Maxwell s Equations:

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A-337 Phone: (915) Maxwell s Equations: Physical Interpretation EE-3321 Electromagnetic Field Theory Outline Maxwell s Equations Physical Meaning of Maxwell s Equations Gauss law Gauss law for magnetic fields Faraday s law Ampere s circuit law Constitutive relations Maxwell's Equations -- Physical Interpretation lide 2 1

2 Forms of Maxwell s Equations Time-Domain Frequency-Domain Dds dv Bds 0 Dds dv Bds 0 Integral Form v v Most general form B E d ds t D H d J ds t E d jb ds H d J jd ds Differential Form D v B 0 B E t D H J t Most common form D v B 0 E jb H J jd Constitutive Relations: D E BH Maxwell's Equations -- Physical Interpretation lide 3 Physical Meaning of Maxwell s Equations: Gauss aw 2

3 Fields & Charges (1 of 2) The electric field induced by charges is best described by the electric flux density D because this quantity is most closely associated with charge. D Q 4 r 2 Electric field lines diverge from positive charges. Electric field lines converge on negative charges. or both! Maxwell's Equations -- Physical Interpretation lide 5 Fields & Charges (2 of 2) The magnitude of the charge is usually conveyed by the density of the field lines. mall charge arge charge Maxwell's Equations -- Physical Interpretation lide 6 3

4 A Note About Field ines It is in some ways misleading to draw electric field lines because it incorrectly implies that the field exists at some points but not others. D exists here...but not here The electric field is a smooth and continuous phenomenon, but that is hard to convey in a diagram. Maxwell's Equations -- Physical Interpretation lide 7 How to Calculate Total Charge Q Method 1: urface integral of electric flux ince the density of field lines convey the amount of charge, it makes sense that we can calculate the total charge Q enclosed within a surface by integrating the flux of the field lines at the surface. D ds Q Dds Dds Q The choice of surface does not matter as long as the surface completely encloses the charge. Usually the shape of the surface is chosen to make the math as simple as possible. Maxwell's Equations -- Physical Interpretation lide 8 4

5 How to Calculate Total Charge Q Method 2: olume integral of electric charge density The total charge Q can be calculated by integrating the electric charge density v within a volume that completely encompasses the charge. v 1 dv Q dv dv v 1 2 v 2 1 Q The choice of volume is the same as the choice of the surface for Method 1. the shape does not matter as along it completely encompasses the charge. It is usually chosen for mathematical convenience. Maxwell's Equations -- Physical Interpretation lide 9 Gauss aw in Integral Form Both methods calculate the same total charge so they can be set equal. D ds Method 1 Method 2 v 1 dv Q Q Q D ds dv v Maxwell's Equations -- Physical Interpretation lide 10 5

6 Apply Divergence Theorem The divergence theorem allows us to write a closed-contour surface integral as a volume integral. Ads A dv Applying this to Gauss law gives us Q D ds dv Q D dv dv v v Maxwell's Equations -- Physical Interpretation lide 11 Gauss aw in Differential Form If the surface and volume describe the same space, the argument of both integrals must be equal. etting these arguments equal gives Gauss law in differential form. Q D dv dv v D v Maxwell's Equations -- Physical Interpretation lide 12 6

7 Physical Meaning of Maxwell s Equations: Gauss aw for Magnetic Fields No Magnetic Charge ince there exists no isolated magnetic charge, the magnetic field cannot have a beginning or an end. The magnetic field can only form loops. B B Maxwell's Equations -- Physical Interpretation lide 14 7

8 Prove Zero Magnetic Charge urface integral of magnetic flux Following what we did for electric fields, we calculate total magnetic charge enclosed within a surface by integrating the flux of the magnetic field lines at the surface. No flux lines through surface. Q m = 0 B 0 Bds Flux adds to zero. Q m = 0 B ds B Maxwell's Equations -- Physical Interpretation lide 15 Gauss aw for Magnetic Fields in Integral Form The result from last slide is Gauss law for magnetic fields. Bds 0 Maxwell's Equations -- Physical Interpretation lide 16 8

