DOCSLIB.ORG

Electromagnetic Fields and Waves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electromagnetic Fields and Waves Lecture 2 Review 2 More on Maxwell Wave Equations Boundary Conditions Poynting Vector Transmission Line A. Nassiri - ANL Maxwell’s equations in differential form ∇.D = ρ Gauss' law for electrostatics ∇.B = 0 Gauss' law for magnetostatics ∂D ∇ × H = J + Ampere' s law dt ∂B ∇ × E = − Faraday' s law dt ∂ρ ∇.J = − Equation of continuity ∂t D = εE Varying E and H fields are coupled B = µH Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 2 Electromagnetic waves in lossless media - Maxwell’s equations Constitutive relations Maxwell ∂D D = εE = ε ε E ∇× H = J + r o dt B = µH = µ µ H ∂B r o ∇×E = − dt J = σE SI Units ∇ = ρ • J Amp/ metre2 .D • D Coulomb/metre2 • H Amps/metre • B Tesla ∇.B = 0 Weber/metre2 Volt-Second/metre2 Equation of continuity • E Volt/metre • ε Farad/metre • µ Henry/metre ∂ρ • σ Siemen/metre ∇.J = − ∂t Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 3 Wave equations in free space • In free space – σ=0 ⇒J=0 – Hence: ∂D ∂D ∇× H = J + = dt dt ∂B ∇×E = − dt – Taking curl of both sides of latter equation: ∂B ∂ ∂ ∇×∇× E = −∇× = − ∇× B = −µ ∇× H ∂t ∂t o ∂t ∂ ∂D = −µo ∂t ∂t ∂ 2E ∇×∇× E = −µoε ∂t 2 Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 4 Wave equations in free space cont. ∂ 2E ∇×∇×E = −µoε ∂t 2 • It has been shown (last week) that for any vector A where is the Laplacian operator 2 Thus: ∇×∇× A = ∇∇.A − ∇ A ∂ 2 ∂ 2 ∂ 2 ∇2 = + + ∂x2 ∂y 2 ∂z 2 2 2 ∂ E ∇∇.E − ∇ E = −µoε ∂t 2 There are no free charges in free space so ∇.E=ρ=0 and we get 2 2 ∂ E ∇ E = µoε ∂t 2 A three dimensional wave equation Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 5 Wave equations in free space cont. • Both E and H obey second order partial differential wave equations: ∂ 2E ∇2E = µ ε o ∂ 2 t2 2 ∂ H ∇ H = µoε ∂t 2 What does this mean – dimensional analysis ? Volts/metre Volts/metre = µoε metre2 seconds2 -2 – µοε has units of velocity – Why is this a wave with velocity 1/ √µοε ? Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 6 Uniform plane waves - transverse relation of E and H • Consider a uniform plane wave, propagating in the z direction. E is independent of x and y ∂E ∂E = 0 = 0 ∂x ∂y In a source free region, ∇.D=ρ =0 (Gauss’ law) : ∂E ∂E ∂E ∇.E = x + y + z = 0 ∂x ∂y ∂z E is independent of x and y, so ∂E ∂E ∂E x = 0, y = 0 ⇒ z = 0 ⇒ E = 0 (E = const is not a wave) ∂x ∂y ∂z z z So for a plane wave, E has no component in the direction of propagation. Similarly for H. Plane waves have only transverse E and H components. Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 7 Orthogonal relationship between E and H: • For a plane z-directed wave there are no variations along x and y: ∂ ∂A ∂ Az y H y ∂H x ∇× A = a x − + ∇× H = −a + a ∂y ∂z x ∂z y ∂z ∂Ax ∂Az a y − + ∂D ∂z ∂x = ∂A ∂A ∂t a y − x z ∂x ∂y ∂ ∂Ex Ey ∂Ez = εa + a + a ∂D x ∂t y ∂t y ∂t ∇× H = J + dt Equating terms: and likewise for : ∇×E = −µo ∂H ∂t ∂H ∂E ∂E ∂H − y = ε x y = µ x ∂z ∂t ∂z o ∂t ∂H ∂E ∂E ∂H x = ε y x = µ y ∂z ∂t ∂z o ∂t Spatial rate of change of H is proportionate to the temporal rate of change of the orthogonal component of E & v.v. at the same point in space Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 8 Orthogonal and phase relationship between E and H: • Consider a linearly polarised wave that has a transverse component in (say) the y direction only: ∂ H y ∂Ex = − − = ε Ey Eo f (z vt) ∂z ∂t ∂H x ∂E = ∂H ∂E ⇒ ε y = −εvE f ′(z − vt) ∂z x = ε y ∂t o ∂z ∂t ⇒ = −ε ′ − + = −ε − H x vEo ∫ f (z vt)dz const vEo f (z vt) = −εvEy ε H x = − Ey µo Similarly ∂ E y ∂H x = µo ε ∂z ∂t ∂ H y = Ex ∂Ex H y µ = µo o ∂z ∂t H and E are in phase and