Physics
Magnetostatics
Magnetostatics is the study of magnetic fields in systems that are not changing with time. It deals with the behavior of magnetic fields in materials and their interactions with electric currents. Key concepts include the magnetic field produced by a current-carrying wire, the magnetic force on a moving charged particle, and the behavior of magnetic materials in external magnetic fields.
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11 Key excerpts on "Magnetostatics"
eBook - ePub- Anupam Garg(Author)
- 2012(Publication Date)
- Princeton University Press(Publisher)
4 Magnetostatics in vacuumOperationally, a magnetic field B may be defined via the Lorentz force law.1 Whatever a magnetic field is, it is something to which a moving charge responds as per this law. By a vague principle of reciprocity, one might expect a moving charge to be a source of magnetic field. This turns out to be correct, but a static magnetic field requires not one moving charge, but a steady current of charges. This makes the mathematical treatment of even the simplest magnetostatic fields quite different from that of electrostatics.In addition, however, magnetic fields are also created by point magnetic dipoles.2 Two such dipoles interact in the same way as two electric dipoles. This provides a major similarity with electrostatics, and we shall adopt this as our point of entry for the study of Magnetostatics. But magnetic dipoles are profoundly different from electric dipoles in that they produce a divergenceless field. The resulting subtleties are studied in section 26.The existence of two types of sources of magnetic field obligates us to study the relationship between them. We shall use the Lorentz force law for this purpose. We shall also see how magnetic fields may be described by potentials.The study of magnetic work and energy is deferred to the next chapter.21 Sources of magnetic field
That like poles of two magnets repel and unlike poles attract has been known since antiquity. In 1785, Coulomb established by careful experiments that the force between two very long and thin magnetic rods can be described in terms of poles residing at points at the ends of the rods, and that this force has exactly the same form as that between two point electric charges. Since, however, the poles of a magnet are always equal and opposite, and a single pole or two unbalanced poles can never be created by breakage or any other treatment of the magnet, a mental synthesis of a magnet must be sought not in terms of isolated magnetic charges, but in terms of infinitesimally separated pairs of equal and opposite charges, i.e., dipoles. By the process of abstraction, we are naturally led to consider point dipoles.3
eBook - PDF- David J. Griffiths(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Of course, there’s no such thing in practice as a truly steady current, any more than there is a truly stationary charge. In this sense, both electrostatics and Magnetostatics describe artificial worlds that exist only in textbooks. However, they represent suitable approximations as long as the actual fluctuations are remote, or gradual—in fact, for most purposes Magnetostatics applies very well to household currents, which alternate 120 times a second! 8 Actually, it is not necessary that the charges be stationary, but only that the charge density at each point be constant. For example, the sphere in Prob. 5.6(b) produces an electrostatic field 1/4π 0 ( Q/r 2 ) ˆ r, even though it is rotating, because ρ does not depend on t . 224 Chapter 5 Magnetostatics Notice that a moving point charge cannot possibly constitute a steady current. If it’s here one instant, it’s gone the next. This may seem like a minor thing to you, but it’s a major headache for me. I developed each topic in electrostatics by starting out with the simple case of a point charge at rest; then I generalized to an arbitrary charge distribution by invoking the superposition principle. This approach is not open to us in Magnetostatics because a moving point charge does not produce a static field in the first place. We are forced to deal with extended current distributions right from the start, and, as a result, the arguments are bound to be more cumbersome. When a steady current flows in a wire, its magnitude I must be the same all along the line; otherwise, charge would be piling up somewhere, and it wouldn’t be a steady current. More generally, since ∂ρ/∂ t = 0 in Magnetostatics, the con- tinuity equation (5.29) becomes ∇ · J = 0. (5.33) 5.2.2 The Magnetic Field of a Steady Current The magnetic field of a steady line current is given by the Biot-Savart law: B(r) = μ 0 4π I × ˆ r r 2 dl = μ 0 4π I d l × ˆ r r 2 .
- Roland H. Good, Terence J. Nelson(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
Chapter V Magnetostatics Magnetostatics is different from electrostatics because the magnetic field has zero divergence whereas the electric field has zero curl. Nevertheless, many of the concepts in the two subjects are parallel and it is interesting to keep the electrostatic ideas in mind while studying Magnetostatics and watch the simularities and the contrasts between the two subjects. If the sources are confined to some finite region, then, outside that region, both fields have zero divergence and zero curl and hence may be treated the same way. They differ there only in the way that the fields are related to the sources. Many of the calculation techniques of electrostatics have parallels in Magnetostatics. In electrostatics, however, equipotential surfaces exist and can be realized with conductors; this leads to many interesting problems but does not have a counterpart in Magnetostatics. To avoid repetition, we emphasize in this chapter the aspects of Magnetostatics that are essentially different from electrostatics. We begin with a discussion of the magnetic dipole field, how it differs from the electric dipole field at the source and how this leads to the Fermi 218 17. THE MAGNETIC DIPOLE contact interaction. Then we discuss magnetized materials and the various ways of making the multipole expansion of the fields of given sources. 17* THE MAGNETIC DIPOLE The electrostatic field is produced by charges and the simplest field is that of a point charge. The magnetostatic field is produced by steady currents and the simplest example is the field of a magnetic dipole. Field of a Magnetic Dipole Consider a closed circuit in which a current / flows. A surface capping the circuit can be chosen with an outward normal related to the direction of the current by the right-hand rule. The magnetic moment of the circuit is defined to be m = I -j & d*.
