Section 2: Maxwell Equations
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Chapter 3 Dynamics of the Electromagnetic Fields
Chapter 3 Dynamics of the Electromagnetic Fields 3.1 Maxwell Displacement Current In the early 1860s (during the American civil war!) electricity including induction was well established experimentally. A big row was going on about theory. The warring camps were divided into the • Action-at-a-distance advocates and the • Field-theory advocates. James Clerk Maxwell was firmly in the field-theory camp. He invented mechanical analogies for the behavior of the fields locally in space and how the electric and magnetic influences were carried through space by invisible circulating cogs. Being a consumate mathematician he also formulated differential equations to describe the fields. In modern notation, they would (in 1860) have read: ρ �.E = Coulomb’s Law �0 ∂B � ∧ E = − Faraday’s Law (3.1) ∂t �.B = 0 � ∧ B = µ0j Ampere’s Law. (Quasi-static) Maxwell’s stroke of genius was to realize that this set of equations is inconsistent with charge conservation. In particular it is the quasi-static form of Ampere’s law that has a problem. Taking its divergence µ0�.j = �. (� ∧ B) = 0 (3.2) (because divergence of a curl is zero). This is fine for a static situation, but can’t work for a time-varying one. Conservation of charge in time-dependent case is ∂ρ �.j = − not zero. (3.3) ∂t 55 The problem can be fixed by adding an extra term to Ampere’s law because � � ∂ρ ∂ ∂E �.j + = �.j + �0�.E = �. j + �0 (3.4) ∂t ∂t ∂t Therefore Ampere’s law is consistent with charge conservation only if it is really to be written with the quantity (j + �0∂E/∂t) replacing j. -
Lecture 5: Displacement Current and Ampère's Law
Whites, EE 382 Lecture 5 Page 1 of 8 Lecture 5: Displacement Current and Ampère’s Law. One more addition needs to be made to the governing equations of electromagnetics before we are finished. Specifically, we need to clean up a glaring inconsistency. From Ampère’s law in magnetostatics, we learned that H J (1) Taking the divergence of this equation gives 0 H J That is, J 0 (2) However, as is shown in Section 5.8 of the text (“Continuity Equation and Relaxation Time), the continuity equation (conservation of charge) requires that J (3) t We can see that (2) and (3) agree only when there is no time variation or no free charge density. This makes sense since (2) was derived only for magnetostatic fields in Ch. 7. Ampère’s law in (1) is only valid for static fields and, consequently, it violates the conservation of charge principle if we try to directly use it for time varying fields. © 2017 Keith W. Whites Whites, EE 382 Lecture 5 Page 2 of 8 Ampère’s Law for Dynamic Fields Well, what is the correct form of Ampère’s law for dynamic (time varying) fields? Enter James Clerk Maxwell (ca. 1865) – The Father of Classical Electromagnetism. He combined the results of Coulomb’s, Ampère’s, and Faraday’s laws and added a new term to Ampère’s law to form the set of fundamental equations of classical EM called Maxwell’s equations. It is this addition to Ampère’s law that brings it into congruence with the conservation of charge law. -
This Chapter Deals with Conservation of Energy, Momentum and Angular Momentum in Electromagnetic Systems
This chapter deals with conservation of energy, momentum and angular momentum in electromagnetic systems. The basic idea is to use Maxwell’s Eqn. to write the charge and currents entirely in terms of the E and B-fields. For example, the current density can be written in terms of the curl of B and the Maxwell Displacement current or the rate of change of the E-field. We could then write the power density which is E dot J entirely in terms of fields and their time derivatives. We begin with a discussion of Poynting’s Theorem which describes the flow of power out of an electromagnetic system using this approach. We turn next to a discussion of the Maxwell stress tensor which is an elegant way of computing electromagnetic forces. For example, we write the charge density which is part of the electrostatic force density (rho times E) in terms of the divergence of the E-field. The magnetic forces involve current densities which can be written as the fields as just described to complete the electromagnetic force description. Since the force is the rate of change of momentum, the Maxwell stress tensor naturally leads to a discussion of electromagnetic momentum density which is similar in spirit to our previous discussion of electromagnetic energy density. In particular, we find that electromagnetic fields contain an angular momentum which accounts for the angular momentum achieved by charge distributions due to the EMF from collapsing magnetic fields according to Faraday’s law. This clears up a mystery from Physics 435. We will frequently re-visit this chapter since it develops many of our crucial tools we need in electrodynamics. -
