of atomic electrons.

Electrons have spin, and spinning charges haveElectrons have spin, and spinning charges have magnetic dipole moments that can be aligned by external magnetic fields.

However, the spin of electrons, and the pairings of electrons within atoms, are manifestations of quantum mechanical effects, so magnetism is not really

understandable simply from our present classical p y p viewpoint.

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Magnetism

• There are the three kinds of magnetism:

Ferromagnetism

This is the onlystrong formof magnetism This is the only strong form of magnetism.

oIt comes from aligned spins of unpaired delectrons in

transition metals, especially nickel, iron and cobalt (hence the name), and rare earths like samarium.

Paramagnetism

Analogous to electric polarization, and much weaker than ferromagnetism; can almost understand

than ferromagnetism; can almost understand classically.

oProduced by interaction of with dipole moments of unpaired s or p electrons. Aluminum, for example, has one p electron in its valence shell, and is paramagnetic. So is molecular oxygen, which has two unpaired spins among its bonding electrons.

Br

Magnetism

Diamagnetism

Comes from the interaction of with induced magnetic dipoles in atoms; also much weaker than

Br

magnetic dipoles in atoms; also much weaker than ferromagnetism.

oContrary to the way electric polarization works, diamagnetism is characterized by magnetization in the direction opposite that of the applied field.

oSeen best in materials in which all the electrons are paired off (Ne, N₂).

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Magnetization

• Whatever the type of magnetism, we can define a

magnetization, in analogy with the electric polarization :Pr moment

dipole magnetic r

• Like , which we interpret in terms of bound charge, can be interpreted in terms of bound currents, which we can characterize by consideration of the magnetic potential for a dipole.

volume unit

moment dipole

magnetic M =

Pr

Mr

( )

⁰ ₂

4 r r ˆ r m

Ar r = r× π µ

mr τ′ d

rr

Magnetization

• In a magnetized object, each volume element carries a dipole moment of , so the total vector potential isMr dτ′

( ) ( )

∫

^′ ^×^r ^′

µ₀ Mr rr ˆd r r

• Recall that , then

• Integrating by parts yields

( ) ( )

∫

^× ^′

=

V r

r τ

π

µ M r d

r

A ⁰ ₂

4 r

r r

r 1

2 =∇′

ˆ

( )

⁼

∫ ( )

^′ ^×^⎜_⎝^⎛∇′ ^⎟_⎠^⎞ ^′

V r τ

π

µ M r d

r

A 1

4

0 r r

r r

mr τ′ d

rr

( ) [ ( ) ] ^{( )}

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ⎥ ′

⎦

⎢ ⎤

⎣

⎡ ′

×

∇′

′−

× ′

∇′

=

∫ ∫

V

Vr τ r τ

π

µ M r d

d r M r

A

r r r r

r r 1

4

0

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Magnetization

• Note that for a vector function and a constant vector we have

( )

r vr r

( ) _∫ ( )

∫

^v^r^×^C^r ^⋅^d^a^r⁼ ^∇^⋅^v^r^×^C^r^d^τ

Cr

• And with a product rule we can transform the integral on the right into

since is a constant vector.

( ) _∫ ( )

∫

^∇

V S

τ d C v a

d C v

( ) _∫ [ ⁽ ⁾ ( ) ] _∫ ⁽ ⁾

∫

^× ^⋅ ⁼ ^⋅ ^∇^× ⁺ ^⋅ ^∇^× ⁼ ^⋅ ^∇^×

V V

S

τ

τ C v d

d C v v C a d C

vr r r r r r r r r

Cr

• Finally, using the cyclic property of triple product on the left integral, we have

( )

∫

^× ⁼ ^⋅ ^∇^×

⋅

V S

τ d v C

v a d

Cr r r r r

Magnetization

• Thus, using

( ) ∫ ( ) ∫

∫

^× ⁼ ^⋅ ^∇^× ^→ ⁻ ^∇^× ⁼ ^×

⋅

S V

V S

a d v d v d

v C

v a d

Cr r r r r r r r

τ τ

on the second integral of the vector potential

we have

( ) [ ( ) ] ^{( )}

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ⎥ ′

⎦

⎢ ⎤

⎣

⎡ ′

×

∇′

′−

× ′

∇′

=

∫ ∫

V

Vr τ r τ

π

µ M r d

d r M r

A

r r r r

r r 1

4

0

( )

^r ⁼

_∫ [

^∇′^×^M

( )

^r^′

]

^d ^′⁺

_∫ [

^M

^{( )}

^r^′ ^×^d^a^′

]

Ar r r r 1 r r r

4 1

4

0

0 τ µ

µ

• This last result, shows that the vector potential of a

magnetized object can be thought as resulted from current densities.