orthogonal Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 9 ε ε H = − E H = E x µ y y x o µo • The ratio of the magnetic to electric fields strengths is: 2 2 Ex + E y E µ = = o = η Note: 2 2 H ε H x + H y E E 1 = = = c which has units of impedance µ B o H µoεo Volts / metre = Ω amps / metre and the impedance of free space is: µ 4π ×10−7 o = = 120π = 377Ω ε 1 − o ×10 9 36π Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 10 Orientation of E and H • For any medium the intrinsic impedance is denoted by η E E η = − y = x and taking the scalar product H x H y E.H = Ex H x + E y H y = ηH y H x −ηH x H y = 0 so E and H are mutually orthogonal Taking the cross product of E and H we get the direction of wave propagation E× H = a z (Ex H y − E y H x ) A×B = a x (Ay Bz − Az By )+ 2 2 = a z (ηH y −ηH x ) a y (Az Bx − Ax Bz )+ 2 E× H = a zηH a z (Ax By − Ay Bx ) Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 11 A ‘horizontally’ polarised wave • Sinusoidal variation of E and H • E and H in phase and orthogonal ε H y = Ex µo Hy Ex E× H Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 12 A block of space containing an EM plane wave • Every point in 3D space is characterised by – Ex, Ey, Ez – Which determine • Hx, Hy, Hz • and vice versa – 3 degrees of freedom λ Ex ε H y = Ex µo E× H ε H x = − E y Hy µo Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 13 Power flow of EM radiation 2 • Energy stored in the EM field in the thin box is: εE u = E 2 dU = dU E + dU H = (uE + uH )Adx µ H 2 2 2 o εE µ H uH = = + o dU Adx 2 2 2 2 ε = εE Adx H = E y µ x Power transmitted through the box is dU/dt=dU/(dx/c).... o λ dx Ex E× H Hy Area A Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 14 Power flow of EM radiation cont. dU = εE 2 Adx dU εE 2 ε E 2 S = = Adx = = W/m2 Adt A(dx c) µo η • This is the instantaneous power flow – Half is contained in the electric component – Half is contained in the magnetic component • E varies sinusoidal, so the average value of S is obtained as: 2π E = E sin (z − vt) o λ E 2 sin 2 (z − vt) S = o η E 2 E 2 S = o RMS(E 2 sin 2 (z − vt))= o η o 2η S is the Poynting vector and indicates the direction and magnitude of power flow in the EM field. Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 15 Example • The door of a microwave oven is left open – estimate the peak E and H strengths in the aperture of the door. – Which plane contains both E and H vectors ? – What parameters and equations are required? • Power-750 W • Area of aperture - 0.3 x 0.2 m • impedance of free space - 377 Ω • Poynting vector: E 2 S = = ηH 2 W/m2 η Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 16 E 2 Power = SA = A = ηH 2 A Watts η Power 750 E = η = 377 = 2,171V/m A 0.3.0.2 E 2170 H = = = 5.75A/m η 377 −7 B = µo H = 4π ×10 ×5.75 = 7.2μTesla Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 17 Constitutive relations -12 • permittivity of free space ε0=8.85 x 10 F/m -7 • permeability of free space µo=4πx10 H/m D = εE = ε rεoE Normally ε (dielectric constant) and µ • r r = µ = µ µ – vary with material B H r oH – are frequency dependant J = σE • For non-magnetic materials mr ~1 and for Fe is ~200,000 • er is normally a few ~2.25 for glass at optical frequencies – are normally simple scalars (i.e. for isotropic materials) so that D and E are parallel and B and H are parallel • For ferroelectrics and ferromagnetics er and mr depend on the relative orientation of the material and the applied field: µ µ µ Bx xx xy xz H x µ − jκ 0 At By = µ yx µ yy µ yz H y µ = jκ µ 0 microwave ij B µ µ µ H z zx zy zz z frequencies: 0 0 µo Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 18 Constitutive relations cont..
Load more

Recommended publications
  • 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.