Meaning of the Maxwell Tensor
Meaning of the Maxwell Tensor We define the Maxwell Stress Tensor as 1 1 1 T = ε E E − δ E E + B B − δ B B ij 0 i j ij k k µ i j ij k k 2 0 2 While it looks scary, the Maxwell tensor is just a different way of writing the fact that electric fields exert forces on charges, and that magnetic fields exert forces on cur- rents. Dimensionally, Momentum T = (Area)(Time) This is the dimensionality of a pressure: Force = dP dt = Momentum Area Area (Area)(Time) Meaning of the Maxwell Tensor (2) If we have a uniform electric field in the x direction, what are the components of the tensor? E = (E,0,0) ; B = (0,0,0) 2 1 0 0 1 0 0 ε 1 0 0 2 1 1 0 E = ε 0 0 0 − 0 1 0 + ( ) = 0 −1 0 Tij 0 E 0 µ − 0 0 0 2 0 0 1 0 2 0 0 1 This says that there is negative x momentum flowing in the x direction, and positive y momentum flowing in the y direction, and positive z momentum flowing in the z di- rection. But it’s a constant, with no divergence, so if we integrate the flux of any component out of (not through!) a closed volume, it’s zero Meaning of the Maxwell Tensor (3) But what if the Maxwell tensor is not uniform? Consider a parallel-plate capacitor, and do the surface integral with part of the surface between the plates, and part out- side of the plates Now the x-flux integral will be finite on one end face, and zero on the other. -
Application Limits of the Airgap Maxwell Tensor
Application Limits of the Airgap Maxwell Tensor Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, Jean Le Besnerais, Jean-Philippe Lecointe To cite this version: Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, et al.. Application Limits of the Airgap Maxwell Tensor. 2019. hal-02169268 HAL Id: hal-02169268 https://hal.archives-ouvertes.fr/hal-02169268 Preprint submitted on 1 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Application Limits of the Airgap Maxwell Tensor Raphael¨ Pile1,2, Guillaume Parent3, Emile Devillers1,2, Thomas Henneron2, Yvonnick Le Menach2, Jean Le Besnerais1, and Jean-Philippe Lecointe3 1EOMYS ENGINEERING, Lille-Hellemmes 59260, France 2Univ. Lille, Arts et Metiers ParisTech, Centrale Lille, HEI, EA 2697 - L2EP - Laboratoire d’Electrotechnique et d’Electronique de Puissance, F-59000 Lille, France 3Univ. Artois, EA 4025, Laboratoire Systemes` Electrotechniques´ et Environnement (LSEE), F-62400 Bethune,´ France Airgap Maxwell tensor is widely used in numerical simulations to accurately compute global magnetic forces and torque but also to estimate magnetic surface force waves, for instance when evaluating magnetic stress harmonics responsible for electromagnetic vibrations and acoustic noise in electrical machines. -
Chapter 6. Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws
Chapter 6. Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws 6.1 Displacement Current The magnetic field due to a current distribution satisfies Ampere’s law, (5.45) Using Stokes’s theorem we can transform this equation into an integral form: ∮ ∫ (5.47) We shall examine this law, show that it sometimes fails, and find a generalization that is always valid. Magnetic field near a charging capacitor loop C Fig 6.1 parallel plate capacitor Consider the circuit shown in Fig. 6.1, which consists of a parallel plate capacitor being charged by a constant current . Ampere’s law applied to the loop C and the surface leads to ∫ ∫ (6.1) If, on the other hand, Ampere’s law is applied to the loop C and the surface , we find (6.2) ∫ ∫ Equations 6.1 and 6.2 contradict each other and thus cannot both be correct. The origin of this contradiction is that Ampere’s law is a faulty equation, not consistent with charge conservation: (6.3) ∫ ∫ ∫ ∫ 1 Generalization of Ampere’s law by Maxwell Ampere’s law was derived for steady-state current phenomena with . Using the continuity equation for charge and current (6.4) and Coulombs law (6.5) we obtain ( ) (6.6) Ampere’s law is generalized by the replacement (6.7) i.e., (6.8) Maxwell equations Maxwell equations consist of Eq. 6.8 plus three equations with which we are already familiar: (6.9) 6.2 Vector and Scalar Potentials Definition of and Since , we can still define in terms of a vector potential: (6.10) Then Faradays law can be written ( ) The vanishing curl means that we can define a scalar potential satisfying (6.11) or (6.12) 2 Maxwell equations in terms of vector and scalar potentials and defined as Eqs. -
PHYS 352 Electromagnetic Waves
