Physics
Magnetic Vector Potential
The magnetic vector potential is a fundamental concept in physics used to describe the magnetic field in terms of a vector quantity. It is related to the magnetic field through the equation B = curl(A), where B is the magnetic field and A is the magnetic vector potential. This concept is particularly useful in the study of electromagnetism and quantum mechanics.
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eBook - ePubElectromagnetics Explained
A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
- Ron Schmitt(Author)
- 2002(Publication Date)
- Newnes(Publisher)
Chapter 6 , you will learn more about the vector potential when we discuss quantum physics.MAGNETIC MATERIALS
Diamagnetism
In Chapter 2 , you learned that different materials behave differently in electric fields. You learned about conductors and dielectrics. Electric fields induce reactions in materials. In conductors, charges separate and nullify the field within the conductor. In dielectrics, atoms or molecules rotate or polarize to reduce the field. Magnetic fields also induce reactions in materials. However, since there are no magnet charges, there is no such thing as a “magnetic conductor.” All materials react to magnetic fields similarly to the way dielectrics react to electric fields. To be precise, magnetic materials usually interact with an external magnetic field via dipole rotations at the atomic level. For a simple explanation, you can think of an atom as a dense positive nucleus with light electrons orbiting the nucleus, an arrangement reminiscent of the planets orbiting the sun in the solar system. Another similar situation is that of a person swinging a ball at the end of a string. In each situation, the object is held in orbit by a force that points toward the orbit center. This type of force is called a centripetal force. The force is conveyed by electricity, gravity, or the string tension, respectively, for the three situations. Referring to Figure 3.17 and using the cross product right hand rule, you find that the force due to the external magnetic field points inward, adding to the centripetal force. The increase in speed increases the electron’s magnetic field, which is opposite to the external field. The net effect is that the orbiting electron tends to cancel part of the external field. Just as the free electron rotates in opposition to a magnetic field, the orbiting electron changes to oppose the magnetic field. This effect is called diamagnetism
eBook - PDF- Akira Tonomura(Author)
- 1998(Publication Date)
- World Scientific(Publisher)
Chapter 10 VECTOR POTENTIALS, REAL OR NOT? Aharonov and Bohm asserted that the AB effect was due to vector potentials. What are vector potentials? In the 1960s we learned at universities that vector potentials were merely a mathematical tool with which electromagnetic problems could be solved easily. Now, however, vector potentials are regarded as the most fundamental physical quantity in unified theories of all fundamental forces in nature. There, vector potentials are called gauge fields. In this chapter, I would like to give you a physical image of vector potentials and the history of vector potentials and then to tell you the story of our efforts until the physical reality or significance of vector potentials was established. What are Vector Potentials? Faraday introduced the concept of magnetic lines of force. The density of magnetic lines of force is called a magnetic field, or magnetic flux density. This quantity has not only a magnitude but also a direction. A physical quantity having both magnitude and direction is called a vector and is conventionally written with a bold and italic letter, such as B. The vector B has the direction of a magnetic line of force and has a magnitude proportional to the density of magnetic lines. What then are vector potentials? The vector potential A is mathematically related to magnetic field B as follows: B = rot A. (15) If you are not familiar with mathematics, you may be puzzled at this equation. But please don't hesitate. It is only a representation of the relation between two quantities. Here rot is the abbreviation of rotation. The meaning of this equation is as follows. If a distribution of A has a rotational component like a vortex, then the magnitude of B is given by the strength of the vortex and the direction of B is along a line perpendicular to the vortex plane. The vector B cannot yet be determined, since the line has two opposite directions.
