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ECE 307: Electricity and Magnetism Fall 2012 Instructor: J.D. Williams, Assistant Professor Electrical and Computer Engineering University of Alabama in Huntsville 406 Optics Building, Huntsville, Al 35899 Phone: (256) 824-2898, email: [email protected] Course material posted on UAH Angel course management website Textbook: M.N.O. Sadiku, Elements of Electromagnetics 5th ed. Oxford University Press, 2009. Optional Reading: H.M. Shey, Div Grad Curl and all that: an informal text on vector calculus, 4th ed. Norton Press, 2005. All figures taken from primary textbook unless otherwise cited. Chapter 5: Electric Fields in Material Space • Topics Covered – – – – – – – – • Properties of Materials Convection and Conduction Currents Conductors Polarization in Dielectrics Dielectric Constant and Strength Linear, Isotropic, and Homogeneous Dielectrics Continuity Equation and Relaxation Time Boundary Conditions Homework: 2, 11,13, 23, 26, 38, 39, 40, 42 8/17/2012 2 All figures taken from primary textbook unless otherwise cited. Convection and Conduction Currents • Current (in amperes) through a given area is the electric charge passing through the area per unit time Current • dQ dt Current density is the amount of current flowing through a surface, A/m2, or the current through a unit normal area at that point Current density • I I J S where I J dS s Depending on how the current is produced, there are different types of current density – – – Convection current density Conduction current density Displacement current density (Chapter 9) • Current generated by a magnetic field 8/17/2012 3 Convection Current Density • Convection current density – Does not involve conductors and does not obey Ohm’s law – Occurs when current flows through an insulating medium such as liquid, gas, or vacuum I Q y v S v Su y t t Where u is the velocity vector of the fluid I Jy vu y S J vu 8/17/2012 4 Conduction Current Density • • Conduction current density – Current in a conductor – Obeys Ohm’s law Consider a large number of free electrons traveling in a metal with mass (m), velocity (u), and scattering time (time betweenelectron collisions), . mu F qE • The carrier density is determined by the number of electrons, n, with charge, e v ne • Conduction current density can then be calculate as ne 2 J vu E E m • • Where is the conductivity of the conductor This relationship between current concentration and electric field is known as Ohm’s Law 8/17/2012 5 Conductors • • • Conductors are materials with an abundance of free moving charges Convention states that when an electric field is applied to a conductor, the positive free charges are pushed along the same direction as the applied field, while the negative charges move in the opposite direction The free charges do two things – They accumulate on the surface of the conductor to form an induced surface charge – The induced charges set up internal induced field Ei, which cancels the externally applied field inside the material • Shielding of a conductor by an induced field generates current within the material Good Conductor: Reduced electric field inside vs. that incident on the material. 8/17/2012 6 Conductors (2) • • Perfect conductor is a conductor in which no electrostatic field may enter, because the induced surface charges match the external field exactly eliminating all fields within the material Such conductors are called equipotential bodies, because the potential is the same everywhere within the conductor based on the fact that E = -Grad(V)=0 – In reality metals are very good conductors in which the electric field below the skin depth of the conductor is indeed zero. However the skin depth is a frequency dependent function that is usually observed only in high frequency applications. If indeed the skin depth is considered in a problem, then the electric field below the skin depth of carrier conduction within the material is zero, and current is generated only on the surface. Perfect Conductor: No electric field inside 8/17/2012 Skin Depth: • The depth beneath the surface of a conductor at which the current drops to e-1 below the current density on the surface. • This term is quite commonly used to determine the depth of high frequency electromagnetic waves incident on a surface or propagating along a metallic wire. 7 Electrical Resistively • • • • Consider a conductor whose ends are maintained at a potential difference ( i.e. the electric field within the conductor is nonzero and a field is passed through the material.) Note that there is no static equilibrium in this system. The conductor is being fed energy by the application of the electric field (bias potential) As electrons move within the material to set up induction fields, they scatter and are therefore damped. This damping is quantified as the resistance, R, of the material. For this example assume: – – a uniform cross sectional area S, and length l. The direction of the electric field, E, produced is the same as the direction of flow of positive charges (or the same as the current, I). E dl V R