Electromagnetic Theory: Magnetic Materials and Electric Dipoles, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY

MAGNETIC MATERIALS

METU - NCC

1

Assist. Prof. Dr.

Özlem Özgün Office^

:^ S- Phone^

:^ 661 2972 E-mail^

:^ ozozgun@metu.edu.tr Web^

:^ http://www.metu.edu.tr/~ozozgun/

FIRST,REMEMBER ELECTRIC FIELDS IN MATERIAL MEDIA!

2

FIELDS IN MATERIAL MEDIA!

Remember Electric Dipole in Electrostatics

p^

qd =

+ d^ -

q + 4 2

2 0

ˆ 0 cos 4

4

R

qd^

p a

V^

R^

R θ πε

⋅ πε

=^

ˆ^30

ˆ 2 cos^

sin

4

R p E^

a^

a R

θθ^ θ

=^ πε

p^ electric dipolemoment vector(Coul-m)

qd =

+ d -^ p

q +

q −

Remember Electric Dipole in Electrostatics Question: How do we find the effect (e.g., E-field, potential)created by a polarized dielectric?

5

2

: dipole moment vector (coul-m) p : polarization vector (coul/m ) P Polarization Vector:^ at point M

pi i lim 0 v

P^

v

volume v^ ∆∆∆∆v^ M

Remember Electric Flux Density and Dielectric Constant

0

D^

E^ P ε= +

For an arbitrary dielectric:For a linear dielectric:

0 e

P^

E

ε χ=

(^0) e χ^ ≥^ Electric susceptibility(dimensionless) 7

(^

0

0

0

(^10) e^

e

D^

E^ P

E

E^

E

ε

ε^

ε χ^

ε^

χ

=^

+^ =

=^

For a linear dielectric:^ : relative permittivity (dielectric consta

nt)

ε^ r

1 ε ≥ r

: absolute permittivity (^0) r ε^ ε ε=

D^

E ε=

(^

P^

E ε^ ε= −

MAGNETIC MATERIALS

8

Magnetic Properties of Materials

Magneticdipole 10

Before

B^ is applied (random orientation)

After^ B

is applied (aligned along

B ) (magnetized)

Magnetic Dipole

(^ ,^ ,) P R^ θ φ

0 =^ ∫ 4^ c

ldI R A

μ^ π

The vector magnetic potential isevaluated for

R >> a as

(^2) I a μ^ π

R

11

2 (^0) ˆ 2

sin I^ a 4 A^ a

R

μ^ φ π^

θ π = The magnetic flux density isevaluated for

R >> a as

(^

2 0 3

ˆ^

2cos

sin

R I^ a

B^

A^

a^

a

R

θ

μ^ π

θ

θ

π = ∇×

=^

Analogy btw Electric and Magnetic Dipoles^ Electric dipole

Magnetic dipole E E^13

B

Magnetization Vector

at point^

lim^0

i i P^

v

m

M^

v =^ ∆ →

∑^ ∆

Vector sum of atomic dipolemoments per unit volume aroundpoint P, as

∆∆ v

^0 ∆∆

2

: dipole moment vector (A.m ) m^ : magnetization vector (A/m) M

14

V

P Magnetized material

∆∆∆∆ V

: magnetization vector (A/m) M Volume density ofmagnetic dipole moment

Generalized Ampere’s Law in magnetized material

16

Volume current densitydue to magnetization

Free volume current density(externally-put)

Generalized Ampere’s Law in magnetized material

17

Permeability concept

19

A measure of how susceptible (or sensitive) the material is toa magnetic field.

Permeability concept

20

Permeability is a material property that describes the ease withwhich a magnetic flux is established in a component