eBook - PDFMagnetic Fields
A Comprehensive Theoretical Treatise for Practical Use
- Heinz E. Knoepfel(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
Figure 2.2-2 Spatial current distribution. 2.2 Magnetic Vector Potential 81 Infigure 2.2-3 the relations between the current density j, the magnetic flux B, and the scalar andvector potentials 4,A are represented schematically. The Magnetic Vector Potential is very attractive for two-dimensional plane problems or axisymmetric magnetic field problems (where the current is purely axial j = jle, or azimuthal j= j4e,) because it then has only onecomponent (Az, or Ah), which is orthogonal to the plane containing the magnetic field vector, as we shall see. For three-dimensional problems, allthree components of the vector potential mustingeneral be computed [from a system ofthree coupled, partial differential equations derived from the vectorial Poisson equation (2.2-13)], and thus thevector potential becomes less useful than the magnetic scalar potential (whenever the latter exists). The vector potential hasthe merit, in anycase, of being applicable both in free space and within current-carrying conductors andavoiding the multiple-valued problem of the scalar potential. n Figure 2.2-3 The magnetostatic equations relating current density j, magnetic induction B = pH, scalar andvector potentials 0, A, with the equations shown (with reference totheir position in the chaptery; the potentials canbe convenient intermediate functions for simplifying the solution [CP isdefined in free space, so there is nogeneral meaningful relation to j in the conductor]. 82 CHAPTER 2 MAGNETIC POTENTIALS Indiscussing the general solution ofthe vector potential problem, we note that the integral form (2.2-21) is the most useful to start from because it is valid in spaces with or without currents, and the integration extends onlyovertheconductor. An important advantage of solving magnetic field problems through the vector potential formalism is that this integral has a simple structure andis relatively easy tohandle.
- K Umashankar(Author)
- 1989(Publication Date)
- WSPC(Publisher)
Based on the duality concepts, various relationships between the vector magnetic field quantity and the corresponding scalar magnetic potential function are established. It can be shown easily that the static magnetic field is just a gradient of the scalar potential function. Particularly in many practical magnetostatic field problems, where the information on the electric source current distribution is not known, the approach based on gradient of the magnetostatic potential is quite useful. It is also observed that the representation of the vector magnetic field in terms of the gradient of a scalar potential is not just sufficient where electric current sources are present. It is well known that the electric current source distribution 305 306 INTRODUCTION TO ENGINEERING ELECTROMAGNETIC FIELDS which produces the vector magnetic field is a vector point function. Hence, a vector potential function, referred to as the Magnetic Vector Potential, is introduced in order to express rigorously the magnetic field distribution in a given medium. Similar to the case of static electric field, it is possible to treat the study of static magnetic field as a boundary value problem satisfying either a differential equation or an integral equation. In many practical problems, generally, the boundary conditions on the potential distribution will be known at certain boundary locations. One can easily derive either the Poisson's differential equation or the Laplace's differential equation for the Magnetic Vector Potential function in a magnetostatic boundary value problem. The known boundary conditions can be utilized as initial conditions to solve these partial differential equations. Once the vector potential distribution is known, then the static magnetic field distribution can be calculated based on the curl of the Magnetic Vector Potential obtained from the Gauss's law.
eBook - PDFGeophysical Field Theory and Method, Part A
Gravitational, Electric, and Magnetic Fields
- Alexander A. Kaufman(Author)
- 1992(Publication Date)
- Academic Press(Publisher)
IV.2 The Vector Potential of the Magnetic Field 407 vector potential caused by volume, surface, and linear currents. (IV.24) The components of the vector potential can be derived directly from this equation. For instance, in Cartesian coordinates we have /-La [f i, dV f i y dS N rr. dey] A =----+ --+ '[J'Y-y 47T v L q p S L qp ;=1 I L qp (IV.25) Similar expressions can be written for the vector potential components in other systems of coordinates. As is seen from Eqs. (IV.25), if a current flows along a single straight line, the vector potential has only one component, which is parallel to this line. It is also obvious that if currents are situated in a single plane, then the vector potential A at every point is parallel to this plane. Later we will consider several examples illustrating the behavior of the vector potential and the magnetic field B, but now let us derive two useful relations for the function A, which simplify to a great extent the task of deriving the system of the magnetic field equations. First, we will determine the divergence of the vector potential A. As follows from Eq. (IV.22), we have P p /-La j(q) divA = div-f --dV 47T v L q p Since differentiation and integration in this expression are performed with respect to different points, we can change the order of operations and then obtain P P ( ) divA = ~ f div~dV 47T v L q p (IV.26) The volume over which the integration is carried out includes all currents, and therefore it can be enclosed by a surface S such that outside 408 IV Magnetic Fields of it currents are absent. Correspondingly, the normal component of the current density at this surface equals zero. jn = 0 on S The integrand in Eq, (IV.26) can be represented as (IV.27) (IV.29) (IV.28) p p j Vj. pl. p 1 V-=-+J'V-=J'V-i-; -; L q p -: because the current density does not depend on the observation point.