[ ] _∫ [ ]

∫

Vr 4 Sr

4π π

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Bound Currents

• Define a volume and surface bound currents as

( )

r M

( )

r , K

( )

r M

( )

r nˆ Jr_b r′ =∇′× r r′ r_b r′ = r r′ ×

• Then, with this definition, the vector potential of a magnetized object becomes

i.e. the vector potential of a magnetized object is the same as that produced by volume current through out

( ) ( ) ( )

∫

^′ ^′⁺ ^′ ^′

=

S

V r K rr da

r d r J

A ^b ^b

r r r r

r r

π τ µ π

µ

4 4

0 0

M Jr_b r

×

∇

=

as that produced by volume current through out the material and a surface current on the

boundary.

• Compare this, with bound charges in polarized dielectrics

M J_b=∇×

nˆ M Kr_b = r×

nˆ

b=Pr⋅ P σ

b

⋅r

−∇

ρ =

Bound Currents

• Example 6.1

Find the field of a uniformly magnetized

sphere. _M^r

rr

Z

sphere. θ

For simplicity, let’s choose the magnetization in the z-direction, then

M

Y

X

θ ϕ

=0

×

∇

= M

Jr_b r

ϕ θ ˆ sin M nˆ M Kr_b= r× =

• The resulting bound surface current is the same as in Example 5.11 of a sphere with surface charge density σ being rotated around the z-axis, which generates a surface current

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Bound Currents

• In example 5.11, a sphere with surface charge density σ being rotated around

the z-axis with constant angular _ω^r rr

Z

velocity ω, so that the surface current is θ

generated

• The resulting magnetic field inside the sphere is

(

ω

)

σω θϕ

σ

σv r Rsin ˆ Kr= r= r×r =

ω

Y

X

θ σ ϕ

ω σ

µ ^r

r R

B=2

• And the magnetic field outside is calculated from the vector potential

ω σ µ R

B ₀

=3

( )

π

(

σ ω

)

θϕ

π ϕ µ θ ωσ

µ ˆ

r R sin ˆ R

r sin R R

r

A ₂ ⁰ ³ ₂

4 0

3 4 4

3 ⎟

⎠

⎜ ⎞

⎝

= ⎛

= r ≥

Bound Currents

• Looking at the analogy of the surface current of example 5.11

(

ω

)

σω θϕ

σ

σv r Rsin ˆ Kr= r= r×r =

Mr

rr

Z

and the surface bound current of the θ

magnetized sphere,

we can identify the magnetization M with Rσω.

• Thus, the magnetic field inside the magnetized sphere, b analog is

( )

ϕ

ϕ θ ˆ sin M nˆ M Kr_b= r× =

M

Y

X

θ ϕ

by analogy, is

• And the field outside is that of a pure dipole since M

B R

Br r r r

0

0 3

2 3

2µσ ω → = µ

=

( ) ( )

A_dipole

r m R r

R R

r

Ar r

=

⎟ =

⎠

⎜ ⎞

⎝

= ⎛

≥ θϕ

π ϕ µ σω θ

π π

µ sin ˆ

ˆ 4 sin 3

4

4 ²

0 2

0 3

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Interpretation of Bound Currents

• The bound current are physical current.

• Consider a uniformly magnetized object in the form of a thin slab of areaaand thicknesst.

thin slab of area aand thickness t.

t

Mr a

• As magnetization is defined to be the magnetic dipole

moment per unit volume, then we can consider that the magnetization is created from many-many small magnetic

• Note that the resulting current only exist in the ribbon surrounding the object.

many many small magnetic dipole in the form of closed loop current.

Interpretation of Bound Currents

• This surface current on the boundary can be expressed in terms of the magnetization .

