Static Magnetic Fields

Chapter 6.

Static Magnetic Fields

Magnetism

Magnetism & EM force

• Magnetism

– Discovered when pieces of magnetic loadestone were found to exhibit a mysterious attractive force.

– Found near the ancient Greek city called Magnesia

• A magnetic field

– Caused by a permanent magnet, by moving charges (current)

• Electromagnetic force : electric force + magnetic force – Electric force : F_e = qE (N)

– Magnetic force : F_m = q u×B (N), B : magnetic flux density [T]

– Total electromagnetic force on a charge q : F = q (E + u×B) Lorentz’s force Equation

Note! Magnetic forces do no work! dW_mag  F_magdl  q u B udt(  )  0!

Magnetostatics

Stationary charges  constant electric fields  Electrostatics Steady current  constant magnetic fields  Magnetostatics

0 0

t



    

 J

Magnetostatics  Steady current 

• Biot-Savart Law  Coulomb’s law

 

'

0 R

2

T=N/(A m) 4

I dl R





   ^a 

B 

 





0 permeability of free space 4 10 N/A





R

( , , )

B x y z

Divergence of B:

 

0 R

2

T=N/(A m) 4

I dl R





   ^a 

B 

Idl ^  JSdl ^  Jd  ^

R

S

( , , ) B x y z

0 R

2

'

4

J d

R

 



   ^a B

0 R

2

'

4 J d

R

 



 

   ^B       ^a  

R R R

2 2

( )

J J J

R R R

   

                

a a a

Divergence and curl are to be taken with respect to (x,y,z) coordinate.

J is a function of (x’,y’,z’) 

R

2

0

R

   

 

 

a

0

  J

   B 0

   B 0





R R

2 d 2 ds 0 sinds R d d r R

R  R   

       

 

 



 

Curl of B:

0 R

2

'

4

J d

R

 



   ^a B

0 R

2

'

4 J d

R

 



 

  ^B      ^a  

R R R

2 2 ( ) 2

J J J

R R R

   

       

   

a a a

0

J



  B

0 3

( ')4 ( ') '

( ) 4  J R  R R d   J R

  ^B    

R 3

2 4 ( )R

R 

  a  _R

(J ) 2d ' 0 R 

 



0 J



  B

    da    dl 

J da 

 ^B   ^B 

0 enc

dl  I

 ^B  



R R

2 2

3 3

3

3 3

R 2

( ) ( ')

' '

' ( ' )

' '

' '

' ' '

0 on the boundary (all current is safely inside)

( ) 0

J J

R R

R R R R

J J

R R

R R R J

R R R R

J d Jda

R R

J

J d

R





   

 

   

       

  

   

 

 

   



  

 



a a

a



Two postulates of magnetostatics in free space

• The divergence and the curl of B in nonmagnetic media –

– No magnetic flow sources (no isolated magnetic charges)

– Divergence of the curl of any vector field = 0

0

0

. (in nonmagnetic media)



 

  B

B J



 

 

0 permeability of free space 4 10 H/m

    ^

0 0

V S

d  d

   

 ^{ B}   ^{B s }

 

   B    J

Comparison of Magnetostatics and Electrostatics

0

0 E

E



  

 



0

0



 

  B

B J



: Gauss’s law

: Ampere’s law Coulomb’s law

Biot-Savart law

( )

F  q E    B

Lorenz force law

Ampere’s circuital law

• Example 6-1.

– An infinite long, straight, solid, nonmagnetic conductor with a circular cross

section of radius b carries a steady current I. Determine the magnetic flux density both inside and outside the conductor.

1 1

1 1

2

1 1

0

1 2 2

0

1 2 1 2

) Inside the conductor ,

,

2

The current through the area enclosed by C ,

2

C

a

B

d rd

d B rd rB

r r I

I I I r r b

b b b

 





 





 





 





 

          

 

B a l a

B l

B a

 

Ampere’s circuital law

• Example 5-1.