eBook - PDFElectromagnetics through the Finite Element Method
A Simplified Approach Using Maxwell's Equations
- José Roberto Cardoso(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
is called magnetization vector , which is represented as (Figure 2.17) 1 � � � B H M + . = m 0 Therefore, B = m 0 ( H + M ). (2.27) In linear medium, the magnetization vector M is directly proportional to the mag- netic intensity vector; thus, M = c m H where χ m is magnetic susceptibility of the medium. Replacing its value in Equation 2.27, we get B = m 0 (1 + c m ) H. Since B H , we get = m m m = 0 ( 1 + c m ). (2.28) 2.3 QUANTITIES ASSOCIATED WITH FIELD VECTORS The resolution of electromagnetic problems needs some quantities associated with electromagnetic felds. Some of these quantities are frequently used than the vector felds, because it is possible to measure them by instruments. For example, voltage � � � � � � � � � � � � � 35 Fundamentals of Electromagnetism drop between two points in a circuit is measured by a voltmeter in some situations. Frequently, we do not need to know the electric or the magnetic feld that creates them. 2.3.1 VOLTAGE BETWEEN T WO POINTS Figure 2.18 shows two points A and B located in a region surrounded by an electric feld E . With the electric feld, we are able to evaluate the work involved in dragging the charge q on the path AB, which displacement direction is opposed to the direction of the electric feld. Thus, it is necessary the presence of an external agent for taking the charge between that points, because the action of the electric feld is opposite to what we propose. Thus, the external agent that drags the charge should apply a force F ext greater than the force F applied by the electric feld on the charge. The force resulting from the composition of F ext and F should have a component tangent to the line of the charge movement. In addition, if no dragging is required to move the charge along the path AB, that is, the movement is very slow, it implies that the time necessary for dragging the charge from A to B is virtually infnite.
eBook - PDF- Edward Purcell(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
6.1 Definition of the Magnetic Field 6.2 Some Properties of the Magnetic Field 6.3 Vector Potential 6.4 Field of Any Current-Carrying Wire 6.5 Fields of Rings and Coils 6.6 Change in B at a Current Sheet 6.7 How the Fields Transform 6.8 Rowland's Experiment 6.9 Electric Conduction in a Magnetic Field: The Hall Effect Problems THE MAGNETIC FIELD 208 214 220 223 226 231 235 241 241 245 208 CHAPTER SIX DEFINITION OF THE MAGNETIC FIELD 6.1 A charge which is moving parallel to a current of other charges experiences a force perpendicular to its own velocity. We can see it happening in the deflection of the electron beam in Fig. 5.3. We dis- covered in Section 5.9 that this is consistent with-indeed, is required by-Coulomb's law with charge invariance and special relativity. And we found that a force perpendicular to the charged particle's velocity also arises in motion at right angles to the current-carrying wire. For a given current the magnitude of the force, which we calculated for the particular case in Fig. 5.20a, is proportional to the product of the particle's charge q and its speed v in our frame. Just as we defined the electric field E as the vector force on unit charge at rest, so we can define another field B by the velocity-dependent part of the force that acts on a charge in motion. The defining relation was introduced at the beginning of Chapter 5. Let us state it again more carefully. At some instant t a particle of charge q passes the point (x, y, z) in our frame, moving with velocity v. At that moment the force on the particle (its rate of change of momentum) is F. The electric field at that time and place is known to be E. Then the magnetic field at that time and place is defined as the vector B which satisfies the vector equation F = qE + 1v X B c (1) Of course, F here includes only the charge-dependent force and not, for instance, the weight of the particle carrying the charge.
