Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India

Program for Hyderabad Area and Andhra Pradesh Faculty Sponsored by IEEE Hyderabad Section, IETE Hyderabad Center, and Vasavi College of Engineering IETE Conference Hall, Osmania University Campus Hyderabad, Andhra Pradesh June 3 – June 11, 2009 Workshop for Master Trainer Faculty Sponsored by IUCEE (Indo-US Coalition for Engineering Education) Infosys Campus, Mysore, Karnataka June 22 – July 3, 2009

5-2 Module 5 Materials and Wave Propagation in Material Media 5.1 Conductors and dielectrics 5.2 Magnetic materials 5.3 Wave equation and solution 5.4 Uniform waves in dielectrics and conductors 5.5 Boundary conditions 5.6. Reflection and transmission of uniform plane waves

5-3 Instructional Objectives 31. Find the charge densities on the surfaces of infinite plane conducting slabs (with zero or nonzero net surface charge densities) placed parallel to infinite plane sheets of charge 32. Find the displacement flux density, electric field intensity, and the polarization vector in a dielectric material in the presence of a specified charge distribution, for simple cases involving symmetry 33. Find the magnetic field intensity, magnetic flux density, and the magnetization vector in a magnetic material in the presence of a specified current distribution, for simple cases involving symmetry

5-4 Instructional Objectives (Continued) 34. Determine if the polarization of a specified electric/magnetic field in an anisotropic dielectric/magnetic material of permittivity/permeability matrix represents a characteristic polarization corresponding to the material 35. Write expressions for the electric and magnetic fields of a uniform plane wave propagating away from an infinite plane sheet of a specified sinusoidal current density, in a material medium 36. Find the material parameters from the propagation parameters of a sinusoidal uniform plane wave in a material medium 37. Find the power flow, power dissipation, and the electric and magnetic stored energies associated with electric and magnetic fields in a material medium

5-5 Instructional Objectives (Continued) 38. Determine whether a lossy material with a given set of material parameters is an imperfect dielectric or good conductor for a specified frequency 39. Find the charge and current densities on a perfect conductor surface by applying the boundary conditions for the electric and magnetic fields on the surface 40. Find the electric and magnetic fields at points on one side of a dielectric-dielectric interface, given the electric and magnetic fields at points on the other side of the interface 41. Find the reflected and transmitted wave fields for a given field of a uniform plane wave incident normally on a plane interface between two material media

Conductors and Dielectrics (EEE, Secs. 4.1, 4.2; FEME, Sec. 5.1)

5-7 Materials Materials contain charged particles that under the application of external fields respond giving rise to three basic phenomena known as conduction, polarization, and magnetization. While these phenomena occur on the atomic or “microscopic” scale, it is sufficient for our purpose to characterize the material based on “macroscopic” scale observations, that is, observations averaged over volumes large compared with atomic dimensions. 8

5-8 Material Media can be classified as (1)Conductors and Semiconductors (2)Dielectrics (3)Magnetic materials – magnetic property Conductors and Semiconductors Conductors are based upon the property of conduction, the phenomenon of drift of free electrons in the material with an average drift velocity proportional to the applied electric field. electric property

5-9 In semiconductors, conduction occurs not only by electrons but also by holes – vacancies created by detachment of electrons due to breaking of covalent bonds with other atoms. The conduction current density is given by Ohm’s Law at a point

5-10 conductors semiconductors The effect of conduction is taken into account explicitly by using J = J c on the right side of Maxwell’s curl equation for H.

5-11 Ohm’s Law  Resistance

5-12 D4.1 (a)For cu, (b)

5-13 (c)From

5-14 Conductor in a static electric field

5-15  S = –  0 E 0  S =  0 E 0  S = –  0 E 0  S =  0 E 0 S0S0 –S0–S0 E = – S00S00 azaz Plane conducting slab in a uniform electric field  S =  0 E 0

5-16 P4.3 (a)

5-17 (b) Write two more equations and solve for the four unknowns. (1) (2)

5-18 Solving the four equations, we obtain

5-19 Dielectrics are based upon the property of polarization, which is the phenomenon of the creation of electric dipoles within the material. Electronic polarization: (bound electrons are displaced to form a dipole) Dipole moment p = Qd

5-20 Orientational polarization: (Already existing dipoles are acted upon by a torque) Ionic polarization: (separation of positive and negative ions in molecules) Direction into the paper.