2 2

2

2

2 2

0

0

2 2

) Outside the conductor:

, ,

2 1 , 2

C

b

B

d rd

d B rd rB

B I r b

r

 





 

  



 









 

  

 

B

a l a

B l

B a a

 

Ampere’s circuital law

• Example 6-2.

– Determine B inside a closely wound toroidal coil with an air core having N turns of coil and carrying a current I.

– The toroid has a mean radius b, and the radius of each turn is a.

– Cylindrical symmetry  B has only a φ component and is constant along any circular path about the axis of the toroid.

 

 

   

0

0

A cicular contour with radius .

2 , -

, - 2

0 for and .

C

C r

d r NI b a r b a

B NI b a r b a

r

r b a r a b



  

 





    

    

    



B a a

B



Ampere’s circuital law

• Example 6.3. Infinitely long solenoid with air core – Direct application of Ampere’s law

– A special case of toroid (b)

0 0

BL   nLI   B  nI

0

0 0

2 2

NI N

B I nI

r b

  

 

 

      

6-3. Vector magnetic potential

• B : divergence-free –

– A : vector magnetic potential (Wb/m)

– not sufficient definition. Its divergence is needed.

  B  0

 

0 T

  B     B A

 

 

0

2 0

2 2 2 2 2

2

0

Simplifying it, we choose =0 (Coulomb gauge).

: vector Poisson's equation

o x

A

A

A









 

        

            



  

B J

A J A A A

A A J A a a a

A

A J











 Ampere’s law

Vector potential:

0 2

0

Amp , , becomes vector Poisson's equation,

( thr Poisson's equations, one for each Cartesian component) e

ee re law 



 

   

 B J

A J

   A 0 ?

0 0

If we add A the gradient of any scalar ,  A  A      B A is also satisfied.

0 0 0

Suppose that A satisfying B  A is not divergenceless (A  0)

2

The new divergence is      A A0   

2

If a function can be found that satisfies the Poisson's equation, 0, It is always possible to make divergenceless.

     A

A

    A 0

Finding A from current density

• Mathematical analogy with the scalar Poisson’s eqn.

   

2

0 0

, , 1

, , 4 ^V

x y z

V V x y z d

R

 

  

  ^

  

    



 ^{, ,} 

 ^{, ,}   ^Wb/m 

4

x y z

x y z d

R

 



  

 



2 2 2

( ) ( ) ( )

R  R  x x  y  y  z z

2

Ampere's law in vector potential, ^{ A  } 

^J , gives

If _ goes to zero at infinite,

by assuming goes to zero at infinite,J The solution of the vector Poisson’s eqn.

For line current:

 

 ^{, ,} 

1

, , ' '

4 4

I x y z I

x y z dl dl

R R

 

 

  

   

A

For surface current:

 

 ^{, ,} 

, , '

4

K x y z

x y z da

R





  

 

A

For bulk current:

Magnetic flux

• Magnetic flux Φ through a given area S

  ^(Wb)

s S

C

d d dl

   ^{B s }    ^{A  s}    ^{A }

Biot-Savart Law

• For thin wire with cross sectional area S, d’ equal to Sdl’

• Magnetic flux from the vector potential

( ,x y z  , )

R J

R



S

( , , ) P x y z

0 0

1 (Wb/m)

4

4

Jd JSdl Idl

I dl

R d R



 





    

 

  ^J  

A 

0 0

4 4

The curl operation implies differentiations with respect to the space coordinates of the field point, and the integral operation is with res

C C

I dl I dl

R R

unprimed

 





     

               

B A  

pect to the primed source coordinate .