- Stephen H. Hall, Howard L. Heck(Authors)
- 2011(Publication Date)
- Wiley-IEEE Press(Publisher)
Magnetic Vector Potential . The Magnetic Vector Potential is used to calculate inductance, which is in turn used to calculate the energy stored in a magnetic field.The basic laws that rule magnetostatics are the time-invariant forms of Ampère’s and Gauss’s laws for magnetism:(2-82a)(2-82b)If the following vector identity is applied (from Appendix A),it implies that can be written in terms of the curl of a vector . This is called the Magnetic Vector Potential , which sometimes helps to simplify calculations such as inductance:(2-83)To calculate the form of , it is first necessary to introduce the most basic law of magnetostatics, the Biot-Savart law , which describes how the field at a given point is produced by the moving charges in the vicinity of that point. During application of this law, we consider currents that are either static or very slowly varying with time. The Biot-Savart law is given by [Inan and Inan, 1998](2-84)where I is the current, a unit vector pointing from the location of the differential current element Id to the point P , and R = the distance between the current element and point P . Note that a primed quantity represents the position vector or the coordinates of the source points and the unprimed quantities represent the position vector or the coordinates of the point where is being evaluated. Note that the cross product in (2-84) indicates that d is perpendicular to both Id and , with the orientation being described by the right-hand rule. Also note that the field generated by Id will fall off with the square of the distance.Note that the current element Id is a small part of a closed current loop and that an arbitrarily shaped loop can be constructed by the superposition of many such closed elemental loops of current, as shown in Figure 2-18 . Subsequently, the Biot-Savart law can be written in terms of an integral:(2-85)Careful observation of (2-85) allows us to calculate the form of the Magnetic Vector Potential . Equation (2-83) states that the magnetic field is the curl of . The trick is to manipulate (2-85) into the form of a curl so that can be found. The right-hand term of (2-85)
- K. Kurokawa(Author)
- 1969(Publication Date)
- Academic Press(Publisher)
C H A P T E R 2 ELECTROMAGNETIC FIELD VECTORS In order to discuss microwave circuits, Maxwell's equations must be studied since these describe the relations between electric and magnetic fields. If the concept of a vector is introduced, the relations between the fields become simpler to describe and easier to understand. Therefore, we shall make extensive use of vectors in this book. This chapter reviews some of the important theorems on vector analysis which will facilitate our later study. In Section 2.1, elements of vectors and vector analysis are presented, and in Section 2.2, Maxwell's equations are explained in terms of the field vectors in order to refresh the reader's understanding of electromagnetic theory. Section 2.3 gives an analysis of plane waves, first in terms of scalar quantities resolving the field vectors into their components, and then in terms of vectors after which the results are compared. 2.1 Vectors A vector is a quantity with magnitude and direction. To visualize it, we usually consider an arrow whose length and direction expresses the magnitude and direction of the vector, respectively, as shown in Fig. 2.1. The space of A Fig. 2.1. Arrow representing vector A. 40 2.1. Vectors 41 this figure is not the actual space in which we live, but rather it is an abstract space where the magnitude and direction of the vector is represented by the length and direction of the corresponding arrow. Vectors are generally functions of time and position existing in actual space, therefore, depending on the position and time, the lengths and the directions of the vector arrows vary in abstract space. Sometimes however, the tail of the arrow is located at the point in the actual space where the vector is considered. An example will be shown in Fig. 2.10. In such a case, the illustrated space has two meanings; one is the actual space, and the other is the abstract space in which we consider the arrows representing vectors.