• Since the volume of the object isat, then the magnetization Mr

Since the volume of the object is at, then the magnetization is

Mat at m

M= m → =

• Where mis the dipole moment which in terms of the current, it is equal to m = Ia.

t

Mr a

nˆ Krb

• Thus, I = Mt, and recall that the surface current is defined to be K_b= I/t, then to have the correct direction for the bound surface current, we must have

nˆ M Kr_b= r×

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Interpretation of Bound Currents

• The volume current results from a non-uniform magnetized object, we consider small volume element, and deduce each component of the volume current density .Jr

• For each small volume element, it is assumed a uniform magnetization.

• Consider the z-component of this magnetization on two adjacent volume elements.

• The volume current component in

( )

y M_z

Z

dz dy

(

y dy

)

M_z +

• The volume current component in the x-direction can be calculated from the surface currents on these adjacent volume elements

( ) ( )

[ ]

dydz

y dz M y M dy y M dz K

I_x _x _z _z ^z

∂

=∂

− +

=

X

Y

Interpretation of Bound Currents

• Then the x-component of the volume current resulted from the z-component of the magnetization is

( )

J = ^I^x =∂^M^z

• There is another contribution on the x-component of the volume current, namely that is resulted from volume elements depicted in the right figure.

( ) ( )

[ ]

M d d

d d M M

d K

I ∂ ^y

( )

J_b _x dydz y

= ∂

=

( )

z M_y

Z

dy

(

z dz

)

M_y +

dz

with volume current density

( ) ( )

[ ]

dydz

dy z dz z M z M dy K

I_x _x _y _y ^y

− ∂

= +

−

=

( )

z

M dydz

J_b _x I^x ^y

∂

−∂

=

X

Y y

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Interpretation of Bound Currents

• Thus the total x-component of the volume current resulted from the magnetization is

( )

J ∂^M^z ∂^M^y

• The other components of the volume current are calculated in the same way and the final result is

• These bound currents are called bound because it is

( )

J_b _x y^z z^y

− ∂

= ∂

M Jr_b r

×

∇

=

resulted from restricted motion of charges, that is the current is resulted from a small current loop, and these charges can not moved freely as charges in a conductor placed on a potential difference.

Auxiliary Field

• We have seen that the effect of magnetization is to established bound currents.

• Thus the magnetic field of a material can be regarded as aThus the magnetic field of a material can be regarded as a result of these boundcurrent and the freecurrent .

• Recall the Ampere’s law for the magnetic field remember that we did not specify what is the current .

• Thus, we can now write

J Br r

µ0

=

×

∇

Br Jr^f

Jr 1

or

(

^B

)

^J ^Jf ^Jb ^Jf ^M

r r

r r r

r = = + = +∇×

×

∇

0

1 µ

Jf

M Br r r

⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛ −

×

∇ 1

µ

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Auxiliary Field

• The quantity in the bracket, is then defined to be the Auxiliary Magnetic field Hr, i.e.

(

^H ^M

)

B M

B

Hr r r r r r

+

1 → µ

• Note the similarity with the electric field

• Then, the Ampere’s Law can now be written in terms of only free current

(

^H ^M

)

B M

B

H ≡ − → = ₀ +

0

µ µ

P E Dr r r

+

=ε₀

or

• Example 6.2

Jf

Hr r

=

×

∇

fenc

I d H⋅ =

∫

C

rl r

Auxiliary Field

• The last result of Ampere’s Law gives us a tool to

determined the magnetic field when a free current exist on a magnetized medium with the magnetization is not known.

• However, one must be careful about the use of , for many of the same reasons that we are cautious about , since does turn out to be somewhat more generally useful than .

Reason: measurability. We can easily measure potential differences and free currents (with voltmeters and ammeters), so these relate best to the fields and , not and .

R l f th b i f l h th it ti i t i l

Hr

Dr Hr Dr

Er Hr

Dr Br Hr

Rule of thumb: is useful whenever the situation issymmetrical enough to use Ampère’s Law: infinite cylinders, planes and solenoids; toroids.

• Remember, that is the fundamental field. There isn’t a Biot-Savart law or Lorentz force law for .

H

Br

Hr

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Auxiliary Field

• Further, as a vectorial quantity, the field is better defined as compare to the , Hr Br

B and J

Br r r

=

⋅

∇

=

×

∇ µ₀ 0

• Consider for example a short cylindrical magnet made of iron with permanent magnetization .