9 Apply Divergence Theorem The divergence theorem allows us to write a closed-contour surface integral as a volume integral. Ads A dv Applying this to Gauss law for magnetic fields gives us Q Bds dv m m Q B dv dv m m Maxwell's Equations -- Physical Interpretation lide 17 Gauss aw for Magnetic Fields in Differential Form If the surface and volume describe the same space, the argument of both integrals must be equal. etting these arguments equal gives Gauss law for magnetic fields in differential form. Q B dv dv m B m However, there is no magnetic charge so m = 0. B 0 m Maxwell's Equations -- Physical Interpretation lide 18 9

10 Physical Meaning of Maxwell s Equations: Faraday s aw Faraday s Experiment Observations: 1. The more (or stronger) the magnetic flux, the higher the voltage reading. Bds emf 2. The more turns of the loop, the higher the voltage reading. emf N N # turns 3. The faster the time rate of change of the magnetic flux, the higher the voltage reading. emf d dt Maxwell's Equations -- Physical Interpretation lide 20 10

11 Calculate Induced oltage (1 of 2) Method 1: By experiment. Faraday performed an experiment and determined that emf d N dt The total magnetic flux accounting for the number of turns is N Bds magnetic flux N magnetic flux linkage Flux linkage is a property of a two-terminal device. It is not equivalent to flux. Flux can exist without the loop. Further, if the loops do not have the same orientation, they will not link to the magnetic field the same. Flux and flux linkage are not the same thing, but often used synonymously because most devices are designed so that each loop links the same to the magnetic field and the math reduces to them being nearly equivalent. Combing the above equations leads an expression for EMF in terms of just the magnetic flux density B. d d B emf N N B ds ds dt dt t t Maxwell's Equations -- Physical Interpretation lide 21 Calculate Induced oltage (2 of 2) Method 2: Use Kirchoff s voltage law The voltage across the terminals of the resistor can be calculated using Kirchoff s voltage law. For electromagnetics, Kirchoff s voltage law becomes a line integral. Assuming the resistor is very small compared to the loop, we get Ed E emf Maxwell's Equations -- Physical Interpretation lide 22 11

12 Faraday s aw in Integral Form Both methods calculate the same voltage so they can be set equal. Method 2 Method 1 B emf Ed ds t Maxwell's Equations -- Physical Interpretation lide 23 Apply toke s Theorem toke s theorem allows us to write a closed-contour line integral as a surface integral. Ad A ds Applying this to Faraday s law in integral form gives us B emf Ed ds t B emf Eds ds t Maxwell's Equations -- Physical Interpretation lide 24 12

13 Faraday s aw in Differential Form If the line and surface describe the same space, the argument of both integrals must be equal. etting these arguments equal gives Faraday s law in differential form. B emf Eds ds t E B t Maxwell's Equations -- Physical Interpretation lide 25 Physical Meaning of Maxwell s Equations: Ampere s Circuit aw 13

14 Ampere s Experiment Observations: 1. There exists a circulating magnetic field H around a conductor carrying a current I. 2. The current I can induce the magnetic field H, or the magnetic field H can induce the current I. 3. The measured current I is in proportion to the circulation of H. I H d Maxwell's Equations -- Physical Interpretation lide 27 Three Types of Current The total current I can be calculated by integrating the flux of the electric current density J through a cross section of the conductor. I J ds total However, recall that there are three types of electric current. J total J J J D D J D displacement current t J J current due to free charges Putting this together gives D I J ds t Maxwell's Equations -- Physical Interpretation lide 28 14

15 Ampere s Circuit aw in Integral Form We have two ways of calculating the total current I in a conductor. etting these equal gives Ampere s Experiment imple integration of current density D I H d J ds t Maxwell's Equations -- Physical Interpretation lide 29 Apply toke s Theorem toke s theorem allows us to write a closed-contour line integral as a surface integral. Ad A ds Applying this to Ampere s law in integral form gives us D I H d J ds t D I Hds J ds t Maxwell's Equations -- Physical Interpretation lide 30 15