The phenomenon of polarization results in a polarization charge in the material which produces a secondary E.

5-22 Plane dielectric slab in a uniform electric field

5-23 Polarization Current (in the time-varying case)

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5-27 The effect of polarization needs to be taken into account by adding the contributions from the polarization charges and the polarization current to the right sides of Maxwell’s equations. For free space,

5-28 For dielectrics, Where P is the polarization vector, or the dipole moment per unit volume. Rearranging, where now,

5-29 Thus, to take into account the effect of polarization, we define the displacement flux density vector, D, as vary with the material, implicitly taking into account the effect of polarization.

5-30 As an example, consider Then, inside the material,

5-31 D4.3 For 0 < z < d, (a)

5-32 (c) (b)

5-33 Isotropic Dielectrics: Anisotropic Dielectrics: D is not parallel to E in general. Only for certain directions (or polarizations) of E is D parallel to E. These are known as characteristic polarizations. D is parallel to E for all E.

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5-35 D4.4 (a)

5-36 (b)

5-37 (c)

5-38 Review Questions 5.1. Distinguish between bound electrons and free electrons in an atom Briefly describe the phenomenon of conduction State Ohm’ law valid at a point, defining conductivity. How is conduction current taken into account in Maxwell’s equations? 5.4. Discuss the formation of surface charge at the boundaries of a conductor placed in a static electric field Briefly describe the phenomenon of polarization in a dielectric material. What are the different kinds of polarization? 5.6. What is an electric dipole? How is its strength defined?

5-39 Review Questions (Continued) 5.7. Discuss the effect of polarization in a dielectric material, involving polarization charge and polarization current What is the polarization vector? How is it related to the electric field intensity? 5.9. Discuss how the effect of polarization in a dielectric material is taken into account in Maxwell’s equations Discuss the revised definition of the displacement flux density and the permittivity concept What is an anisotropic dielectric material? When can an effective permittivity be defined for an anisotropic dielectric material?

5-40 Problem S5.1. Finding the electric field due to a point charge in the presence of a conductor

5-41 Problem S5.1. Finding the electric field due to a point charge in the presence of a conductor (Continued)

5-42 Problem S5.2. Finding D, E, and P for a line charge surrounded by a cylindrical shell of dielectric material

5-43 Problem S5.3. Expressing an electric field in terms of the characteristic polarizations of an anisotropic dielectric

5.2 Magnetic Materials (EEE, Sec. 4.3; FEME, Sec. 5.2)

5-45 Magnetic Materials are based upon the property of magnetization, which is the phenomenon of creation of magnetic dipoles within the material. Diamagnetism: A net dipole moment is induced by changing the angular velocities of the electronic orbits. Dipole moment m = IA a n

5-46 Paramagnetism: Already existing dipoles are acted upon by a torque. Other: Ferromagnetism, antiferromagnetism, ferrimagnetism

5-47 The phenomenon of magnetization results in a magnetization current in the material which produces a secondary B.

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5-49 Magnetization Current

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5-52 The effect of magnetization needs to be taken into account by adding the contributions from the magnetization current to the right sides of Maxwell’s equations. For nonmagnetic materials,

5-53 For magnetic materials, where M is the magnetization vector, or the magnetic dipole moment per unit volume.

5-54 Rearranging, where now

5-55 Thus, to take into account the effect of magnetization, we define the magnetic field intensity vector, H, as  r and  vary with the material, implicitly taking into account the effect of magnetization.

5-56 As an example, consider Then inside the material,

5-57 D4.6 For 0 < z < d, (a)

5-58 (c) (b)

5-59 Materials and Constitutive Relations Conductors Dielectrics Magnetic materials  E and B are the fundamental field vectors. Summarizing, D and H are mixed vectors taking into account the dielectric and magnetic properties of the material implicity through  and , respectively.

