Physics
Lorentz Force Law
The Lorentz Force Law describes the force experienced by a charged particle moving through an electric and magnetic field. It states that the total force on the particle is the sum of the electric force and the magnetic force acting on it. Mathematically, the Lorentz Force Law is expressed as F = q(E + v x B), where F is the force, q is the charge, E is the electric field, v is the velocity, and B is the magnetic field.
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11 Key excerpts on "Lorentz Force Law"
eBook - PDF- Barton Zwiebach(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
The Lorentz Force Law ( 3.5 ) is a useful guide to the construction of the lower-dimensional theory. Suppose that there is no magnetic field. Then, in order to keep the z component of momentum equal to zero we must have E z = 0; the z component of the electric field must go. The case of the magnetic field is more surprising. Assume that the electric field is zero. If the velocity of the particle is a vector in the ( x , y ) plane, a component of the magnetic field in the plane would generate, via the cross product, a force in the z direction. On the other hand, a z component of the magnetic field would generate a force in the ( x , y ) plane! 48 Electromagnetism and gravitation We conclude that B x and B y must be set equal to zero, while we can keep B z . All in all, E z = B x = B y = 0 . (3.11) The left-over fields E x , E y , and B z can only depend on x and y . In the three-dimensional world with coordinates t , x , and y , the z index of B z is not a vector index. Therefore, in this reduced world, B z behaves like a Lorentz scalar (more precisely, it is an object called a pseudo-scalar). In summary, we have a two-dimensional vector E and a scalar field B z . We can test the consistency of this truncation by taking a look at the x and y components of ( 3.1 ): ∂ E z ∂ y − ∂ E y ∂ z = − 1 c ∂ B x ∂ t , ∂ E x ∂ z − ∂ E z ∂ x = − 1 c ∂ B y ∂ t . (3.12) Since the right-hand sides are set to zero by our truncation, the left-hand sides should vanish as well. Indeed, they do. Each term on the left-hand sides equals zero, either because it contains an E z , or because it contains a z derivative. You may examine the consistency of the remaining equations in Problem 3.3 . While setting up three-dimensional electrodynamics was not too difficult, it is much harder to guess what five-dimensional electrodynamics should be. As we will see next, the mani-festly relativistic formulation of Maxwell’s equations immediately gives the appropriate generalization to other dimensions.
eBook - PDF- David J. Griffiths(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
5.1.2 Magnetic Forces In fact, this combination of directions is just right for a cross product: the magnetic force on a charge Q, moving with velocity v in a magnetic field B, is 2 F mag = Q(v × B). (5.1) This is known as the Lorentz Force Law. 3 In the presence of both electric and magnetic fields, the net force on Q would be F = Q[E + (v × B)]. (5.2) I do not pretend to have derived Eq. 5.1, of course; it is a fundamental axiom of the theory, whose justification is to be found in experiments such as the one I described in Sect. 5.1.1. Our main job from now on is to calculate the magnetic field B (and for that matter the electric field E as well; the rules are more complicated when the source charges are in motion). But before we proceed, it is worthwhile to take a closer look at the Lorentz Force Law itself; it is a peculiar law, and it leads to some truly bizarre particle trajectories. Example 5.1. Cyclotron motion. The archtypical motion of a charged particle in a magnetic field is circular, with the magnetic force providing the centripetal acceleration. In Fig. 5.5, a uniform magnetic field points into the page; if the charge Q moves counterclockwise, with speed v, around a circle of radius R, the magnetic force points inward, and has a fixed magnitude Qv B —just right to sustain uniform circular motion: Qv B = m v 2 R , or p = QBR, (5.3) 2 Since F and v are vectors, B is actually a pseudovector. 3 Actually, it is due to Oliver Heaviside. 5.1 The Lorentz Force Law 213 y Q R x z v B F FIGURE 5.5 B v FIGURE 5.6 where m is the particle’s mass and p = mv is its momentum. Equation 5.3 is known as the cyclotron formula because it describes the motion of a particle in a cyclotron—the first of the modern particle accelerators. It also suggests a simple experimental technique for finding the momentum of a charged particle: send it through a region of known magnetic field, and measure the radius of its trajectory.