    [Show full text]
  • Chapter 3 Electromagnetic Waves & Maxwell's Equations
    Chapter 3 Electromagnetic Waves & Maxwell’s Equations Part I Maxwell’s Equations Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861 Maxwell’s Equations: Integral Form Gauss's law Gauss's law for magnetism Faraday's law of induction (Maxwell–Faraday equation) Ampère's law (with Maxwell's addition) Maxwell’s Equations Relation of the speed of light and electric and magnetic vacuum constants 1 c = 2.99792458 108 [m/s] 0 0 permittivity of free space, also called the electric constant permeability of free space, also called the magnetic constant Gauss’s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is This result for the electric flux is known as Gauss’s Law. Magnetic Gauss’s Law The net magnetic flux through any closed surface is equal to zero: As of today there is no evidence of magnetic monopoles See: Phys.Rev.Lett.85:5292,2000 Ampère's Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source: B ds 0I I is the total current inside the loop. ds B i1 Direction of integration i 3 i2 Faraday’s Law The change of magnetic flux in a loop will induce emf, i.e., electric field B E ds dA A t Lenz's Law Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Problem with Ampère's Law Maxwell realized that Ampere’s law is not valid when the current is discontinuous.
    [Show full text]
  • Poynting Vector and Power Flow in Electromagnetic Fields
    Poynting Vector and Power Flow in Electromagnetic Fields: Electromagnetic waves can transport energy as a result of their travelling or propagating characteristics. Starting from Maxwell's Equations: Together with the vector identity One can write In simple medium where and are constant, and Divergence theorem states, This equation is referred to as Poynting theorem and it states that the net power flowing out of a given volume is equal to the time rate of decrease in the energy stored within the volume minus the conduction losses. In the equation, the following term represents the rate of change of the stored energy in the electric and magnetic fields On the other hand, the power dissipation within the volume appears in the following form Hence the total decrease in power within the volume under consideration: Here (W/mt2) is called the Poynting vector and it represents the power density vector associated with the electromagnetic field. The integration of the Poynting vector over any closed surface gives the net power flowing out of the surface. Poynting vector for the time harmonic case: Using the convention, the instantaneous value of a quantity is the real part of the product of a phasor quantity and when is used as reference. Considering the following phasor: −푗훽푧 퐸⃗ (푧) = 푥̂퐸푥(푧) = 푥̂퐸0푒 The instantaneous field becomes: 푗푤푡 퐸⃗ (푧, 푡) = 푅푒{퐸⃗ (푧)푒 } = 푥̂퐸0푐표푠(휔푡 − 훽푧) when E0 is real. Let us consider two instantaneous quantities A and B such that where A and B are the phasor quantities. Therefore, Since A and B are periodic with period , the time average value of the product form AB, 푇 1 퐴퐵 = ∫ 퐴퐵푑푡 푎푣푒푟푎푔푒 푇 0 푇 1 퐴퐵 = ∫|퐴||퐵|푐표푠(휔푡 + 훼)푐표푠(휔푡 + 훽)푑푡 푎푣푒푟푎푔푒 푇 0 1 퐴퐵 = |퐴||퐵|푐표푠(훼 − 훽) 푎푣푒푟푎푔푒 2 For phasors, and , where * denotes complex conjugate.
    [Show full text]
  • 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.
    [Show full text]
  • How to Introduce the Magnetic Dipole Moment
    IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 33 (2012) 1313–1320 doi:10.1088/0143-0807/33/5/1313 How to introduce the magnetic dipole moment M Bezerra, W J M Kort-Kamp, M V Cougo-Pinto and C Farina Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, CEP 21941-972, Rio de Janeiro, Brazil E-mail: [email protected] Received 17 May 2012, in final form 26 June 2012 Published 19 July 2012 Online at stacks.iop.org/EJP/33/1313 Abstract We show how the concept of the magnetic dipole moment can be introduced in the same way as the concept of the electric dipole moment in introductory courses on electromagnetism. Considering a localized steady current distribution, we make a Taylor expansion directly in the Biot–Savart law to obtain, explicitly, the dominant contribution of the magnetic field at distant points, identifying the magnetic dipole moment of the distribution. We also present a simple but general demonstration of the torque exerted by a uniform magnetic field on a current loop of general form, not necessarily planar. For pedagogical reasons we start by reviewing briefly the concept of the electric dipole moment. 1. Introduction The general concepts of electric and magnetic dipole moments are commonly found in our daily life. For instance, it is not rare to refer to polar molecules as those possessing a permanent electric dipole moment. Concerning magnetic dipole moments, it is difficult to find someone who has never heard about magnetic resonance imaging (or has never had such an examination).
    [Show full text]
  • 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.