Part 1: Fundamentals These are notes for the first part of PHYS 352 Electromagnetic Waves. This course follows on from PHYS 350. At the end of that course, you will have seen the full set of Maxwell's equations, which in vacuum are ρ @B~ r~ · E~ = r~ × E~ = − 0 @t @E~ r~ · B~ = 0 r~ × B~ = µ J~ + µ (1.1) 0 0 0 @t with @ρ r~ · J~ = − : (1.2) @t In this course, we will investigate the implications and applications of these results. We will cover • electromagnetic waves • energy and momentum of electromagnetic fields • electromagnetism and relativity • electromagnetic waves in materials and plasmas • waveguides and transmission lines • electromagnetic radiation from accelerated charges • numerical methods for solving problems in electromagnetism By the end of the course, you will be able to calculate the properties of electromagnetic waves in a range of materials, calculate the radiation from arrangements of accelerating charges, and have a greater appreciation of the theory of electromagnetism and its relation to special relativity. The spirit of the course is well-summed up by the \intermission" in Griffith’s book. After working from statics to dynamics in the first seven chapters of the book, developing the full set of Maxwell's equations, Griffiths comments (I paraphrase) that the full power of electromagnetism now lies at your fingertips, and the fun is only just beginning. It is a disappointing ending to PHYS 350, but an exciting place to start PHYS 352! { 2 { Why study electromagnetism? One reason is that it is a fundamental part of physics (one of the four forces), but it is also ubiquitous in everyday life, technology, and in natural phenomena in geophysics, astrophysics or biophysics. -
Displacement Current and Ampère's Circuital Law Ivan S
ELECTRICAL ENGINEERING Displacement current and Ampère's circuital law Ivan S. Bozev, Radoslav B. Borisov The existing literature about displacement current, although it is clearly defined, there are not enough publications clarifying its nature. Usually it is assumed that the electrical current is three types: conduction current, convection current and displacement current. In the first two cases we have directed movement of electrical charges, while in the third case we have time varying electric field. Most often for the displacement current is talking in capacitors. Taking account that charge carriers (electrons and charged particles occupy the negligible space in the surrounding them space, they can be regarded only as exciters of the displacement current that current fills all space and is superposition of the currents of the individual moving charges. For this purpose in the article analyzes the current configuration of lines in space around a moving charge. An analysis of the relationship between the excited magnetic field around the charge and the displacement current is made. It is shown excited magnetic flux density and excited the displacement current are linked by Ampere’s circuital law. ъ а аа я аъ а ъя (Ива . Бв, аав Б. Бв.) В я, я , я , яя . , я : , я я. я я, я я . я . К , я ( я я , я, я я я. З я я я. я я. , я я я я . 1. Introduction configurations of the electric field of moving charge are shown on Fig. 1 and Fig. 2. First figure represents The size of electronic components constantly delayed potentials of the electric field according to shrinks and the discrete nature of the matter is Liénard–Wiechert and this picture is not symmetrical becomming more obvious. -
PHYS 110B - HW #4 Fall 2005, Solutions by David Pace Equations Referenced As ”EQ
PHYS 110B - HW #4 Fall 2005, Solutions by David Pace Equations referenced as ”EQ. #” are from Griffiths Problem statements are paraphrased [1.] Problem 8.2 from Griffiths Reference problem 7.31 (figure 7.43). (a) Let the charge on the ends of the wire be zero at t = 0. Find the electric and magnetic fields in the gap, E~ (s, t) and B~ (s, t). (b) Find the energy density and Poynting vector in the gap. Verify equation 8.14. (c) Solve for the total energy in the gap; it will be time-dependent. Find the total power flowing into the gap by integrating the Poynting vector over the relevant surface. Verify that the input power is equivalent to the rate of increasing energy in the gap. (Griffiths Hint: This verification amounts to proving the validity of equation 8.9 in the case where W = 0.) Solution (a) The electric field between the plates of a parallel plate capacitor is known to be (see example 2.5 in Griffiths), σ E~ = zˆ (1) o where I define zˆ as the direction in which the current is flowing. We may assume that the charge is always spread uniformly over the surfaces of the wire. The resultant charge density on each “plate” is then time-dependent because the flowing current causes charge to pile up. At time zero there is no charge on the plates, so we know that the charge density increases linearly as time progresses. It σ(t) = (2) πa2 where a is the radius of the wire and It is the total charge on the plates at any instant (current is in units of Coulombs/second, so the total charge on the end plate is the current multiplied by the length of time over which the current has been flowing). -
Metamaterials for Electromagnetic and Thermal Waves