5-60 Review Questions Briefly describe the phenomenon of magnetization in a material. What are the different kinds magnetization? What is a magnetic dipole? How is its strength defined? Discuss the effect of magnetization in a magnetic material, involving magnetization current What is the magnetization vector? How is it related to the magnetic flux density? Discuss how the effect of magnetization in a magnetic material is taken into account in Maxwell’s equations Discuss the revised definition of the magnetic field intensity and the permeability concept Summarize the constitutive relations for a material medium.

5-61 Problem S5.4. Finding H, B, and M for a wire of current surrounded by a cylindrical shell of magnetic material

5.3 Wave Equation and Solution (EEE, Sec. 4.4; FEME, Sec. 5.3)

5-63 Maxwell’s equations for a material medium

5-64 Waves in Material Media

5-65 Combining, we get Define Then Wave equation

5-66 Solution:

5-67  = attenuation constant, Np/m  = phase constant, rad/m attenuation

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5-71 Summarizing, Conversely,

5-72 Characteristics of Wave Propagation The quantity  characterizes attenuation, which is a function of frequency. The quantity v p characterizes phase velocity, which is a function of frequency, giving rise to dispersion. The complex nature of results in phase difference between the electric and magnetic fields.

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5-74 Three-dimensional depiction of wave propagation

5-75 E5.1 Solution:

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5-79 Power Flow and Energy Storage Poynting Vector A Vector Identity For

5-80 Poynting’s Theorem for Material Medium Power dissipation density Electric stored energy density Magnetic stored energy density Source power density

5-81 Interpretation of Poynting’s Theorem Poynting’s Theorem for the material medium says that the power delivered to the volume V by the current source J 0 is accounted for by the power dissipated in the volume due to the conduction current in the medium, plus the time rates of increase of the energies stored in the electric and magnetic fields, plus another term, which we must interpret as the power carried by the electromagnetic field out of the volume V, for conservation of energy to be satisfied.

5-82 In the case of the infinite plane sheet of current, work is done by an external agent (source) for the current to flow, and represents the power density (per unit volume) associated with this work.

5-83 Review Questions Summarize Maxwell’s equations for a material medium What is the propagation constant in a material medium? Discuss the significance of its real and imaginary parts What is the intrinsic impedance of a material medium? Discuss the significance of its complex nature Discuss the consequence of the frequency dependence of the phase velocity in a material medium Discuss the solution for the electromagnetic field due to to an infinite plane current sheet of sinusoidally time- varying current density embedded in a material medium How would you obtain the electromagnetic field due to a current sheet of nonsinusoidally time-varying current density embedded in a material medium?

5-84 Review Questions (Continued) State and discuss Poynting’s theorem for a material medium What are the power dissipation density, the electric stored energy density, and the magnetic stored energy density associated with an electromagnetic field in a material medium?

5-85 Problem S5.5. Finding the magnetic field and material parameters from a specified electric field

5-86 Problem S5.6. Finding E and H for a current sheet of nonsinusoidal current density in a material medium

5-87 Problem S5.7. For showing that the time average powers delivered to, and dissipated in, a medium are the same

5.4 Uniform Plane Waves in Dielectrics and Conductors (EEE, Sec. 4.5; FEME, Sec. 5.4)

5-89 Special Cases: Case 1. Perfect dielectric Behavior same as in free space except that  0   and  0  .

5-90 For materials with nonzero conductivity, there are two special cases, depending on the relationship between and . The quantity /  is known as the loss tangent. From it can be seen that /  is the ratio of the amplitudes of the conduction current density and the displacement current density in the material.

5-91 Case 2. Imperfect Dielectric Behavior essentially like in a perfect dielectric except for attenuation.

5-92 Case 3. Good Conductor Behavior much different from that in a dielectric.

5-93 Characteristics of wave propagation in a good conductor Skin effect: Concentration of fields near the skin of the conductor. Skin depth,  : Distance in which fields are attenuated by the factor e  1, that is,  = 1. Ex: For copper,

5-94 Even at a low frequency of 1 MHz,  = mm. This explains the phenomenon of shielding by good conductors, such as copper, aluminum, etc.

5-95. Furthermore, the lower the frequency, the smaller is the ratio. Coupled with the fact that this property makes low frequencies more suitable for communication with underwater objects. Ex: For sea water, The value of the ratio of the two wavelengths for f = 25 kHz is 1/134.