v I E dS s 8/17/2012 V l I V J E S l l V l R c I S S E 8 Electrical Power • Power is defined either as the rate change of energy (Joules) or force times velocity P v E u dv E Jdv Joule’s Law 2 dP wp EJ E dv Power density v • v For a conductor with uniform cross section P Edl JdS VI I 2 R L 8/17/2012 S 9 Polarization in Dielectrics • • • • • The main difference between a conductor and a dielectric is the availability of free electrons in the outermost atomic shells to conduct current Carriers in a dielectric are bound by finite forces and as such, electric displacement occurs when external forces are applied Such displacements are produced when an applied electric field, E, creates dipoles within the media that polarize it Polarized media are evaluated by summing the original charge distribution and the dipole moment induced One may also define the polarization, P, of the material as the dipole moment per unit volume n n P lim v 0 • k 1 v lim v 0 p k k 1 v See slides 39-40 for more on E fields, electrostatic potential, and dipoles Two types of dielectrics exist in nature: polar and nonpolar – – 8/17/2012 qk d k Nonpolar dielectrics do not posses dipole moments until a strong electric field is applied Polar dielectrics such as water, posses permanent dipole moments that further align (if possible) in the presence of an external field 10 Polarization in Dielectrics (2) • Potential due to a dipole moment p ar p (r r ' ) V 4 o R 2 4 o r r ' 3 p ar dv V 4 o R 2 v Where, 2 R r r ' ( x x' ) 2 ( y y ' ) 2 ( z z ' ) 2 ( x x ' ) a ( y y ' ) a ( z z ' ) a R ar 1 x y z ' 3 2 2 2 2 3/ 2 R R R ( x x' ) ( y y ' ) ( z z ' ) 2 Where the ’ operator is with respect to (x’,y’,z’) P 'P P ar 1 P ' ' R2 R R R P 'P 1 dv' V ' 4 o R R v P a 'n 'P V dS ' dv' 4 o R 4 o R v v ps P an pv P When polarization occurs, an equivalent volume charge density, pv, is formed throughout the dielectric, while an equivalent surface charge density, ps, is formed over the surface. 8/17/2012 11 Polarization in Dielectrics(3) • When polarization occurs, an equivalent volume charge density, pv, is formed throughout the dielectric, while an equivalent surface charge density, ps, is formed over the surface. ps P an pv P • For nonpolar dielectrics with no added free charge Qtotal psdS pvdv 0 S • v For cases in which the dielectric contains free charge density, v t v pv oE Hence v o E pv oE P ( o E P) D 8/17/2012 This redefines our Electric Displacement definition from chapter 4 on slide 24 to include polarized media. Our previous definition is the special case in which the polarization of the material is zero 12 The Dielectric Constant • • • It is important to note that up to this point, we have not committed ourselves to the cause of the polarization, P. We dealt only with its effects. We have stated that the polarization of a dielectric results from an electric field which lines up the atomic or molecular dipoles. In many substances, experimental evidence shows that the polarization is proportional to the electric field, provided that E is not too strong. These substances are said to have a linear, isotropic dielectric constant This proportionality constant is called the electric susceptibility, e. The convention is to extract the permittivity of free space from the electric susceptibility to make the units dimensionless. Thus we have P o e E • • • From the previous slide D o E P o (1 e ) E D o r E D E The dielectric constant (or relative permittivity) of the material, r, is the ratio of the permittivity to that of free space If the electric field is too strong, then it begins to strip electrons completely from molecules leading to short term conduction of electrons within the media. This is called dielectric breakdown. The maximum strength of the electric field that a dielectric can tolerate prior to which breakdown occurs is called the dielectric strength. 13 8/17/2012 Linear, Isotropic, and Homogeneous Dielectrics • • • • • • • In linear dielectrics, the permittivity, , does not change with applied field, E. Homogenous dielectrics do not change their permittivity from point to point within the material Isotropic dielectrics do not change their dielectric constant with respect to direction within the material Most commercial dielectrics are linear over some range, but may not be homogenous over large areas, and may not be isotropic. Inhomogeneity is most commonly due to local concentrations of one type of material verses another in an alloy, or simply from machine tolerance error on the thickness of a dielectric from point to point. These are commonly processing issues that need to be evaluated by the engineer when choosing the appropriate material and manufacturing process for the job. Isotropy is a material property. Many materials, such as single crystals, plasmas and magneto active materials possess anisotropic dielectric constants. These may be taken advantage of for specific engineering applications. For linear, homogeneous anisotropic materials: 8/17/2012 Dx xx D y yx Dz zx xy xz E x yy yz E y zy zz E z Note that these same concepts can be used to expand on anisotropic