eBook - PDF- Saad Osman Bashir(Author)
- 2015(Publication Date)
- IntechOpen(Publisher)
Even Graneau’s exploding wires and Hering’s pump cause difficulties, when trying to use Lorentz’ force law in order to explain the effects that have been registered [6-11]. Therefore it is most exciting to explain one of the most frequent applications of Lorentz Force Law, the attractive force exerted between two parallel conductors carrying a DC current. Confessedly, © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. others have made efforts in this respect already near 200 years ago, most famous of them Ampère [12]. Successors, like Grassmann, have made serious efforts to make the Lorentz force (in his early pre-Lorentz formulation) appear to be in accordance with Ampère’s results [13]. In a more recent paper, this claim has been discarded through a mathematical analysis of Grassmann’s derivation [14]. Additionally, in order to introduce a new theory, it must be able not only to explain experiments that a recognized theory cannot, but also to explain the experiments that it apparently is successful in explaining. One crucial phenomenon is that of light, or electromagnetic radiation. In fact, it has been possible to explain this, too, using basically Coulomb’s law [15-18]. One may mention also electromagnetic induction [3-5]. The traditional methods have the benefit of being able to predict certain experiments, but not all. A new method must therefore in order to be better both done the first thing, but also be able to explain more evidence. By going back to the most basic well-corroborated law, Coulomb’s law, one would expect a possible solution, provided one is very careful and applies mathematical method in a very strict fashion.
- G.G. Emch(Author)
- 2000(Publication Date)
- North Holland(Publisher)
As in our approach to the Newtonian theory of gravitation (see Chapter One), we first concentrate our attention on the idealization where the scale of the phenomena to be considered is such t h a t the notion of test-particle makes sense. These are objects described by their trajectories x : t E I H x(t) E IR’ (with I IR), their inertial mass m, and their electric charge e. For these particles, the fact of experience referred to above is now quantifyable: these particles behave as if they were individually acted upon by a force, called the Lorentzforce, of the form where E(., t) and B(., t ) are vector fields on lR3 t h a t may depend explicitly on the time t. It is a part of the definition of a test particle t h a t its presence disturbs the fields E and B so little t h a t this disturbance can be ignored. In any experimental situation, prescribed by non-vanishing electric and mag- netic fields E and B, the Lorenta force (1) can be used to compare the charges of different test particles; this is done routinely-albeit at great costs-in both the preparation and the analysis of the final products of the scattering situa- tions t h a t constitute the heart of experimental elementary particle physics. Since enough experiments of this kind have been performed consistently, one can use (1) to probe E and B and verify t h a t these fields satisfy the so-called Maxwell equations. In an emphatically not accidental manner, these equations are the simplest field equations one can write in a relativistic theory: E and 3.1. PHENOMENOLOGY 89 B are the components of a 2-form 3 on space time, t h a t satisfy d 3 = 0 and defines J = d i v 3 ; see Sections 5.1 and 6.2. We shall take here the more traditional description and write the Maxwell equations in the form curl B - E = J div E = p curl E + B = 0 div B = 0 One can view here (2a) and (2b) as definitions of J and p, and note t h a t they imply (3) div J + p = O .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
The second equation corresponds to the two remaining equations, Gauss's law for magnetism (for α = 0 ) and Faraday's Law (for α = 1,2,3). This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors. ________________________ WORLD TECHNOLOGIES ________________________ Properties of the field Reciprocal behavior of electric and magnetic fields The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator. The Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor. Light as an electromagnetic disturbance Maxwell's equations take the form of an electromagnetic wave in an area that is very far away from any charges or currents (free space) – that is, where ρ and are zero. It can be shown, that, under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation: James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's Circuital law. Relation to and comparison with other physical fields Fundamental Forces In physics, fundamental interactions (sometimes called interactive forces ) are the ways that the simplest particles in the universe interact with one another. An interaction is fundamental when it cannot be described in terms of other interactions. The four known fundamental interactions, all of which are non-contact forces, are electromagnetism, strong interaction, weak interaction (also known as strong and weak nuclear force) and gravitation.