- S. B. Lal Seksena, Kaustuv Dasgupta(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Prerequisite knowledge Basics of vector algebra Electrostatic field (Chapter 3) 5.1 Introduction In chapter 3, we have discussed about four different force fields. Magnetic force field is one of those force field exists in the universe. The Magnetostatic deals with the magnetic field. Electrical engineers are concerned about the magnetic field as much as electrical field. These two fields are associated with each other. Like electric field, magnetic field is also inverse square law field. We define different parameters of magnetic field which are analogous to electric field. These two fields are inter-related. Magnetic field plays a very important role to deliver electrical power to our useful mechanical system. This develops the importance to study both the electrical and magnetic field together. Electromagnetism is the branch of science and technology where we study the effects of magnetic field on the electric field and vice-versa. We have studied the alternating current in chapter 4. Any alternating current is associated with an alternating magnetic field. Alternating magnetic field again induces another electric field. In this chapter we shall study the properties and nature of magnetic field and also its connection to electric field. When an electric charge is in motion it produces the magnetic field. 5.2 Different Physical Parameters Related to Magnetic Field 5.2.1Magnetic field We have already defined the electric field and field intensity with respect to a positive unit electric charge. When an electric charge is in motion it creates a magnetic field. The magnetic field also exerts a force on the charge. So a charge Q with velocity V – will experience force – F both by the electric field B – and magnetic field B – . The combined force which will be acted upon the charge Q is known as Lorentz force. We know the force experienced by charge Q in electric field E – .
eBook - PDFEngineering Electromagnetics
Pergamon Unified Engineering Series
- David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
7 Magnetostatic Fields INTRODUCTION By now you should be thoroughly familiar with concepts of vectors, fields and the nature of electrostatic fields as caused by charges. Recalling the definition of E, the electric field at a point, P, is the hypothetical force on a unit charge located at P. In like manner, the magnetic flux density, B, is a force field defined as the force, F, at any point which would be exerted on a unit magnetic pole. The nature of a magnetic pole must be determined from our basic equations for such fields, Maxwell's Equations, and just to keep you from excruciating suspense, let me say that the magnetic pole does not exist. However, as we shall discover magnetic dipoles do exist, and are in a sense the sources of magnetic fields. A much more practical quantity can, however, be considered the source of magnetic fields; namely, the current or current density, J, discussed in the previous chapter. Because of the indirect connection between current and mag-netic field, the nature of magnetic fields is in many ways different from electro-static fields. Magnetic fields in one sense are dependent on electrostatic fields or electric phenomena, since a current is basically a moving collection of charges. This relativistic approach to magnetic fields is a perfectly valid one, and has been exploited in some recent texts, including Elliott 1 . The more standard approach will be employed here, as beginning students probably do not understand rela-tivity sufficiently well to comprehend that approach. MAGNETOSTATIC FIELD EQUATIONS As before in electrostatics, we place two restrictions on the fields to be considered: (1) there is no time dependence, hence terms involving time de-rivatives are eliminated, and (2) only magnetic fields (B, H) are considered. This 1 Elliott, R. S., Electromagnetics, McGraw-Hill, 1967. 215 216 Magnetostatic Fields is possible as the static electric and magnetic fields can be separated and con-sidered independently.
- Luis Manuel Braga da Costa Campos(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
26 Magnetostatics, Currents, and Permeability The magnetostatic field (Chapter 26) like the electrostatic field (Chapter 24) is specified by three related vectors, viz. the magnetic field, induction, and polarization (Section 26.1); there are differences in the constitutive properties relating them that are specified by the magnetic permeability and susceptibility (Section 26.2). These differences ultimately arise from Maxwell’s equations: the electric (magnetic) field is created by electric charges (cur-rents), and is analogous to the potential flow due to source/sinks (vortices), leading (Section 26.3) to a Coulomb (Biot–Savart force) that acts along (across) the relative position vector. In follows that magnetic multipoles interchange the field lines and equipotentials of elec-tric multipoles (Section 26.4), and reverse the alternating and identical images on planes (Section 26.5), corners (Section 26.6), and cylinders (Section 26.7). Bearing in mind these differences of detail, the general methods are similar, for example, for a cylindrical interface between media with distinct magnetic permeabilities (Section 26.8) and the generation of the field by multipole distributions with finite or infinite extent (Section 28.9). The two building blocks to be combined in the electromagnetic phenomena like waves and circuits are best studied separately in electrostatics (Chapter 24) and Magnetostatics (Chapter 26). The latter have analogies and differences, for example, the magnetic field is always solenoidal and the electric field is irrotational in steady conditions.