Biot-Savart Law

• Magnetic flux from the vector potential (cont’d)

 

 

0

2 2 2 1/ 2

Using

1 1

4 +

1 ( ) ( ) ( )

1 1 1 1

( ) ( ) ( )

=

(

C

x y z

x y z

f f f

I dl dl

R R

x x y y z z

R

R x R y R z R

x x y y z z







     

     

              

  

 

       

  

       

                      

  

    





G G G

B A

a a a

a a a



R

3/ 2 3 2

2 2 2

) ( ) ( ) R R

x x y y z z

   

  

      

 

a

R

Biot-Savart Law

• Biot-Savart Law

0 0 R

2

0 R 0

2 3

1 (T)

4 4

4 or 4

C C

C

I I dl

R dl R

I dl I dl

d d d

R R

 

 

 

 

 



   

       

 

   

 



B a

a R

B B B B

 



A current-carrying straight line

• Example 6-4.

– Find magnetic flux density B from A

 

2 2

2 2

0 0

2 2

2 2

0

2 2

2 2

0 0

2 2 2 2

0

)

4 4 ln

= ln4

1

4 ln 2

When , (infin

2

z

L L

z z

L L

z

z z

r

dl dz

a R z r

I dz I

z z r

z r

I L r L

L r L

A A

r r

I L r L IL

r L r L r L r

r L I

r



 

 

 

 







 

 





 

  

  

    

       

 

 

 

   

 

 

  

        

 



a

A a a

A a

B A a a

B a a

B a itely long wire carrying current )I

A current-carrying straight line

• Example 6-4.

 

 

0 0 0

3 2 2 3/ 2 2 2

)

4 4 2

z

r z

z r z

L L

dl dz

b r z

dl z r z rdz

I dl I rdz IL

d R z r r L r



 

  

 



  

  

        

  

 

             

a

R a a

R a a a a

B B R a a

A square loop (example 6-5)

• Find B at the center of a planar square loop, with side w carrying a direct current I.

• Exercise

– 8(cm) × 6 (cm) rectangular conducting loop and 5 (A) direct current

– B at the center of the loop ?

0 0

The magnetic flux density at the center of the square loop is equal to four times that caused by a single side of length

/ 2

4 2 2

z 2 z

w L r w

I I

w w

 

 

 

  

B a a

A circular loop (example 6.6)

• Find B at a point on the axis

 

   

2 2

2

2 2

0 0

2 2 3/ 2 2 2

, ,

component is canceled by the contribution of the element located diametrically opposite to

4 2

z r

z r r z

r

z z

dl bd z b R z b

dl bd z b bzd b d

dl

I b d Ib

z b z b







  

  



      

         





  

 

a R a a

R a a a a a

a

B

a a

 

0^

T



A circular loop (example 6.7)

• Uniform magnetic field – Helmholtz coil

– MRI Magnet

The magnetic dipole

• Example 6.7

 

 

0

1

x

x y

2 / 2

0 0

x 0 / 2

1 1

2 2 2

1

= 4

at is not the same as at the . In fact, at is -

sin cos '

sin sin

= or

4 2

2 cos

cos sin cos 90

C

I dl R

dl P P

dl bd

I b Ib

d d

R R

R R b bR

R R

  

 

 





  

     

 



  









   

   



  

   



 

A

a a a a

a a

A a a



2 2 2 2 2

1

2 1/ 2 2 1

sin sin

2 cos 2 sin sin

1 1 2

1 sin sin

R

R R b bR R b bR

b b

R R R R

 

  

 





      

 

    

 

x

cos sin sin R  R  ^

The magnetic dipole

• Example 6.7

 

 

2 2 2 2

1/ 2

1

1

0 / 2

/ 2 2 0

2

When , / can be neglected in comparison with 1

1 1 2

1 sin sin

1 1

1 sin sin 1 1 when 1.

= sin 1 sin sin

2 , = sin

4



 



 

 

    



 









 

   

 

       

 

   

 



A a

A a

B

p

R b b R

b

R R R

b x px x

R R R

Ib b

R R d

R Ib

R

 

2 0

3 2 cos sin

4 ^

  

 A Ib a_R  R a

The magnetic dipole

• Similarity between electric dipole and magnetic dipole at distant points.

