ELEG 648 Summer 2012 Lecture #1 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Tel: (302)

Vector Analysis Review: = unit vector 1. Dot Product (projection) 2. Cross Product

Orthogonal Coordinate Systems:

dl dl 1 dl 2 dl 3

Cartesian Coordinate Systems: x y z

Cartesian Coordinate Systems (cont):

Cylindrical Coordinate Systems: x y z  r z (r,  z)

Spherical Coordinate Systems: x y z  R (R  ) 

Vector Coordinate Transformation:

Gradient of a Scalar Field: Assume f(x,y,z) is a scalar field The maximum spatial rate of change of f at some location is a vector given by the gradient of f denoted by Grad(f) or

Divergence of a Vector Field: Assume E(x,y,z) is a vector field. The divergence of E is defined as the net outward flux of E in some volume as the volume goes to zero. It is denoted by

Curl of a Vector Field: Assume E(x,y,z) is a vector field. The curl of E is measure of the circulation of E also called a “vortex” source. It is denoted by

Laplacian of a Scalar Field: Assume f(x,y,z) is a scalar field. The Laplacian is defined asand denoted by

Examples: 1. Given the scalar function Find the magnitude and direction of the maximum rate of chance at location (xo,yo,zo) 2. Determine 3. Determine 3. The magnetic field produced by a long wire conducting a constant current Is given by Find

Basic Theorems: 1. Divergence Theorem or Gauss’s Law 2. Stokes Theorem

Examples: 1. Verify the Divergence Theorem for on a cylindrical region enclosed by r=5, z=0 and z=4 r = 5 z = 0 z = 4

Odds and Ends: 1. Normal component of field 2. Tangential component of field

Maxwell’s Equations in Differential Form Faraday’s Law Ampere’s Law Gauss’s Law Gauss’s Magnetic Law

Faraday’s Law S C

Ampere’s Law

Gauss’s Law

Gauss’s Magnetic Law “all the flow of B entering the volume V must leave the volume”

CONSTITUTIVE RELATIONS  r  o = permittivity (F/m)  o = x (F/m)  r  o = permeability (H/m)  o = 4  x (H/m)  = conductivity (S/m)

POWER and ENERGY JiJi  E, H V S n take Using the vector identity Integrate eq4 over the volume V in the figure Applying the divergence theorem

POWER and ENERGY (continued) Stored magnetic power (W) Stored electric power (W) Supplied power (W) Dissipated power (W) What is this term?

POWER and ENERGY (continued) Stored magnetic power (W) Stored electric power (W) Supplied power (W) Dissipated power (W) What is this term? P s = power exiting the volume through radiation W/m 2 Poynting vector

TIME HARMONIC EM FIELDS Assume all sources have a sinusoidal time dependence and all materials properties are linear. Since Maxwell’s equations are linear all electric and magnetic fields must also have the same sinusoidal time dependence. They can be written for the electric field as: is a complex function of space (phasor) called the time-harmonic electric field. All field values and sources can be represented by their time-harmonic form. Euler’s Formula

PROPERTIES OF TIME HARMONIC FIELDS Time derivative: Time integration:

TIME HARMONIC MAXWELL’S EQUATIONS Employing the derivative property results in the following set of equations:

TIME HARMONIC EM FIELDS BOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES The constitutive properties and boundary conditions are very similar for the time harmonic form: Constitutive Properties General Boundary Conditions PEC Boundary Conditions

TIME HARMONIC EM FIELDS IMPEDANCE BOUNDARY CONDITIONS If one of the material at an interface is a good conductor but of finite conductivity it is useful to define an impedance boundary condition:  1,      2,      1 >>  2

POWER and ENERGY: TIME HARMONIC Time average magneticenergy (J) Time average electric energy (J) Supplied complex power (W) Dissipated real power (W) Time average exiting power

CONTINUITY OF CURRENT LAW vector identity time harmonic

SUMMARY Frequency Domain Time Domain

Electromagnetic Properties of Materials Primary Material Properties  r  o = permeability (H/m)  o = 4  x (H/m)  r  o = permittivity (F/m)  o = x (F/m)  = conductivity (S/m) Electrical Properties Magnetic Properties Secondary Material Properties Electrical Properties Index of refraction Electric susceptibility Magnetic Properties Magnetic susceptibility

Electric Properties of Materials + E ext - + lili QiQi No external field Applied external field Bulk material (N molecules) Electric dipole moment of individual atom or molecule: Net dipole moment or polarization vector: E ext

Electric Properties of Materials (continued) Bulk material (N molecules) E ext What are the assumptions here? Static permittivity or relative permittivity

Electric Properties of Materials (continued) Conductivity x y zE E J J=current density=q v v z where q v =volume charge density and v z = charge drift velocity When subjected to an external electric field E the charge velocity is increased and is given by Where  e is called the electron mobility. The current density is thus given by Where  is called the conductivity. Its units are S/m Where  is called the resistivity. Material Conductivity (S/m) Silver 6.1 x 10 7 Glass 1.0 x Sea Water 4

Electric Properties of Materials (continued) 1.Orientational Polarization: molecules have a slight polarization even in the absence of an applied field. However each polarization vector is orientated randomly so the net P vector is zero. Such materials are known as polar; water is a good example. 2.Ionic Polarization: Evident in materials ionic materials such as NaCl. Positive and negative ions tend to align with the applied field. 3.Electronic Polarization: Evident in most materials and exists when an applied field displaces the electron cloud of an atom relative to the positive nucleus.