eBook - PDF- Rajeev Bansal(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Recording and storing various data are most commonly accomplished using y Deceased. — Milica and Zoya Popovic´ Montre ´al, Quebec Boulder, Colorado 89 magnetic storage components, such as computer disks and tapes. Most household appliances, as well as industrial plants, use motors and generators, the operation of which is based on magnetic forces. The goal of this chapter is to present: Fundamental theoretical foundations for Magnetostatics, most importantly Ampere’s law Some simple and commonly encountered examples, such as calculation of the magnetic field inside a coaxial cable A few common applications, such as Hall element sensors, magnetic storage, and MRI medical imaging. 3.2. THEORETICAL BACKGROUND AND FUNDAMENTAL EQUATIONS 3.2.1. Magnetic Flux Density and Lorentz Force The electric force on a charge is described in terms of the electric field vector, E . The magnetic force on a charge moving with respect to other moving charges is described in terms of the magnetic flux density vector , B . The unit for B is a tesla (T). If a point charge Q [in coulombs (C)] is moving with a velocity v [in meters per second (m/s)], it experiences a force [in newtons (N)] equal to F ¼ Q v B ð 3 : 1 Þ where ‘‘ ’’ denotes the vector product (or cross product) of two vectors. The region of space in which a force of the form in Eq. (3.1) acts on a moving charge is said to have a magnetic field present. If in addition there is an electric field in that region, the total force on the charge (the Lorentz force) is given by F ¼ Q E þ Q v B ð 3 : 2 Þ where E is the electric field intensity in volts per meter (V/m). 3.2.2. The Biot^Savart Law The magnetic flux density is produced by current-carrying conductors or by permanent magnets. If the source of the magnetic field is the electric current in thin wire loops, i.e. current loops, situated in vacuum (or in air), we first adopt the orientation along the loop to be in the direction of the current in it.
eBook - PDF- Rajeev Bansal(Author)
- 2004(Publication Date)
- CRC Press(Publisher)
For example, the approximate direction of the North Magnetic Pole is detected with a magnetic device—a compass. Recording and storing various data are most commonly accomplished using y Deceased. 89 magnetic storage components, such as computer disks and tapes. Most household appliances, as well as industrial plants, use motors and generators, the operation of which is based on magnetic forces. The goal of this chapter is to present: Fundamental theoretical foundations for Magnetostatics, most importantly Ampere’s law Some simple and commonly encountered examples, such as calculation of the magnetic field inside a coaxial cable A few common applications, such as Hall element sensors, magnetic storage, and MRI medical imaging. 3.2. THEORETICAL BACKGROUND AND FUNDAMENTAL EQUATIONS 3.2.1. Magnetic Flux Density and Lorentz Force The electric force on a charge is described in terms of the electric field vector, E . The magnetic force on a charge moving with respect to other moving charges is described in terms of the magnetic flux density vector , B . The unit for B is a tesla (T). If a point charge Q [in coulombs (C)] is moving with a velocity v [in meters per second (m/s)], it experiences a force [in newtons (N)] equal to F ¼ Q v B ð 3 : 1 Þ where ‘‘ ’’ denotes the vector product (or cross product) of two vectors. The region of space in which a force of the form in Eq. (3.1) acts on a moving charge is said to have a magnetic field present. If in addition there is an electric field in that region, the total force on the charge (the Lorentz force) is given by F ¼ Q E þ Q v B ð 3 : 2 Þ where E is the electric field intensity in volts per meter (V/m). 3.2.2. The Biot^Savart Law The magnetic flux density is produced by current-carrying conductors or by permanent magnets. If the source of the magnetic field is the electric current in thin wire loops, i.e.
