Physics
Jefimenko's Equations
Jefimenko's Equations are a set of equations in electromagnetism that describe the electric and magnetic fields in terms of the charge and current distributions. They provide a way to calculate the fields at any point in space and time based on the distribution of charges and currents. These equations are useful for understanding and predicting electromagnetic phenomena.
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eBook - PDFPhysics Of Space Plasmas
An Introduction
- George K Parks(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
2 Equations and Definitions 2.1 Introduction AB shown in the previous chapter, space is endowed with a rich variety of electrodynamic phenomena. The problem to be studied in space, briefly stated, is how ionized particles interact among themselves and with electro-magnetic fields. The formal equations that are required for this study are now presented, with a short discussion of how some of the electrodynamic quantities are measured in space. 2.2 Maxwell Equations James Clerk Maxwell's unified theory of electromagnetic fields, published in 1864, is fundamental to this subject. The sources of electric and magnetic fields are distributions of charges and currents, which can be either discrete or continuous. With a charge density p and a vector current density J, 19 20 Chapter 2. Equations and Definitions the electrodynamic Maxwell equations in the mks (meter/kilogram/second) system are aB (2.1) 7 X E = --at an 7xH=J+8t (2.2) 'V·B=O (2.3) V'·D=p (2.4) By traditional usage, E and H are electric and magnetic field intensities, Dis the electric displacement and B the magnetic induction vector. E, B, D and H are vector functions defined in space and time (x, y, z, t) that obey the usual continuity and derivative rules of ordinary calculus. The relationships of these field vectors in vacuum are B=Ji.uH (2.5) and D= t:oE (2.6) Here, fo and Jl.o are constants equal to 8.85 X w-12 farads/meter and 411 x w-7 henrys/meter, respectively. The charge and current densities at any point in space and time are defined by (2.7) and (2.8) The summation is carried over a suitably chosen small volume element ~V. Qk and vk are the charge and velocity vectors, respectively, of the kth par-ticle. 2.3 Lorentz Equation of Motion The electromagnetic fields originate in charges p and currents J, and to describe the motion of these charges in space, the set of Maxwell equa-tions must be supplemented with the Lorentz equation of motion. For a
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Research World(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter-1 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter-4 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications tech-nologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- University Publications(Publisher)
____________________ WORLD TECHNOLOGIES ____________________ Chapter- 5 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications tech-nologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations
eBook - PDF- Luca Dal Negro(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
2.1 Maxwell’s Equations Maxwell’s equations, together with the Lorentz force law, are the foundation of clas- sical electrodynamics and describe how electric and magnetic fields are generated and modified by charges, currents, and by each other. The Maxwell-Lorentz equations describe the joint dynamics of the electromagnetic field and of a set of charged particles in vacuum. Maxwell’s equations relate the electric field E (r,t) [V/m] 1 and the magnetic flux density or magnetic induction 1 The symbols in square brackets indicate the units in which the preceding quantities are measured. 24 Electromagnetics Background B (r,t) [Wb/m 2 ] to the total charge density 2 ρ tot (r,t) [C/m 3 ] and current density J tot (r,t) [A/m 2 ]. In international system (SI) units, Maxwell’s equations are as follows: ∇ · E(r , t) = ρ tot (r , t) 0 ∇ · B(r , t) = 0 ∇ × E(r , t) = − ∂ B(r , t) ∂t ∇ × B(r , t) = 0 μ 0 ∂ E(r , t) ∂t + μ 0 J tot (r , t) (2.1) (2.2) (2.3) (2.4) In the preceding equations, 0 = 8.854×10 −12 [F/m] is the free-space electric per- mittivity, μ 0 = 4π × 10 −7 [H/m] is the free-space magnetic permeability, and the two constants are related to the speed of light in free space c 0 by the relation 0 μ 0 = 1/c 2 0 . 2.2 What Do They Mean? The divergence of a vector field at a given point measures the total outgoing flux (per unit volume) in the neighborhood of that point whereas the curl measures the total field rotation (per unit surface) around that point. Therefore, according to equation (2.1), the positive (negative) electric charge density at a given point acts as the source (sink) of the electric field. On the other hand, equation (2.2) tells us that there are no sources (or sinks) for the magnetic flux density. Equations (2.1) and (2.2) are known as the Gauss’s electric and magnetic laws, respectively. Equation (2.3) is Fara- day’s induction law, which establishes how a time-varying magnetic flux generates an electric field with rotation.
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