- Rajat Acharya(Author)
- 2017(Publication Date)
- Academic Press(Publisher)
Chapter 2Maxwell's equation and EM waves
Abstract
This chapter deals with the Maxwell's equations. This is the basic technical prerequisite for the rest of the book. It starts with the statement of the equations and explains the significance of each with different forms that these equations can take. Using these equations and imposing definite conditions, both the static, electric and magnetic fields are derived and discussed to explain different aspects of them. Then the dynamic case of generation of electromagnetic fields and waves is addressed. Both the electric and magnetic field component of the wave and the scalar and vector potentials are thoroughly discussed. Finally, the Poynting theorem is explained and the Poynting vector is defined.Keywords
Electric fields; Magnetic fields; Electromagnetic wave; Vector potential; Poynting vector2.1 Electric and Magnetic Fields
In this book, we shall deal with the satellite signals which are basically electromagnetic (EM) waves. These waves traverse through space as simultaneous electric and magnetic fields. Each of these fields gets generated from the variations of the other field and they are mutually orthogonal in direction. To understand the signals, it is necessary at the outset to clearly define and explain the origin of the electric and magnetic fields constituting the wave. It is important to realize how these fields are sustained and how they behave, as they propagate through free space or through a medium. In this chapter, we shall learn about the EM wave propagation and its behaviour in free space.By the term ‘Field’, in a strict sense, we actually mean any physical quantity which can be specified at all points in the physical space of our interest. It may take different values at different points in this region and may vary with time (Galbis and Maestre, 2012
- J. Felipe de Almeida(Author)
- 2021(Publication Date)
- Editora Dialética(Publisher)
V and measured in SI at Newton times meter per Coulomb (Joule/Coulomb), which is defined as Volt, and the electric field unit is Volt per meter (V/m). In terms of field intensity, we can write. (3.36)Example 3.11 : Consider the electric E = x i –y j . Determine the work to move a charge Q =1 C, from (1, 0, 0), across the (0, 0, 0) to (0, 1, 0).Solution : We have that d w=–Q E· d r . However, d r =dx i + dy j + dz k and to the x -axis implies to dy = dz =0. Then, d w=–x i · dx i =–xdx . Similarly, to the y -axis: dy = dz =0 and d w=y j · dy j =ydx . Hence,When the source of the origin of E is positioned in space, that is, it involves the three dimensions x , y and z , the electric potential depends on these same coordinates, i.e. V (x , y , z ). Thus, the coordinates of the electric field are obtained by the following equation of partial derivatives:. (3.37)These intensity values represent the spatial components of the electric field vector E = E x i + E y j + E x k . So,. (3.38)Which can be written more compactly using the operator Ñ , i.e.(3.39)The electric potential is a scalar function of a certain position, and therefore, its location in space is made by the position vector r . This result is called the electrical potential gradient (also written as grad V ) and is extremely valid for measurements of electrical interactions.One way to interpret geometrically Equation (3.39) is based on the fundamental calculus theorem. For instance,(3.40)Consider now the displacement of a charge in any path and subjected to the action of a field of electric forces E . Consider also that, the displacement Dl is made along a path l joined from the position of a point determined by r 1 (A ) to another (B ) by r 2
eBook - PDFElectric Machines
Theory and Analysis Using the Finite Element Method
- Dionysios Aliprantis, Oleg Wasynczuk(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
(1.254) Recall that the physical constant ǫ 0 ≈ 8.854 · 10 −12 F/m (farads per meter) (1.255) is the permittivity of free space. We also know that the curl of the electrostatic field is zero: ∇ × #» E = 0 . (1.256) 60 1 Review of Vector Calculus and Electromagnetic Fields Using Helmholtz’s theorem, we can deduce that the vector potential is zero, and thereby we express the electrostatic field as #» E (r) = −∇ϕ(r) , (1.257) where the scalar potential is measured in V, and is given by ϕ(r) = 1 4πǫ 0 V ρ(r ′ ) ‖r − r ′ ‖ dv ′ . (1.258) This integral provides the electrostatic potential at a point r ∈ V . (See also Exam- ple 1.29, where the same result was obtained by integration of Poisson’s equation.) The potential can be thought of as the superposition of contributions from all elementary charges dq(r ′ ) = ρ(r ′ ) dv ′ , with r ′ ∈ V . Now, consider the magnetostatic field in vacuum, where the role of #» F is played by the magnetic field density #» B , which is measured in T. According to the fundamental postulates of electromagnetism, the divergence of the B-field is always zero: ∇ · #» B = 0 , (1.259) whereas the curl satisfies Amp` ere’s law: ∇ × #» B = µ 0 #» J . (1.260) Here, the physical constant µ 0 = 4π10 −7 H/m (1.261) is the permeability of free space. The source of the field is the current density #» J , which is measured in A/m 2 . Therefore, Helmholtz’s theorem yields #» B (r) = ∇ × #» A(r) , (1.262) with the MVP #» A(r) = µ 0 4π V #» J (r ′ ) ‖r − r ′ ‖ dv ′ , (1.263) which may be measured in T·m or Wb/m (webers 31 per meter). This volume integral provides the MVP at a point r ∈ V , where r ′ is the variable of integration. Integration is performed component-wise for each of three separate scalar potentials. For instance, the x-component of the current, J x , only affects the x-component of the MVP, A x . The potential can be thought of as the superposition of contributions from all elementary current elements #» J (r ′ ) dv ′ , with r ′ ∈ V .
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