Metamaterials for electromagnetic and thermal waves 1 1,2 1,3 1 Erin Donnelly , Antoine Durant , Célia Lacoste , Luigi La Spada 1School of Engineering and the Built Environment, Edinburgh Napier University, Edinburgh, United Kingdom 2IUT1, Université Grenoble Alpes, Grenoble, France 3INP-ENSEEIHT, University of Toulouse, Toulouse, France [email protected] Abstract— In the last decade, electromagnetic is of great interest in any industrial applications. Moreover, metamaterials, thanks to their exotic properties and the unsolved issues are still present: significant presence of possibility to control electromagnetic waves at will, become losses, narrow bandwidth, material properties difficult to crucial building blocks to develop new and advanced technologies in several practical applications. Even though achieve, complexity in both design and manufacturing. until now the metamaterial concept has been mostly associated Therefore, in this paper, we model, design and fabricate a with electromagnetism and optics; the same concepts can be multi-functional metamaterial able to control amplitude and also applied to other wave phenomena, such as phase of both electromagnetic and thermal wave. thermodynamics. For this reason, here the aim is to realize a The paper is organized as follows: (i) in the modeling multi-functional metamaterial able to control and manipulate section, by using field theory, we evaluate the metamaterial simultaneously both electromagnetic and thermal waves. fundamental parameters: electric permittivity ε and thermal -
PHYS 332 Homework #1 Maxwell's Equations, Poynting Vector, Stress
PHYS 332 Homework #1 Maxwell's Equations, Poynting Vector, Stress Energy Tensor, Electromagnetic Momentum, and Angular Momentum 1 : Wangsness 21-1. A parallel plate capacitor consists of two circular plates of area S (an effectively infinite area) with a vacuum between them. It is connected to a battery of constant emf . The plates are then slowly oscillated so that the separation d between ~ them is described by d = d0 + d1 sin !t. Find the magnetic field H between the plates produced by the displacement current. Similarly, find H~ if the capacitor is first disconnected from the battery and then the plates are oscillated in the same manner. 2 : Wangsness 21-7. Find the form of Maxwell's equations in terms of E~ and B~ for a linear isotropic but non-homogeneous medium. Do not assume that Ohm's Law is valid. 3 : Wangsness 21-9. Figure 21-3 shows a charging parallel plate capacitor. The plates are circular with radius a (effectively infinite). Find the Poynting vector on the bounding surface of region 1 of the figure. Find the total rate at which energy is entering region 1 and then show that it equals the rate at which the energy of the capacitor is increasing. 4a: Extra-2. You have a parallel plate capacitor (assumed infinite) with charge densities σ and −σ and plate separation d. The capacitor moves with velocity v in a direction perpendicular to the area of the plates (NOT perpendicular to the area vector for the plates). Determine S~. b: This time the capacitor moves with velocity v parallel to the area of the plates. -
Energy, Momentum, and Symmetries 1 Fields 2 Complex Notation
Energy, Momentum, and Symmetries 1 Fields The interaction of charges was described through the concept of a field. A field connects charge to a geometry of space-time. Thus charge modifies surrounding space such that this space affects another charge. While the abstraction of an interaction to include an intermedi- ate step seems irrelevant to those approaching the subject from a Newtonian viewpoint, it is a way to include relativity in the interaction. The Newtonian concept of action-at-a-distance is not consistent with standard relativity theory. Most think in terms of “force”, and in particular, force acting directly between objects. You should now think in terms of fields, i.e. a modification to space-time geometry which then affects a particle positioned at that geometric point. In fact, it is not the fields themselves, but the potentials (which are more closely aligned with energy) which will be fundamental to the dynamics of interactions. 2 Complex notation Note that a description of the EM interaction involves the time dependent Maxwell equa- tions. The time dependence can be removed by essentially employing a Fourier transform. Remember in an earlier lecture, the Fourier transformation was introduced. ∞ (ω) = 1 dt F (t) e−iωt F √2π −∞R ∞ F (t) = 1 dt (ω) eiωt √2π −∞R F Now assume that all functions have the form; F (~x, t) F (~x)eiωt → These solutions are valid for a particular frequency, ω. One can superimpose solutions of different frequencies through a weighted, inverse Fourier transform to get the time depen- dence of any function. Also note that this assumption results in the introduction of complex functions to describe measurable quantities which in fact must be real.