Magnetic Properties of Materials No magnetic field: random oriented magnetic dipoles Net magnetic dipole moment or magnetization vector: IiIi MiMi IiIi MiMi IiIi MiMi IiIi MiMi Applied external magnetic field B ext IiIi MiMi IiIi MiMi IiIi MiMi IiIi MiMi Magnetic dipoles randomly oriented resulting in zero net magnetization vector: Magnetic dipoles tend to align with external magnetic field resulting in non-zero net magnetization vector:

Magnetic Properties of Materials (continued) What are the assumptions here? Static permeability or relative permeability Applied external magnetic field B ext IiIi MiMi IiIi MiMi IiIi MiMi IiIi MiMi

Magnetic Properties of Materials (continued) 1.Diamagnetic: Net magnetization vector tends to appose the direction of the applied field resulting in a relative permeability slightly less than 1.0 Examples: silver (  r =0.9998) 2.Paramagnetic: Net magnetization vector tends to align in the direction of the applied field resulting in a relative permeability slightly greater than 1.0 Examples: Aluminum (  r = ) 3Ferromagnetic: Net magnetization vector tends to align strongly in the direction of the applied field resulting in a relative permeability much greater than 1.0 Examples: Iron (  r =5000).

Classification of Materials 1.Homogenous or Inhomogenous: If the material properties are independent of spatial location then the material is homogenous, otherwise it is called inhomogenous 2.Isotropic or Anisotropic: If the material properties are independent of the polarization of the applied field then the material is isotropic, otherwise it is called anisotropic. 3.Linear or non-Linear: If the material properties are independent on the magnitude and phase of the electric and magnetic fields, otherwise it is called non-linear Inhomogenous anisotropic

Classification of Materials t=t 1 t=t 2 t=t 3 t=t 4 t=t 5 t=t 6 A material’s atoms or molecules attempt to keep up with a changing electric field. This results in two things: (1) friction causes energy loss via heat and (2) the dynamic response of the molecules will be a function of the frequency of the applied field (i.e. frequency dependant material properties) 4.Dispersive or non-dispersive: If the material properties are independent of frequency then the material is non-dispersive, otherwise it is called dispersive

Electric Properties of Materials Frequency Behavior (Complex Permittivity) is called the complex permittivity loss term Dielectric constant

Frequency Behavior of Sea Water

Electric Properties of Materials Frequency Behavior (Complex Permittivity) Displacement current Source current Effective electric conduction current

Electric Properties of Materials Frequency Behavior (Complex Permittivity)

Displacement current Source current Effective electric conduction current Good Dielectric Good Conductor

Wave Equation Vector Identity Time Dependent Homogenous Wave Equation (E-Field)

Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field) Source Free Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field) Lossless

Wave Equation Source-Free Time Dependent Homogenous Wave Equation (H-Field) Source Free Source Free and Lossless

Wave Equation: Time Harmonic Source Free Lossless Time DomainFrequency Domain Source Free Lossless “Helmholtz Equation”

General Solution: Point Source x y z  r (r  )  point source Solution: Inward traveling spherical waveOutward traveling spherical wave 0

General Solution: Point Source y z  r (r  )  point source Wave speed In free space: Same as the speed of light!

General Solution Case: Time Harmonic Rectangular Coordinates Wave Number

Separation of Variable Solutions Assume Solution of the form: function of x function of y function of z constant

Separation of Variable Solutions function of x function of y function of z constant

Separation of Variable Solutions Solutions:  purely real  purely imaginary  complex Traveling and standing waves Evanesent waves Exponentially modulated traveling wave or

Separation of Variable Solutions: Examples case a.  x,  y,  z all real (forward traveling waves) Plane Waves case b.  x real (forward traveling wave),  x (real standing wave),  z imaginary (evanesent wave) Surface Waves z x y

Separation of Variable Solutions: Examples case c.  x real (forward standing wave),  x (real standing wave),  z real (traveling wave) Guided Waves Unknown constants A 1, B 1, C 1, D 1,  x,  y,  z Found by applying boundary conditions and dispersion relation. Namely: z x y b h PEC Walls  Stay tuned we will solve the complete solution for modes in a rectangular waveguide in a later lecture.