7 Key excerpts on "Divergence of Magnetic Field"

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  eBook - ePub

  Liquid Crystal Displays

  Fundamental Physics and Technology

  • Robert H. Chen(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  To say that the Maxwell equations merely describe electromagnetic phenomena ill serves their greatness. It should be understood that the reality of the electromagnetic fields themselves depends on the verity of the Maxwell equations. That is, the existence of the electromagnetic field and the truth of the Maxwell equations are inexorably intertwined and mutually dependent, and which begets the other is a question for epistemological discourse.

  Vector Analysis

  The point of departure of the exposition of the Maxwell equations, was to describe physical reality gleaned from observations of Nature and controlled experimentation. Further elucidation of the electromagnetic theory requires a brief foray into the realm of the vector calculus, a mathematical formalism the bases of which lay in theories of fluid mechanics and electromagnetics, but nonetheless also stands independently as a mathematical discipline in and of itself.

  To start, it is necessary to establish a spatial coordinate system (x,y,z ) upon which the respective unit vectors i,j,k lie collinearly, and accordingly the electric and magnetic field vectors can be resolved into their spatial coordinates lying along the spatial unit vector directions, so that the E and B fields can be written as,

  A basic vector calculus operator can transform a scalar into a gradient vector, by means of an operator called the gradient

  and from that operation, one can see that the effect of the gradient operator is to display a scalar function in all its directional glory, and in particular the gradient will indicate the direction of the greatest slope of the function (that is why it is called a “gradient”).

  The divergence ( ), recalling from the derivation of the Maxwell equations above, when acting upon a vector, produces the scalar product , as for example acting on B ,

  From the above equation, one can see that the divergence operator reveals the vector’s overall change with respect to the x ,y .z coordinate directions. Further, if the operator operates on itself in a scalar product fashion, a new second derivative operator

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  Transmission Lines and Wave Propagation

  • Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Equations 10-1, 10-6, 10-10, and 10-11, which involve line integrals and surface integrals, may be designated as Maxwell's equations in the integral form. Their application to many problems may be expedited by restating them as differential equations. Two differential functions of a vector field are introduced for this purpose, the divergence and the curl. (These are defined in Sec. A-lc, and their forms in rectangular and cylindrical coordinates are listed in Sec. A-2.)

  a. Continuity of Electric Flux

  Electric flux is postulated to emanate from positive charges and to terminate on negative charges. The algebraic sum of such charges within the designated surface of integration in Eq. 10-1 was indicated by Σq . The net outwardly directed flux, , is equal to the algebraic sum of the enclosed charge. For macroscopic applications, charge may be viewed in a mathematical sense which differs from the physical form.

  1 Electric Charge and Mathematical Functions

  Experimentally, electric charge has been found to be concentrated in electrons and protons, "particles" of submicroscopic, yet nonzero, size. If one accepts the restriction that classical electromagnetic theory will be used only to describe macroscopic phenomena, actual charge distributions may be replaced by forms which are easier to work with mathematically.

  The following are idealized charge distributions:

  1. A finite charge q concentrated in a geometric point.
  2. A finite amount of charge per unit length

     ρL

     concentrated in a geometric line.
  3. A finite charge per unit area

     ρs

     concentrated in a geometric surface.
  4. A continuous distribution of charge density ρ throughout a volume.

  Charge density under item 4 constitutes a scalar field.

  In the traditional language of vector analysis, built primarily on hydrodynamics, the positive charges are sources of electric flux, the negative charges, sinks.

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  eBook - ePub

  Essentials of MRI Safety

  • Donald W. McRobbie(Author)
  • 2020(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Appendix 1 . Here we describe their main consequences for MRI safety.

  Electrical charge and electric fields

  Gauss’s Law (Maxwell’s first equation) describes how electrical charges produce static electric fields E. Electric fields start at a positive charge and are directed towards their conclusion at negative charges (Figure 2.1 ). We are not going to use Gauss’s Law much, although it has relevance in minimizing unwanted electric fields in coil design, and at some tissue boundaries where charge may accumulate.

  Figure 2.1

  Electric field lines begin at a source of positive charge and terminate at a negative charge: (a) single point positive charge; (b) positive and negative point charges; (c) capacitor with a potential difference V between the plates.