eBook - PDF- Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
A major breakthrough came in 1820 when Hans Christian Oersted discovered experimentally that a magnetic needle was deflected by a current in a wire. This event bridged the gap between the science of electricity and magnetism. Scientists immediately realized that electric currents are also sources of magnetic fields. Within a short time after Oersted’s discovery, Biot and Savart experi-mentally formulated an equation to determine the magnetic flux density 177 178 5 Magnetostatics Figure 5.1 Magnetic lines of flux surrounding a bar magnet at a point produced by a current-carrying conductor. We now view the Biot–Savart law as the magnetic equivalent of Coulomb’s law. By 1825 Andr´ e Marie Amp` ere had discovered the existence of magnetic force between current-carrying conductors and formulated a set of qualitative relationships based upon a series of experiments. These discoveries lead to the development of electric machines we use in our daily lives. This chapter is devoted to the study of Magnetostatics ; i.e., the mag-netic fields produced by steady currents. We begin our discussion with the Biot-Savart law and use it as a basic tool to calculate the magnetic field set up by any given distribution of currents. 5.2 The Biot–Savart law ................................. It has been found experimentally that the magnetic flux density produced at a point P from an element of length d of a filamentary wire carrying a steady current I , as shown in Figure 5.2, is d B = k Id × a R R 2 In this equation, d B is the elemental magnetic flux density in teslas (T), where one tesla is equal to one weber per square meter (Wb/m 2 ), d is an element of length in the direction of the current, a R is the unit vector pointing from d to point P , the point P is at a distance R from the current element d , and k is the constant of proportionality. Figure 5.2 Magnetic flux density at point P produced by current element at Q
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
Laws of Magnetostatics 10.1 Introduction Magnetic dipoles are the source of magnetic fields; a coil in which a current flows rep- resents a magnetic dipole. Hence, a law which allows to evaluate a magnetic field generated by a flowing current must be determined. This law can be inferred by correlating some ex- perimental observations and by imposing a strong analogy between an electric field and its associated electric force and their magnetic equivalents. The experimental results, obtainable by using a magnetic needle or an exploratory probe, useful in order to deduce this law are: • a rectilinear wire in which a current flows generates a magnetic field whose field lines are circumferences concentric with the wire; • the magnitude of the magnetic field reduces when moving away from the wire; • the superposition principle holds true, that is, if several currents generate a magnetic field in the space, the magnetic field in point P, one and only one due to the principle of the unicity of the field in a point, is given by the vector sum of the magnetic fields generated by each current in absence of other currents in that point. The analogy must be established between: • Coulomb’s law, which expresses the force exerted by an electrostatics field, and the force exerted by a magnetic field on a current; • the formal expression of an electrostatic field generated by a point-like charge and the one of a magnetic field. The first step is the definition of an infinitesimal element, exerting or undergoing a magnetic interaction, being equivalent to a point-like charge for an electric interaction. Hence, consider two wires in which a current flows and ideally isolate an infinitesimal section of wire d s directed as the current on each of them. Section d s 1 , which belongs to the first wire, generates a magnetic field B 1 in the whole space and therefore also in point P in which a wire section d s 2 , which belongs to the second wire, lies.
eBook - PDF- Y K Lim(Author)
- 1986(Publication Date)
- WSPC(Publisher)
(3.42). 3.7 Magnetostatic Field of Magnetized Matter The magnetostatic field is described by the time-independent Maxwell's equations ?*H = J , (3.46) e *7l*9.*i 0 m P*nWm 95 V-B = 0 . (3.47) It may be produced either by stationary currents or by stationary magnetized or ferromagnetic bodies. We shall first consider the case where current is everywhere zero and where the magnetic field is produced entirely by stationary and rigid magnetized matter. The first equation now becomes V x H = 0 , (3.48) which means that H can be expressed in terms of a magnetic scalar potential $ : H = - 7* . (3.49) m v ' In a material medium the magnetic intensity is related to the induction through Eq. (1.46), which may be taken here as the definition of the magnetization M: M = | --H . (3.50) While M is usually related to H, for ferromagnetic materials, however, magnetization may exist even in the absence of external excitation. To include such a possibility we shall assume that the magnetization M may contain a permanent part, which has no relation to H. Substituting the expression (3.49) for H in Eq. (3.50) we obtain K o m o Equation (3.47) then gives V 2 * m = -p ffl , (3.51) with p m = - V'M . (3.52) Equation (3.51) has the form of Poisson's equation in electrostatics and we may by analogy define p as the ' From Eq. (3.49) we also have the relation and we may by analogy define p as the magnetio charge density. 96 V -H = P m , (3.53) r m analogous to Eq. (3.2). Across a surface of discontinuity the boundary conditions for H is H 2 t =H l t . (3.54) Another equation is obtained from the boundary condition for B, B 2n = B ln which gives H 2n-H ln = -( M 2n-M ln)=° m • ^ where a may be called the surface magnetic charge density. The similarity between the forms of Eqs. (3.52) and (3.30) and Eqs. (3.55) and (3.31) shows that the so-called magnetic charges are induced or equivalent in nature, similar to the equivalent polari-zation charges.
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