    [Show full text]
  • Electromagnetic Energy
    Physics 142 Electromagnetic Enmergy Page !1 Electromagnetic Energy A child of five can understand this; send someone to fetch a child of five. — Groucho Marx Energy in the fields can move from place to place We have discussed the energy in an electrostatic field, such as that stored in a capacitor. We found that this energy can be thought of as distributed in space, with an energy per unit volume (energy density) at each point in space. We obtained a formula for this 1 2 quantity, ! ue = 2 ε0E (if there are no dielectric materials). This formula applies to any E- field. We also found that magnetic fields possess energy distributed in space, described 2 by the magnetic energy density ! um = B /2µ0 . If there are both electric and magnetic fields, the total electromagnetic energy density is the sum of ! ue and ! um . These specify how much electromagnetic field energy there is at any point in space. But we have not yet considered how this energy moves from place to place. Consider the energy flow in a flashlight. We know that energy moves from the battery to the bulb, where it is converted into heat and light. It is tempting to assume that this energy flows through the conductors, like water in a pipe. But if we look carefully we find that the electromagnetic energy density in the conductors is much too small to account for the amount of energy in transit. Nearly all of the energy gets to the bulb by flowing through space near the conductors. In a sense they guide the energy but do not carry much of it.
    [Show full text]
  • Gauss' Theorem (See History for Rea- Son)
    Gauss’ Law Contents 1 Gauss’s law 1 1.1 Qualitative description ......................................... 1 1.2 Equation involving E field ....................................... 1 1.2.1 Integral form ......................................... 1 1.2.2 Differential form ....................................... 2 1.2.3 Equivalence of integral and differential forms ........................ 2 1.3 Equation involving D field ....................................... 2 1.3.1 Free, bound, and total charge ................................. 2 1.3.2 Integral form ......................................... 2 1.3.3 Differential form ....................................... 2 1.4 Equivalence of total and free charge statements ............................ 2 1.5 Equation for linear materials ...................................... 2 1.6 Relation to Coulomb’s law ....................................... 3 1.6.1 Deriving Gauss’s law from Coulomb’s law .......................... 3 1.6.2 Deriving Coulomb’s law from Gauss’s law .......................... 3 1.7 See also ................................................ 3 1.8 Notes ................................................. 3 1.9 References ............................................... 3 1.10 External links ............................................. 3 2 Electric flux 4 2.1 See also ................................................ 4 2.2 References ............................................... 4 2.3 External links ............................................. 4 3 Ampère’s circuital law 5 3.1 Ampère’s original
    [Show full text]
  • Non-Ionizing Radiation, Part 2: Radiofrequency Electromagnetic Fields Volume 102
    NON-IONIZING RADIATION, PART 2: RADIOFREQUENCY ELECTROMAGNETIC FIELDS VOLUME 102 This publication represents the views and expert opinions of an IARC Working Group on the Evaluation of Carcinogenic Risks to Humans, which met in Lyon, 24-31 May 2011 LYON, FRANCE - 2013 IARC MONOGRAPHS ON THE EVALUATION OF CARCINOGENIC RISKS TO HUMANS 1. EXPOSURE DATA 1.1 Introduction a wave (Fig. 1.1) and is related to the frequency according to λ = c/f (ICNIRP, 2009a). This chapter explains the physical principles The fundamental equations of electromag- and terminology relating to sources, exposures netism, Maxwell’s equations, imply that a time- and dosimetry for human exposures to radiofre- varying electric field generates a time-varying quency electromagnetic fields (RF-EMF). It also magnetic field and vice versa. These varying identifies critical aspects for consideration in the fields are thus described as “interdependent” and interpretation of biological and epidemiological together they form a propagating electromagnetic studies. wave. The ratio of the strength of the electric-field component to that of the magnetic-field compo- 1.1.1 Electromagnetic radiation nent is constant in an electromagnetic wave and is known as the characteristic impedance of the Radiation is the process through which medium (η) through which the wave propagates. energy travels (or “propagates”) in the form of The characteristic impedance of free space and waves or particles through space or some other air is equal to 377 ohm (ICNIRP, 2009a). medium. The term “electromagnetic radiation” It should be noted that the perfect sinusoidal specifically refers to the wave-like mode of case shown in Fig.
