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Constitutive Relations 1/14/2021 Electromagnetics: Electromagnetic Field Theory Constitutive Relations Outline • Electric Response of Materials • Magnetic Response of Materials • Classification of Materials Slide 2 1 1/14/2021 Electric Response of Materials D E P 0 Vacuum response Material response Note: is the free space permittivity and multiples so that has the same units as . 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 Slide 3 Electric Polarization In general, the relation between the applied electric field and the electric polarization 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 nn11 e mV Slide 4 2 1/14/2021 Relation Between Permittivity & Susceptibility The permittivity is related to the electric susceptibility 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 re1 Vacuum response Dielectric response Slide 5 Magnetic Response of Materials B H M 0 Vacuum response Material response Note: 0 is the free space permeability and multiples so that has the same units as . 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 Slide 6 3 1/14/2021 Magnetic Polarization In general, the relation between the applied magnetic field and the magnetic polarization 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 nn11 m mA Slide 7 Relation Between Permeability & Susceptibility The permeability is related to the magnetic susceptibility through 11 B 0rHH 0 0m H 01 m H 1 rm1 Vacuum response Magnetic response Slide 8 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. Slide 9 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 Slide 10 5 1/14/2021 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 Slide 11 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 Slide 12 6 1/14/2021 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 Slide 13 7.
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