Physics
Magnetic Scalar Potential
The magnetic scalar potential is a concept in physics that describes a scalar field used to simplify the mathematical description of magnetic fields. It is related to the vector potential and is used in the context of magnetostatics to represent the magnetic field in a region of space. The magnetic scalar potential is particularly useful in solving problems involving magnetic fields and their interactions.
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4 Key excerpts on "Magnetic Scalar Potential"
eBook - PDFMagnetic Fields
A Comprehensive Theoretical Treatise for Practical Use
- Heinz E. Knoepfel(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
2.1 Magnetic Scalar Potential A relation between magnetic fields and current distribution can be conveniently established by introducing the Magnetic Scalar Potential 0. The advantage of using this 55 56 CHAPTER 2 MAGNETIC POTENTIALS scalar potential inthe free space external tothe current-carrying conductor, instead of AmNre’s vectorial equations, is that the magnetic problem can besolved byonedifferential equation instead of asystem of three simultaneous differential equations regarding the three components ofthe magnetic field vector. In addition, the solution methods developed for potential problems in other fields of physics (i.e., in electrostatics) can be conveniently applied. Another use of the Magnetic Scalar Potential, discussed at the end of this section, is its application to describe the magnetic field pattern in and around amagnetized material. Potential in empty space Definitions and solutions Just as in electrostatics, where the electric field in empty space E can beexpressed byapotential U through E=-VU , (2.1-1) it follows, from the magnetostatic or quasistationary equations (1.2-1, 18) in empty space, that the magnetic vector field H can be expressed as the gradient of ascalarpotential Figure 2.1-1 Current circular loopwithits dipole-type magnetic field lines (which will be calculated in section 2.3 with the help of the magnetic vector potential). Therotation surfaces (lines) of constant magnetic potential 0 are normal tothe H lines. 2.1 Magnetic Scalar Potential 57 H=-V@. (2.1-2) Infact, taking the curl of this equation leads to V x H = 0 because of(A.3-1 l), as also obtained from (1.2-1,18) inempty space wherej=O. Since also pV.H =O from (1.1-3), the magnetic field problem reduces to apotential problem described by Laplace's equation A@=O , (2.1-3) together with the appropriate boundary conditions (2.1-7, 9, 11).
- K Umashankar(Author)
- 1989(Publication Date)
- WSPC(Publisher)
Chapter: 9 MAGNETIC UECTOR POTENTIAL 9.0 INTRODUCTION The study of the static magnetic field intensity and the static magnetic flux density has exposed various static field phenomena and their associated relation to their constant electric current sources. As discussed in the previous two chapters, various mathematical interpretations of the basic Biot-Savart law, Gauss law and Ampere law have helped to understand corresponding physics of interaction associated with fields. It should be noted that the magnetic field and its density are introduced in the previous chapters, purely as vector point functions. Further, it may be recalled from earlier discussions on the static electric field based on the Coulomb's law that the static electric field was interpreted as a type of force defined as the force per unit charge. This type of electric force or field is capable of performing work, and thus expends electric energy in a given medium. Similarly, based on duality concept, the magnetic force or field is also capable of performing work, and thus expends work or energy to a virtual magnetic pole in a given medium. For example, if a magnetic field is externally impressed in a medium containing distribution of electric current sources, then the electric current sources experience a force of interaction, and tend to move in the medium unless they are held stationary by applying external forces. A complete discussion on the forces of interaction due to magnetic fields is discussed in later sections. In this chapter, various mathematical details of the concept of work done and the corresponding magnetic field energy stored, are discussed in terms of a scalar magnetostatic potential distribution. 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.
eBook - PDF- Freeman J Dyson, David Derbes(Authors)
- 2011(Publication Date)
- World Scientific(Publisher)
Scattering by a Static Potential 189 This is all we can say at present about radiative corrections to scattering by an electrostatic potential. 8.1 The Magnetic Moment of the Electron The scattering by an electrostatic potential, the two terms in (617) were lumped together. Both gave contributions of the same order of magnitude, α ( q 2 /μ 2 ) in the cross section. What then is the meaning of the special form of the second term in (617)? This term has no infra-red divergence and therefore should be particularly simple to interpret experimentally. Consider scattering of a slow electron by a slowly-varying 84 magnetic field. The potentials (534) can then be taken to be a pure vector potential, so that e 4 ( q ) = 0 (651) The matrix elements of γ 1 , γ 2 , γ 3 between positive energy electron states are of the order ( v/c ). Hence M 0 given by (543) is of order ( v/c ). The first term in (617) is thus of the order α ( v/c ) 3 while the second is of order α ( v/c ). Therefore the second term in (617) is the main term in considering magnetic effects, and the first term can be ignored. The meaning of the second term must be a change in the magnetic properties of a non-relativistic electron. As we saw in discussing the Dirac equation, (Eqs. (99) and (100)), an electron by virtue of its charge ( -e ) behaves in non-relativistic approxima-tion as if it had a magnetic moment M = -e ~ 2 mc (652) This moment has an energy of interaction with an external Maxwell field ( E , H ) given by H M = -M ( σ · H -i α · E ) (653) the term which appears in the non-relativistic Schr¨ odinger equation (100). Now suppose that the electron possesses an additional magnetic moment δM which does not arise from its charge. Such an additional moment is called “anomalous”. To give the electron an anomalous moment, we only need to add arbitrarily a term proportional to (653) to the Hamiltonian.
- Ozgur Ergul(Author)
- 2021(Publication Date)
- Wiley(Publisher)
¯ H (¯ r ) = ˆ a φ 3 J 0 a 2 ρ . (4.125) R 12 R 21 a 12 a 21 ^ -dl 1 dl 2 dl 2 dl 1 ^ I 1 I 2 I 1 I 2 F m, 21 F m, 12 -Figure 4.15 Magnetic force between two arbitrarily oriented differential electric currents. Recall that there is no antisymmetry in general. 4.3.3 Magnetic Potential Energy Similar to the electric potential energy, which is related to the electric force, electric scalar potential, and electric fields, one can define the magnetic potential energy: i.e. a form of energy stored by means of magnetic fields. While there are alternative ways to approach the physical interpretation of the magnetic potential energy, we construct analogies with the electric potential energy and related concepts. First, we may consider two infinitesimal electric currents: I 1 dl 1 and I 2 dl 2 located in a vacuum, and the magnetic force between them, (Figure 4.15 ) as ¯ F m, 12 = μ 0 4 π I 2 dl 2 × I 1 dl 1 × ˆ a 12 ( R 12 ) 2 (4.126) ¯ F m, 21 = μ 0 4 π I 1 dl 1 × I 2 dl 2 × ˆ a 21 ( R 21 ) 2 , (4.127) where R 12 = R 21 = | ¯ r 1 − ¯ r 2 | is the distance between currents, ˆ a 12 is the unit vector directed from the location of I 1 to the location of I 2 , and ˆ a 21 = − ˆ a 12 . As discussed in Section 2.7.2 , | ¯ F m, 12 | may not be the same as | ¯ F m, 21 | : i.e. the magnetic force between infinitesimal currents is not necessarily antisymmetric. 38 In addition, it is possible to have ¯ F m, 12 = 0 or ¯ F m, 21 = 0 by 38 Also recall that this is due to the unphysical nature of an isolated infinitesimal current. 226 Introduction to electromagnetic waves with Maxwell’s equations aligning the differential currents without separating them by an infinite distance. Obviously, the magnetic force is quite different from the electric force, for which the distance is the major factor without any orientation concerns.
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