1/14/2021

Electromagnetics: Electromagnetic Field Theory Constitutive Relations

Outline

• Electric Response of Materials • Magnetic Response of Materials • Classification of Materials

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Electric Response of Materials    D  E P 0  Vacuum response Material response

Note: 𝜀 is the free space permittivity and multiples 𝐸 so that 𝜀𝐸 has the same units as 𝑃.

Electric Susceptibility, e  The electric susceptibility e is a measure of how easily bound charges are displaced due to an applied electric field. PE 0e

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Electric Polarization 𝑃

In general, the relation between the applied electric field 𝐸 and the electric polarization 𝑃 is nonlinear so it can be expressed as a polynomial.

n 1 2323  e  electric susceptibility P  0eE 0eEE  0e 

Linear Nonlinear response 1 response e no units These terms are usually ignored. They tend to only become significant when  2 mV the electric field is very strong. e 2 3 22 e is pronounced "chi two" e mV 3 e is pronounced "chi three"   n nn11 e mV

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Relation Between Permittivity & Susceptibility

The permittivity 𝜀 is related to the electric susceptibility 𝜒 through   E 1 E 1 1 E DEP 0 00e 0e The constitutive relation can also be written in terms of the relative permittivity  . r DE   0r 1  re1 

Vacuum response Dielectric response

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Magnetic Response of Materials    B  H M 0  Vacuum response Material response

Note: 0 is the free space permeability and multiples 𝐻 so that 𝜇𝐻 has the same units as 𝑀.

Magnetic Susceptibility, m

The magnetic susceptibility m is a measure of how easily magnetic dipoles are aligned due to an applied magnetic field.  M  0mH

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Magnetic Polarization 𝑀

In general, the relation between the applied magnetic field 𝐻 and the magnetic polarization 𝑀 is nonlinear so it can be expressed as a polynomial.

1 2323  M   H  HH    n  magnetic susceptibility 0m  0m 0m  m Nonlinear response Linear 1 response  no units These terms are usually ignored. They m tend to only become significant when the 2 mA magnetic field is very strong. m  2 is pronounced "chi two" 3 22 m m mA 3 m is pronounced "chi three"   n nn11 m mA

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Relation Between Permeability & Susceptibility

The permeability 𝜇 is related to the magnetic susceptibility 𝜒 through   11  B 0rHH  0  0m H  01  m H

1 rm1 

Vacuum response Magnetic response

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4 1/14/2021

Types of Magnetic Materials

• Diamagnetic

• Negative magnetic susceptibility (m < 0) • Tends to oppose an applied magnetic field. • All materials are diamagnetic, but usually very week. • Copper, silver, gold • Paramagnetic

• Small positive susceptibility (m > 0 but small) • Material is magnetizable and is attracted to an applied magnetic field. • Does not retain magnetization when the external field is removed. • Ferromagnetic • Large positive susceptibility • Like paramagnetic, but they retain their magnetism to some degree when the external field is removed. • Iron, nickel, cobalt, and some aloys.

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Anisotropy

The dielectric response of a material arises due to the electric field displacing charges. Due to structural and bonding effects at the atomic scale, charges are often more easily displaced in some directions than others. This gives rise to anisotropy where the electric field may experience an entirely different permittivity depending on what direction it is oriented.

D E  x xx xy xz  x   ij = how much of Ej DE   Dy  yx yy yzE y   D  contributes to i Dz zx zy zz E z tensor  

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Types of Anisotropy (1 of 2) Isotropic Media This is the typical approximation made in electromagnetics. The D  00E aa  permittivity tensor reduces to a scalar.   Dbb 00 E    D   E Dcc00 E Uniaxial Media Anisotropic materials are said to be birefringent. Daa o 00 E       o  ordinary permittivity D  00 E eo bbo  Positive birefringence:  0  e  extraordinary permittivity Dcc00 e E Negative birefringence:  0 Biaxial Media When orientation is not important, it is convention to order the tensor Daa 00 E a elements according to D  00 E bbb   abc Dccc00 E

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Types of Anisotropy (2 of 2)

Doubly Anisotropic    D  EBH and  

Chiral Materials

Daa jE b0  a D  jE 0  bba  b Gyroelectric Dccc00 E

BjHaa b0  a Bj  0  H bba  bGryomagnetic BHccc00 

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Ordinary and Bi‐ Materials

Ordinary Materials Bi‐ Materials Isotropic Materials Bi‐Isotropic Materials   D   E D  EH Materials   B  H B  EH Isotropic   magnetoelectric coupling coefficient Anisotropic Materials Bi‐Anisotropic Materials   D   E Materials D EH

     B   H B T EH  Anisotropic

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