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Properties of the field Reciprocal behavior of electric and magnetic fields The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field Faraday's Law may be stated roughly ________________________ WORLD TECHNOLOGIES ________________________ as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator. The Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor. Light as an electromagnetic disturbance Maxwell's equations take the form of an electromagnetic wave in an area that is very far away from any charges or currents (free space) – that is, where ρ and are zero. It can be shown, that, under these conditions, the electric and magnetic fields satisfy the electro-magnetic wave equation: James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's Circuital law. Relation to and comparison with other physical fields In physics, fundamental interactions (sometimes called interactive forces ) are the ways that the simplest particles in the universe interact with one another. An interaction is fundamental when it cannot be described in terms of other interactions. The four known fundamental interactions, all of which are non-contact forces, are electromagnetism, strong interaction, weak interaction (also known as strong and weak nuclear force) and gravitation. With the possible exception of gravitation, these interactions can usually be described, in a set of calculational approximation methods known as perturbation theory, as being mediated by the exchange of gauge bosons between particles. However, there are situations where perturbation theory does not adequately describe the observed phenomena, such as bound states and solitons.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
Properties of the field Reciprocal behavior of electric and magnetic fields The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field Faraday's Law may be stated roughly ________________________ WORLD TECHNOLOGIES ________________________ as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator. The Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor. Light as an electromagnetic disturbance Maxwell's equations take the form of an electromagnetic wave in an area that is very far away from any charges or currents (free space) – that is, where ρ and are zero. It can be shown, that, under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation: James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's Circuital law. Relation to and comparison with other physical fields In physics, fundamental interactions (sometimes called interactive forces ) are the ways that the simplest particles in the universe interact with one another. An interaction is fundamental when it cannot be described in terms of other interactions. The four known fundamental interactions, all of which are non-contact forces, are electromagnetism, strong interaction, weak interaction (also known as strong and weak nuclear force) and gravitation. With the possible exception of gravitation, these interactions can usually be described, in a set of calculational approximation methods known as perturbation theory, as being mediated by the exchange of gauge bosons between particles. However, there are situations where perturbation theory does not adequately describe the observed phenomena, such as bound states and solitons.
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.
eBook - PDFBetween Quantum and Cosmos
Studies and Essays in Honor of John Archibald Wheeler
- Alwyn Van der Merwe, Wojciech Hubert Zurek, Warner Allen Miller, Alwyn Van der Merwe, Wojciech Hubert Zurek, Warner Allen Miller, Alwyn Van der Merwe, Wojciech Zurek, Warner Miller(Authors)
- 2017(Publication Date)
- Princeton University Press(Publisher)
The radiation emitted by the perturbed electrons along the direction of the external magnetic field (B) consists of two circularly polarized components with fre- quencies a>±eB/(2m t ,c), just as observed by Zeeman in his early experiments. In fact, Zeeman's discovery led to the determination of the charge (— e) to mass (me ) ratio for the electron. The results of the Lorentz theory may be interpreted in a simple way (Larmor, 1899) by using Larmor's theorem (1897) which, for the present purposes, may be stated as follows: The classical motion of a particle of mass m and charge q in a weak magnetic field B is equivalent to the motion of the particle with respect to a coordinate system rotating uniformly with angular frequency il L = q^Hlmc). The significance of Larmor's theorem lies in the fact that it relates the measurements of an inertial observer in the presence of a weak magnetic field to those of a uniformly rotating observer. Let the magnetic field be along the x 3 -axis, so that the Larmor frequency of the noninertial observer is in the negative .redirection with magnitude Ql = eB/(2me c). Electromagnetic radiation of frequency ω traveling along the x 3 -axis can be decomposed into two components each of frequency ω and with opposite states of circular polarization. The rotating observer will therefore measure two components with circular polarization: a positive 372 Between Quantum and Cosmos helicity component of frequency ω + Q l and a negative helicity component of frequency ω —Q l . The inverse process is also expected to hold, and this would be consistent with the reciprocity between the emission and absorption of radiation. An interesting historical account of these researches is contained in Ref.
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
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