conduction and resistance properties as well 14 Nonpolar Molecules in a Poled Dielectric + - + - + - Field at the center of the cavity is Em=Ex+Ed+Es+E’ Ex is the primary field Ed is the depolarizing field due to polarization charge Es is the polarization charge on the cavity surface, S E’ is due to dipoles inside the cavity surface S s 1 P cos r da Em a z P 0 o o 4 o r 3 Molecular Polarizability, Pm=Em s o s o 2 1 P az P d cos ( cos ) sin d o 4 o 0 0 1 1 1 Take only the direction along P az P PE P o 3 o 3 o For N molecules per unit volume, the polarization is P=NPm 8/17/2012 1 P N E P 3 o P e o E e o E 1 N E e o E 3 o 3 e N 3 e e ( r 1) 3 1 o r N r 2 Clausius-Mossotti eqn. 15 Polarization Vector in a Coaxial Cable l = q E a +q a +q b b D oE P q 1 q q o ˆ ˆ ˆ P o 1 P D1 2 2 2 r One can show that the volume charge density in the dielectric is zero Assume that the there is a total charge Q distributed across the conduct length of the inner shell D dS D2L Qenc qdl qL L q D ˆ 2 E q ˆ 2 1 1 q o 1 0 v P P 2 Thus, the surface charge densities due to polarization of the dielectric must be equal and opposite at surfaces a and b q o q o s ,a P nˆ 1 ˆ ˆ 1 2 2 q o q o s ,b P nˆ 1 ˆ ˆ 1 2 2 8/17/2012 16 Problem from N. Ida, Engineering Electromagnetics, 2ed, Springer, 2003 Continuity Equation • • Remembering that all charge is conserved, the time rate of decrease of charge within a given volume must be equal to the net outward flow through the surface of the volume Thus, the current out of a closed surface is Applying Stokes Theorem • dQ I J dS enclosed v dv dt t S v v J d S J dv S v v t dv Continuity Equation J v t For steady state problems, the derivative of charge with respect to time equals zero, and thus the gradient of current density at the surface is zero, showing that there can be no net accumulation of charge. 8/17/2012 17 Relaxation Time Constant • • Utilizing the continuity equation and material properties such as permittivity and conductivity, one can derive a time constant (in seconds) by which to measure the relaxation time associated with the decay of charge from the point at which it was introduced within a material to the surface of that material We start with Ohm’s and Gauss’ Laws J E E v v J E v t v v 0 dt • • • v t v t ln v ln vo v vo e t / T Time Constant (s) Tr r The relaxation time is the time it takes a charge placed in the interior of a material to drop by e-1 (=36.8%) of its initial value. For good conductors Tr is approx. 2*10-19 s. For good insulators Tr can be days 8/17/2012 18 Electrostatic Boundary Conditions • • • • • So far we have considered electric fields in a single medium If the field exist in two mediums – The fields within each medium obey the same theorems previously stated – An additional set of boundary conditions exist to match the two fields at the interface We shall consider boundary conditions separating – Dielectric media with two different permittivities – Conductors and dielectric media – Conductors and free space (one of the dielectric constants is equal to 1) To complete this analysis we will need both of Maxwell’s Equations for Electrostatics D v E 0 We will also need to break the electric field intensity into two orthogonal components (tangential and normal) E Et En 8/17/2012 19 Dielectric-Dielectric Boundary • Two different dielectrics characterized by 1 and 2. Around the patch abcd that encloses Apply E 0 E dl the boundary of both dielectrics E dl 0 h h h h E2 n E2t w E2 n E1n 2 2 2 2 E1t w E2t w E1t E2t w E1t w E1n h 0 E1t E2t D1t 1 E1t E2t D2t 2 Tangential E undergoes no change and is continuous across the boundary condition Tangential D on the other hand is discontinuous across the interface 8/17/2012 20 Dielectric-Dielectric Boundary (2) • Two different dielectrics characterized by 1 and 2. Apply D v D dS Qenc To a pillbox that encloses the boundary of both dielectrics S Q s S D1n S D2 n S s D1n D2 n s 0 D1n D2 n E1n 1 D1n D2 n E2 n 2 Normal D undergoes no change and is continuous across the boundary condition Normal E on the other hand is discontinuous across the interface 8/17/2012 21 Dielectric-Dielectric Boundary (3) E1t E2t D1n D2 n 8/17/2012 22 Conductor-Dielectric Boundary • Perfect conductor with infinite conductivity (therefore no volume charge density, potential or electric field inside the conductor) and a dielectric, 2. Apply Apply D dS Qenc E dl 0 h h h h 0w E1n E2 n Et w E2 n E1n 2 2 2 2 D h 0, Et 0 t 2 S Q s S Dn S 0S s Dn 2 En E2 t 0 D2 n 2 E2 n E1 0 8/17/2012 23 Snell’s Law of Refraction • • Consider the boundary of two dielectrics, 1 and 2 We can determine the refraction of of the electric field across the interface using the dielectric boundary conditions provided E dl 0 h 0 E1t E2t E1 sin 1 E1t E2t E2 sin 2 E1 sin 1 E2 sin 2 D dS Qenc S s 0 E1 sin 1 E2 sin 2 D1n D2 n E1 1 cos 1 E2 2 cos 2 E1n 1 D1n D2 n E2 n 2 E1 1 cos 1 D1n D2 n E2 2 cos 2 E1 1 cos 1 E2 2 cos 2 • tan 1 1 tan 2 2 Thus an interface between two dielectrics produces bending of flux lines as a result of unequal polarization charges that accumulate on the opposite sides of the interface 8/17/2012 24