Physics
Magnetostatic in Matter
Magnetostatics in matter refers to the study of magnetic fields in materials that are not in motion. It involves the analysis of the behavior of magnetic fields in different types of matter, including ferromagnetic, paramagnetic, and diamagnetic materials. The study of magnetostatics in matter is important in understanding the behavior of magnetic materials and their applications in various fields.
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6 Key excerpts on "Magnetostatic in Matter"
eBook - ePub- Anupam Garg(Author)
- 2012(Publication Date)
- Princeton University Press(Publisher)
16 Magnetostatics in matterOur purpose in this chapter is to obtain a theory for the spatially averaged or macroscopic magnetic field in matter and to develop models for the response of certain types of materials to applied magnetic fields. While the mathematical development parallels that of electrostatics in dielectrics, the physics is quite different. The biggest difference is that, unlike the electric case, the magnetic susceptibility may be either positive or negative, i.e., the magnetic response can either oppose or reinforce the applied field. The two cases are known as diamagnetic and paramagnetic , respectively. This difference can be traced to the lack of any work done by the magnetic field on a moving charge. Second, except for ferromagnets and superconductors, the response is much weaker than in the electric case. Thus, there is often little penalty for confusing the magnetic field B with the magnetizing field H , and many authors even use the symbols interchangeably. This leads to much confusion, which we shall avoid with only partial success.100 Magnetic permeability and susceptibility
In magnetostatics, where we study only steady-state situations, the macroscopic equations to be solved are Ampere’s law,and the law that B is solenoidal,As in the electrostatic case, one needs a constitutive relation between B and H to solve eqs. (100.1) and (100.2). For many materials a linear relationship holds to very good approximation over a large range. If the material is isotropic, we write1The dimensionless number μ (in the Gaussian system), or μ/μ 0 (in SI), is known as the permeability of the medium. It has the same value in the two systems. In solid or liquid crystals, it must be replaced by a tensor.We also define the magnetic susceptibility in parallel with the electric susceptibility:This is related to the permeability via It is also a dimensionless number, butFor the majority of materials for which eq. (100.3) is a good approximation,χmis very small, of order 10−5 . As mentioned above, unlikeχe,χmcan be either positive (paramagnets ) or negative (diamagnets ). Because of the smallness ofχm, if either a para- or diamagnetic body is placed in an external B
eBook - PDF- Wilhelm Jost(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Maxwell's equations relate these fields to the charge density g(r y t) and the current density j(r, t). We confine ourself to the static case; i.e., all charges and all currents are constant in time. In this case, Maxwell's equations separate into two independent sets, so that electrostatics and magnetostatics can be considered distinct phenomena. Electrostatics: Magnetostatics: curl E(r) = 0 div D(r) = 4nQ(r). curl H(r) = 4 ^ j ( r ) div B(r) = 0. (2.1) (2.2) (2.3) (2.4) In addition, there is, in each case, another equation, called a constitutive relation, connecting E and D, and H and B, respectively, characterizing the properties of the matter under consideration. T h e fact that we confine ourselves to the static case is not only due to the simplification of the mathematics, but also has an important thermo-dynamic basis. In the case of fields varying in time, the electric energy is always converted into magnetic energy and vice versa. This process, however, gives rise to dissipation of energy, which again affects the minimum and maximum principles of the thermodynamic potentials. Strictly speaking, the methods of equilibrium thermodynamics are only sufficient to describe matter in static fields, and, if currents are present, they have to be nondissipative. T h e description of the behavior of matter in electric and magnetic fields is more complicated than a description of systems in gravitational fields because gravitational fields as considered in Section I constitute external fields in the sense that they are determined by external sources only and are not modified by the matter present. In electric and magnetic fields, matter, due to its being polarized and magnetized, is an additional source of the field which modifies the externally applied field. Further- 5. T h e r m o d y n a m i c s of M a t t e r i n F i e l d s 411 more, this modification depends on the shape of the specimen situated in the field.
- 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*.
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.
- 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 – .
- João Pedro A. Bastos, Nelson Sadowski(Authors)
- 2003(Publication Date)
- CRC Press(Publisher)
2.8. Magnetostatic Fields The group of equations describing the magnetism in low frequency domain are rotH=J divB=0 In magnetostatics the quantities are independent of the time and we have Electromagnetic modeling by finite element methods 58 rotH=J (2.33) divB=0 (2.34) while the equation rotE=0 (2.35) does not play any role in this situation. The constitutive relations are B=μH J=σE At first look, magnetostatic looks quite limited since the majority of devices have variable current sources, and/or have movement. However, when the structure is built in a way that we can neglect ∂B/∂t in conductive materials, it is possible to treat it as a magnetostatic one. In other words, it is possible to study the structure at each position as a static one and, afterwards, compose the successive results in order to obtain the dynamic behavior of it. In addition we will present here the different types of magnetic materials, the expression of magnetic field energy and the concept of inductances. Although in some instances it will be necessary to use the notion of time, the results we obtain are static in nature. 2.8.1. Maxwell’s Equations in Magnetostatics 2.8.1a. The Equation rotH=J This equation defines qualitatively and quantitatively the generation of H in terms of J. We recall that the same relation in integral form is ∫ S (rotH)·ds=∫ S J·ds (2.36) where S is a surface on which H and J are defined. Using Stokes’ theorem, the left-hand side of the expression can be written as (2.37) where C is a contour, enclosing the surface S. The right-hand side of Eq. (2.36) represents the flux of the vector J crossing the surface S. This flux is the conduction current crossing S. That is (2.38) Maxwell equations, electrostatics, magnetostatics and magnetodynamic fields 59 which indicates that the circulation of H along a contour C encircling a surface S is equal to the current crossing this surface.
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