16 Ampere s Circuit aw in Differential Form If the line and surface describe the same space, the argument of both integrals must be equal. etting these arguments equal gives Ampere s circuit law in differential form. D I Hds J ds t D H J t Maxwell's Equations -- Physical Interpretation lide 31 isualization of Ampere s Circuit aw in Differential Form D H J t Maxwell's Equations -- Physical Interpretation lide 32 16

17 Physical Meaning of Maxwell s Equations: Constitutive Relations Electric Response of Materials Note: 0 is the free space permittivity and multiples E so that 0 E has the same units as P. D 0E P acuum response Material response Maxwell's Equations -- Physical Interpretation lide 34 17

18 Electric Polarization 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 e P 0e E 0 e E 0e E inear response Nonlinear response electric susceptibility no units These terms are usually ignored. They tend to only become significant when the electric field is very strong. 2 e is pronounced "chi two" 2 e is pronounced "chi three" Maxwell's Equations -- Physical Interpretation lide 35 Electric Permittivity & usceptibility The permittivity is related to the electric susceptibility through 1 1 D E P E E 1 E e 0 e The constitutive relation can also be written in terms of the relative permittivity r. D E 0 r 1 r 1 e acuum response Dielectric response Maxwell's Equations -- Physical Interpretation lide 36 18

19 Magnetic Response of Materials Note: 0 is the free space permeability and multiples H so that 0 H has the same units as M. B 0H M acuum response Material response Maxwell's Equations -- Physical Interpretation lide 37 Magnetic Polarization 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. n m 1 23 M H H 2 H 3 0 m 0 m 0 m inear response Nonlinear response magnetic susceptibility no units These terms are usually ignored. They tend to only become significant when the electric field is very strong. 2 m is pronounced "chi two" 2 m is pronounced "chi three" Maxwell's Equations -- Physical Interpretation lide 38 19

20 Magnetic Permeability & usceptibility The permeability is related to the magnetic susceptibility through 1 1 B E H H 1 H 0 r 0 0 m 0 m 1 r 1 m acuum response Magnetic response Maxwell's Equations -- Physical Interpretation lide 39 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 mall 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 arge positive susceptibility ike paramagnetic, but they retain their magnetism to some degree when the external field is removed. Iron, nickel, cobalt, and some allows. Maxwell's Equations -- Physical Interpretation lide 40 20

21 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 tensor Dx xx xy xz Ex D E y yx yy yz y D z zx zy zz E z ij = how much of E j contributes to D i Maxwell's Equations -- Physical Interpretation lide 41 Types of Anisotropy (1 of 2) Isotropic Media Da 0 0Ea D b 0 0 E b D c0 0 E c Uniaxial Media Da o 0 0Ea D 0 0 E b o b D c0 0 e E c Biaxial Media Da a 0 0Ea D b 0 b 0 E b D c0 0 ce c This is the typical approximation made in electromagnetics. The permittivity tensor reduces to a scalar. D E Anisotropic materials are said to be birefringent. e o o ordinary permittivity Positive birefringence: 0 e extraordinary permittivity Negative birefringence: 0 When orientation is not important, it so convention to order the tensor elements according to a b c Maxwell's Equations -- Physical Interpretation lide 42 21

22 Types of Anisotropy (2 of 2) Doubly Anisotropic D E and B H Chiral Materials Daa jb 0 Ea D b jb a 0 E b D c 0 0 ce c Ba a jb 0 Ha B b jb a 0 H b B c 0 0 ch c Gyroelectric Gryomagnetic Maxwell's Equations -- Physical Interpretation lide 43 Ordinary and Bi- Materials Isotropic Materials Anisotropic Materials Ordinary Materials Isotropic Materials D E B H Anisotropic Materials D B E H Bi- Materials Bi-Isotropic Materials D E H B EH magnetoelectric coupling coefficient Bi-Anisotropic Materials D E H B E H T Maxwell's Equations -- Physical Interpretation lide 44 22