• Simplest form of magnetic dipole

 

 

3 0

2 0

3

2 cos sin , 4

2 cos sin 4

R

R

p q

R Ib R





 



  

  

 

E a a p d

B a a

q

q d

   

   

2

0 2 2

2

0

2 2

0

sin A m

4

Wb/m V

4 4

z z z

R R

I b I b IS m

R

R V R



 

 

 

 

    

   

A a m a a a

m a p a

A





m

I

Magnetization

• All matters are composed of atoms, each with a positive charged nucleus and a number of orbiting electrons.

• In addition, both electrons and the nucleus of an atom rotate (spin) on their own axes with certain magnetic dipole moments.

• In the absence of an external magnetic field, the magnetic dipoles of the atoms of most materials (excepts permanent magnets) have random orientations, resulting in no net magnetic moment. The application of an external magnetic fields cause both an alignment of the magnetic moments of spinning electrons and an induced

magnetic moment due to a change in orbital motion of electrons

e-

m

nucleus

because and

nucleus electron nucleus electron electron

m  m M  M  

1 2

q

 M

m L

Magnetization vector, M

• To obtain a formula for determining the change in the magnetic flux density caused by the presence of a magnetic material, we let m_k be the magnetic dipole moment of an atom.

• The magnetic dipole moment dm of dv´ is dm =Mdv´will produce a vector potential.

• The total A is the volume integral of dA and the contribution of magnetization to B is A

 

0 1

lim 1 A/m , # of atoms within

n

k k

n





 





  

   

 

M m

0

4

d R d

R

 



 



A

0

4

R

V

d

R

 





 

     ^{M a}

B A

Equivalent current densities

• The total A

0 0 0

2 2

0 0

1

4 4 4

1 1

Using the vector identity,

4 4

Using vector identity,

R R

V V V

V V

V

S

d d d

R R R

R R R

d d

R R

d d

     

  

   

 



  

 

 

      

     

 

   

  

          

      

     

  

    

  

 

 

M a M a

A M

M M M

M M

A

F



2

1 _R

R R

 

  

  a

0 0

4 4

n V

S

d ds

R R

  





  

   

  ^M   ^{M a}

A 

current





n ms

 

M a J





m A/m

M  J

Magnetization current density

• Magnetization surface current density

• Magnetization volume current density

Magnetization current density

 

ms

 

A/m J M a





m

  A/m

J M



  P 

P ^{b  P n}

P

: out of paper M

J

J

On the surface of the material, 0.

If is uniform (space invariant) inside, the current of the neighboring atomic dipoles will cancelled averywhere.

No net current in the interior ( 0), or,

ms n

ms

 a 



J M

M

J

0 if is space invariant.

m

M  J  M

Physical interpretation of Magnetic current

 

surface current: _ms /

m at m Ia

I Mt

J I t M

 

 

  

M For uniformly magnetized material,

all the internal currents cancel.

m

0

J

  M 

However, at the edge there is a current.

 

J

M a

Physical interpretation of Magnetic current







  A/m J

M

 

 

( ) ( )

x z z

z x

z

m x

I Mt

I M y dy M y dz I M dydz

y J M

 y



   

  



  



For nonuniformly magnetized material, the internal currents no longer cancel.

 

M M

J

y z

 

 

 

 

( ) ( )

x y y

y x

y

m x

I Mt

I M z dz M z dy I M dzdy

z J M

 z



 

     

   



  



Magnetization current density

• Example 6.8 – cylindrical magnet bar – Axial magnetization : _{M  azM0}

magnet

b L

 

 

0 0

0 2

0 2 2 3/ 2

Constant magnetization no magnetic volume current

To find at 0, 0, , We consider a differential length dz with a current and use Eq. (6-38) or to obain

2

ms z r

z

M M

P z M dz

Ib d

z b









  

 

 



J a a a

B a

B a B

 

 

 

2

0 0

2 2 3/ 2

0 0

2 2 2 2

2

2

z

z

M b dz z z b

M z z L

d

z b z L b







  

  

 

  

    

 



a

B B a

m 0

M  J 

Magnetization current density

• Example 6.8 – cylindrical magnet bar

 

0 0

2 2 2 2

2

 ^   ^ 

 

    

 

B a

M z z L

z b z L b

• Because the application of an external magnetic field causes both an alignment of the internal dipole moments and an induced magnetic moment in a magnetic material, the resultant magnetic flux density in the presence of a magnetic material will be different from its value in free space.