5-96 For copper, Even at a frequency of Hz, this is equal to , a very low value. In fact, if we note that we see that Hence, for the same electric field, the magnetic field inside a good conductor is much larger than that inside a dielectric.

5-97 Case 4. Perfect Conductor  No waves can penetrate into a perfect conductor. No time-varying fields inside a perfect conductor.

5-98 Review Questions What is loss tangent? Discuss its significance What is the condition for a material to be a perfect dielectric? How do the characteristics of wave propagation in a perfect dielectric medium differ from those of wave propagation in free space? What is the condition for a material to be an imperfect dielectric? What is the significant feature of wave propagation in an imperfect dielectric as compared to that in a perfect dielectric? What is the condition for a material to be a good conductor? Give two examples of materials that behave as good conductors for frequencies up to several gigahertz What is skin effect? Discuss skin depth, giving some numerical values.

5-99 Review Questions (Continued) Why are low-frequency waves more suitable than high- frequency waves for communication with underwater objects? What is the consequence of the low intrinsic impedance of a good conductor as compared to that of a dielectric medium having the same ε and μ Why can there be no time-varying fields inside a perfect conductor?

5-100 Problem S5.8. Plotting field variations for an infinite plane sheet current source in a perfect dielectric medium

5-101 Problem S5.9. Calculating parameters for good conductor materials to satisfy specified conditions

5.5 Boundary Conditions (EEE, Sec. 4.6; FEME, Sec. 5.5)

5-103 Why boundary conditions? Medium 1 Medium 2 Inc. wave Trans. wave Ref. wave

5-104 Maxwell’s equations in integral form must be satisfied regardless of where the contours, surfaces, and volumes are. Example: Medium 1Medium 2 C2C2 C1C1 C3C3

5-105 Boundary Conditions                Medium 1,z  0 Medium 2,z  0 z  0 x y z D n1 a n J n1 J n2 B n2 D n2 E t1 E t2 B n1 H t1 H t2 J S

5-106 Example of derivation of boundary conditions Medium 1 Medium 2

5-107 or,

5-108 Summary of boundary conditions

5-109 Perfect Conductor Surface (No time-varying fields inside a perfect conductor. Also no static electric field; may be a static magnetic field.) Assuming both E and H to be zero inside, on the surface,

5-110

5-111 Dielectric-Dielectric Interface

5-112

5-113 D4.11At a point on a perfect conductor surface, (a) and pointing away from the surface. Find. D 0 is positive.

5-114 and pointing toward the surface. D 0 is positive. (b)

5-115 E5.2 (a)

5-116 (b) (c)

5-117 Review Questions What is a boundary condition? How do boundary conditions arise and how are they derived? Summarize the boundary conditions for the general case of a boundary between two arbitrary media, indicating correspondingly the Maxwell’s equations in integral form from which they are derived Discuss the boundary conditions on the surface of a perfect conductor Discuss the boundary conditions at the interface between two perfect dielectric media.

5-118 Problem S5.10. For application of boundary conditions on a perfect conductor surface

5-119 Problem S5.11. Applying boundary conditions at a dielectric interface in the presence of a point charge

5-120 Problem S5.11. Applying boundary conditions at a dielectric interface in the presence of a point charge (Continued)

5-121 Problem S5.12. For application of boundary conditions on a the surface of a magnetic material

Reflection and Transmission of Uniform Plane Waves (EEE, Sec. 4.7; FEME, Sec. 5.6)

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5-129 Review Questions Discuss the determination of the reflected and transmitted wave fields of a uniform plane wave incident normally onto a plane boundary between two material media Define reflection and transmission coefficients for a uniform plane wave incident normally onto a plane boundary between two material media Discuss the reflection and transmission coefficients for the special case of two perfect dielectric media What is the consequence of a wave incident on a perfect conductor?

5-130 Problem S5.13. Normal incidence of uniform plane waves on interface between free space and water

5-131 Problem S5.14. Eliminating reflection of uniform plane waves from a dielectric slab between two media

5-132 Problem S5.14. Eliminating reflection of uniform plane waves from a dielectric slab between two media (Continued)

The End