    [Show full text]
  • Temporal Analysis of Radiating Current Densities
    Temporal analysis of radiating current densities Wei Guoa aP. O. Box 470011, Charlotte, North Carolina 28247, USA ARTICLE HISTORY Compiled August 10, 2021 ABSTRACT From electromagnetic wave equations, it is first found that, mathematically, any current density that emits an electromagnetic wave into the far-field region has to be differentiable in time infinitely, and that while the odd-order time derivatives of the current density are built in the emitted electric field, the even-order deriva- tives are built in the emitted magnetic field. With the help of Faraday’s law and Amp`ere’s law, light propagation is then explained as a process involving alternate creation of electric and magnetic fields. From this explanation, the preceding math- ematical result is demonstrated to be physically sound. It is also explained why the conventional retarded solutions to the wave equations fail to describe the emitted fields. KEYWORDS Light emission; electromagnetic wave equations; current density In electrodynamics [1–3], a time-dependent current density ~j(~r, t) and a time- dependent charge density ρ(~r, t), all evaluated at position ~r and time t, are known to be the sources of an emitted electric field E~ and an emitted magnetic field B~ : 1 ∂2 4π ∂ ∇2E~ − E~ = ~j + 4π∇ρ, (1) c2 ∂t2 c2 ∂t and 1 ∂2 4π ∇2B~ − B~ = − ∇× ~j, (2) c2 ∂t2 c2 where c is the speed of these emitted fields in vacuum. See Refs. [3,4] for derivation of these equations. (In some theories [5], on the other hand, ρ and ~j are argued to arXiv:2108.04069v1 [physics.class-ph] 6 Aug 2021 be responsible for instantaneous action-at-a-distance fields, not for fields propagating with speed c.) Note that when the fields are observed in the far-field region, the contribution to E~ from ρ can be practically ignored [6], meaning that, in that region, Eq.
    [Show full text]
  • Poynting's Theorem and the Wave Equation
    Chapter 18: Poynting’s Theorem and the Wave Equation Chapter Learning Objectives: After completing this chapter the student will be able to: Use Poynting’s theorem to determine the direction and magnitude of power flow in an electromagnetic system. Use Maxwell’s Equations to derive a general homogeneous wave equation for the electric and magnetic field. Derive a simplified wave equation assuming propagation in a vacuum and an electric field polarized in only one direction. Use Maxwell’s Equations to derive the speed of light in a vacuum. You can watch the video associated with this chapter at the following link: Historical Perspective: John Henry Poynting (1852-1914) was an English physicist who did work in electromagnetic energy flow, elasticity, and astronomy. He coined the term “Greenhouse Effect.” Both the Poynting Vector and Poynting’s Theorem are named in his honor. Photo credit: https://upload.wikimedia.org/wikipedia/commons/5/5f/John_Henry_Poynting.jpg, [Public domain], via Wikimedia Commons. 1 18.1 Poynting’s Theorem With Maxwell’s Equations, we now have the tools necessary to derive Poynting’s Theorem, which will allow us to perform many useful calculations involving the direction of power flow in electromagnetic fields. We will begin with Faraday’s Law, and we will take the dot product of H with both sides: (Copy of Equation 16.24) (Equation 18.1) Next, we will start with Ampere’s Law and will take the dot product of E with both sides: (Copy of Equation 17.13) (Equation 18.2) Now, let’s subtract both sides of Equation 18.2 from both sides of Equation 18.1: (Equation 18.3) We can now apply the following mathematical identity to the left side of Equation 18.3: (Equation 18.4) This substitution yields: (Equation 18.5) Distributing the E across the right side gives: (Equation 18.6) 2 Now let’s concentrate on the first time on the right side.
    [Show full text]
  • Section 2: Maxwell Equations
    Section 2: Maxwell’s equations Electromotive force We start the discussion of time-dependent magnetic and electric fields by introducing the concept of the electromotive force . Consider a typical electric circuit. There are two forces involved in driving current around a circuit: the source, fs , which is ordinarily confined to one portion of the loop (a battery, say), and the electrostatic force, E, which serves to smooth out the flow and communicate the influence of the source to distant parts of the circuit. Therefore, the total force per unit charge is a circuit is = + f fs E . (2.1) The physical agency responsible for fs , can be any one of many different things: in a battery it’s a chemical force; in a piezoelectric crystal mechanical pressure is converted into an electrical impulse; in a thermocouple it’s a temperature gradient that does the job; in a photoelectric cell it’s light. Whatever the mechanism, its net effect is determined by the line integral of f around the circuit: E =fl ⋅=d f ⋅ d l ∫ ∫ s . (2.2) The latter equality is because ∫ E⋅d l = 0 for electrostatic fields, and it doesn’t matter whether you use f E or fs . The quantity is called the electromotive force , or emf , of the circuit. It’s a lousy term, since this is not a force at all – it’s the integral of a force per unit charge. Within an ideal source of emf (a resistanceless battery, for instance), the net force on the charges is zero, so E = – fs. The potential difference between the terminals ( a and b) is therefore b b ∆Φ=− ⋅ = ⋅ = ⋅ = E ∫Eld ∫ fs d l ∫ f s d l .
    [Show full text]