• Since , hence inside the magnet , thus the magnetic field is

Mr

=0 Jrf

=0 Hr M B M

B

Hr ≡ 1 r r → r=µ r M H

and J

Hr r_f r r

⋅

−∇

=

⋅

∇

=

×

∇ µ0

• However, if we continue this argument to calculate the

magnetic field outside, we will conclude that , which is absurd.

• Further, note that in this problem there is no symmetry.

M B M

B

H ₀

0

µ ⁻ ^→ ⁼µ

≡

0 =0

= M_free_space Br r

µ

Boundary Condition

• The boundary conditions for the field can be obtained from the boundary conditions of

Hr Br

=0

− ^⊥

⊥

below

above B

B

• In the presence of a magnetic materials, a more useful boundary condition is

below above

( )

^K ⁿ^ˆ

B

Br^||_above− r_below^|| = r× µ0

(

^⊥ ^⊥

)

⊥

⊥ − _below=− _above− _below

above H M M

H

nˆ K H

Hr^|| r^|| r ×

• Which are obtained from

n K H

H^||_above− _below^|| = _f×

M H

and J

Hr r_f r r

⋅

−∇

=

⋅

∇

=

×

∇

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Linear Magnetic Media

• Note that for paramagnetic and diamagnetic material, when is removed, disappears. Thus, there should be a linear relation between magnetization and the magnetic

Br

Mr field.

• As before, when the definition of led to that of the electric susceptibility χ_eand the dielectric constant ε, we can define

Dr

B M

E

Pr _er r χ_mr χ µ

ε

0 0

= 1

→

=

• However, this definition is wrong since with the above definition, χ_mwould have a dimension, unlike χ_e. Thus the correct definition is

H Mr =χ_m r

Linear Magnetic Media

• With the above definition, then for a linear media we have

(

^H ^M

) ⁽ ⁾

^H ^H

Br r r _m r r

µ χ

µ

µ + = + =

= ₀ ₀1

where the material’s permeability is

• Note the similarity

(

_m

)

_r χ_m

µ µ µ χ

µ

µ= 1+ → = =1+

0 0

( )

^E ^E

P E

Dr=ε₀r+ r=ε₀1+χ_e r=εr

(

χe

)

εr ε χe

ε

ε = ₀1+ → = =1+

• Further, for linear material, the volume bound current is proportional to the free current

(note the rather vague similarity)

(

e

)

r χe

χ ε

0 0

( )

m m f

b M H J

Jr =∇× r =∇× χ r =χ r _f

e e

b ρ

χ

ρ χ _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

− +

= 1

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Linear Magnetic Media

• Example 6.3

Ferromagnetism

• In linear magnetic materials, the alignment of magnetic dipole of the atom is maintained by the presence of the external field.

• In Ferromagneticmaterial, in which the response of the material on the external field is non linear, the alignment is frozen.

• In a ferromagnetic materials, the dipole of nearby atoms interact so that they like to align in the same direction, and this alignment is virtually 100%.g y

• For ordinary ferromagnetic material, the alignment of these dipoles happens in many small patches, called domains.

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Ferromagnetism

• The magnetization of these domains are oriented randomly, hence this is the reason that ordinary ferromagnetic material sometimes does not shows a magnetic phenomena.

• With a strong magnet, we can rotate the dipole at the domain boundary so that this boundary dipole be aligned with the external magnetic field, i.e. the domain becomes larger.

Ferromagnetism

• When we magnetized a ferromagnetic material with a current, the resulting magnetization depends on the historyof the material.

I (=H) M (= B)

saturation

permanent permanent

magnet

• This loop is called Hysteresis loop.

saturation

permanent magnet

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Summary of Magnetic Materials

• For the three forms of magnetism:

Ferromagnetism: χ_m> 0and very large.

For instance, pure, annealed iron has µ_r_r≈10000

Paramagnetism: χ_m> 0but << 1.

Typically, χ_m≈10^-6→ µ_r≈1.

Diamagnetism: χ_m< 0and |χ_m| << 1→ µ_r≈1.

Typically somewhat smaller than is typical in paramagnetism.

• The consequence is that µ_ris so close to unity for anything that isn’t ferromagnetic, is the reason that one

could easily avoid noticing magnetism in everyday life.

FI 2201 Electromagnetism Magnetic Fields in Matter - Multisite ITB