6-7. B&H and Ampere’s law

B_ext B_ind B_total= B_ext+B_ind

 

0

0

0

1

m







  

    

 

     

 

B J J

B J M

B M J





Ampere's law

A/m

dl I

  

    H J

 H

(magnetic field intensity)

(free current)

 

0

 A/m

 B 

H M

B&H and Permeability

• When the magnetic properties of the medium is linear and isotropic,

• B&H

• Permeability

– The permeability of most materials is very close to that of free space, ₀

– For ferromagnetic materials such as iron, nickel and cobalt,

, : magnetic susceptibility

m m

 



M H

    



0 0 1 0 Wb/m

, 1

m r

r m

     

  

     

   

B H M H H H

B H

Absolute permeability Relative permeability

r

1

 

Analogous Relation

• Electrostatics and Magnetostatics

Electrostatics Magnetostatics

E B

D H

 1/

P - M

 J

V A

 

 

6-9. Magnetic Materials

• Magnetic materials

– Diamagnetic, if _r  1

• _m ~ -10^-5

• the orbital motion of the electrons

• Copper, germanium, silver, gold – Paramagnetic, if _r 1

• _m ~ 10^-5

• Magnetic dipole moments of the spinning electrons

• Aluminum, magnesium, titanium and tungsten – Ferromagnetic, if _r >>1

• _m >> 1 ( 100~ 100,000)

• Magnetized domains (strong coupling forces

between the magnetic dipole moments of the atoms

• Nickel, cobalt, iron (pure), mumetal

Magnetic Materials

Ferromagnetic material

• Hysteresis loops

Residual flux density (permanent magnets)

Coercive field intensity

H

I V B

M

Magnetic recording

Ferromagnetic material

• Hysteresis loss per unit volume per cycle – The area of the hysteresis loop

– The energy lost in the form of heat in overcoming the friction encountered during the domain-wall motion and domain rotation – “Soft” material with tall and narrow hysteresis.

• Curie temperature

– Demagnetization temperature (100 ~ 770 ◦C)

• Ferrites

– Ceramic like compounds with very low conductivity – Low eddy current losses at HF

– Extensive uses

• HF and microwave application such as FM antennas, HF transformers and phase shifters

• Computer magnetic core and disk memory devices and MRAM

6-10. Boundary conditions

• The normal component of B

• The tangential component

1 2

 

1 2

1 1 2 2

T

For linear and isotropic media, and =

n n

n n

B B

H H

 

 

  

 

1 1 2 2

B

B H B H



 

 

   

1 2

1 2

0

2 1 2

w lim

surface current density normal to the contour A/m

C S

abcda i

t t sn

h

sn

n s

dl d

H dl J h

H H J h J

J abcda

 

   

      

   

  

 



H J H J s

H w H w

a H H J

 

  



2 1_n 2_{n s} (C/m )

D D

 

   D  

1 2

0 E _t E _t

  E 

6-11. Inductances and Inductors

• Mutual flux:

 

2

12 1 2

1 1 12 1

S d

Wb

I I

 

   



B



2 2

 

12 2 12 12 12 1

In case has turns, the flux linkage due to is Wb

C N

N L I

 

     

Mutual inductance between two circuits:

 

2

12 2

12 1 2

1 1

H; henry

S

L N d

I I

    ^{B  s}

Self-inductance of loop C₁ :

 

1

11 1

11 1 1

1 1

S

H

L N d

I I

    ^{B  s}

Inductor: a conductor arranged in an appropriate shape (such as a conducting wire wound as a coil) to supply a certain amount of self-inductance