  Magnetic fields

  Maxwell’s second equation states that the “divergence of B is zero.” This means that there is no magnetic equivalent of electrical charge – no “magnetic monopoles”. Magnetic sources are not like electrostatic ones, but exist as dipoles with a north and south pole (just like the Earth). Magnetic field lines have no beginning or end, but form complete loops from north pole to south (Figures 2.2 , 1.23 ). The nature of the B0 fringe field depends upon this.

  Figure 2.2

  Magnetic field lines from a permanent bar magnet.

  Electromagnetic induction

  Maxwell’s third equation is also known as Faraday’s Law of Induction. We have met dB/dt already in Chapter 1 , so clearly this equation is going to have significant implications for us. It states that a time‐varying magnetic field induces an electric field; also, that the electric field lines form complete loops unlike static electric fields (Figure 2.3

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  Electrical Engineering

  Fundamentals

  • Viktor Hacker, Christof Sumereder(Authors)
  • 2020(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  5  The magnetic field

  5.1  The term “field”

  Magnetism45 is a physical phenomenon that manifests itself as a force between magnets, magnetised or magnetisable objects and mobile electric charges, like e.g. current-carrying conductors. This force is conveyed through a magnetic field46 (vector field47 ) that, on the one hand, is created by these objects and, on the other hand, affects them. Magnetic fields occur with any movement of electric charges. A “field” is generally defined as a space where physical laws apply to certain circumstances.

  In permanent magnets, magnetism is caused by Ampère’s molecular currents (electrons rotating around the nucleus create a very small spin current and electrons rotate around themselves – electron spin). In permanent magnets the magnetic effects do not cancel each other out. Demagnetising them requires a considerable amount of energy.

  If a magnetic field spreads in a material body, the magnetic properties of the substance influence the intensity of the field. The flux density B does not display the same field strength H as in a vacuum. This is due to the atomic structure of the substances. The electrons rotating around their own axis (electron spin) and the nucleus generate spin currents which create magnetic fields perpendicular to the circular orbit (elementary fields). The elementary fields usually cancel each other out without an additional external magnetic field.

  The magnetic and electric expansion happens at the speed of light and is one of the properties of space. Not only space filled with matter, but also empty space has physical properties.

  • The field strength is the force (amount and direction) that the field exerts on the standard body: the vector field. The field strength is a vector.
  • Field lines are used to visually describe a field; they are only a mental tool and not a physical reality.
  • Magnetic fields arise from mobile charges. The field lines encircle the current

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  eBook - ePub

  Electromagnetics 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

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  eBook - ePub

  Basics of Electromagnetics and Transmission Lines

  • G. Jagadeeswar Reddy, T. Jayachandra Prasad(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  1.20 D ¯ Q = Q 4 π a ¯ R R 2 where, R ¯ = (4,0,3) − (4,0,0) =. (0,0,3) = − 5 π 4 π 3 a ¯ z × 10 − 3 (9) 3 / 2 = − 5 4 3 a ¯ z × 10 − 3 27 = − 5 a ¯ z × 10 − 3 36 = − 0.139 a ¯ z × 10 − 3 C/m 2 a ¯ ρ = ρ ¯ | ρ ¯ | ρ ¯ = (4, 0, 3) − (0, 0[--=PLGO-SEPARATOR. =--], 0) = 4 a ¯ x + 3 a ¯ z D ¯ L = ρ L 2 π ρ a ¯ ρ = 3 π 2 π × 10 − 3 4 a ¯ x + 3 a ¯ z 25 = 0.24 a ¯ x + 0.18 a ¯ z mC/m 2 D ¯ = D ¯ Q + D ¯ L = 240 a ¯ x + 42 a ¯ z μC/m 2 1.7 Divergence of a. Vector Divergence: The divergence of a vector Ā at a given point is the outward flux in a volume as volume shrinks about the point. It can be represented as d i v A ¯ = ∇ ⋅ A ¯ = lim Δ v → 0 ∮ S A ¯ ⋅ d s ¯ Δ v (1.7.1) Where ∇ is the del operator or gradient operator. ∇ can be operated on a vector or scalar. It has got different meanings when it is operating on a vector and scalar. If it is operating on a scalar V then it can be written as ∇V which is called as scalar gradient. If it is operating on a vector Ā with dot product then it is ∇. Ā and it is called as divergence of vector Ā and If it is operating on a vector Ā with cross product then it is ∇ × Ā and it is called as curl of vector Ā. Fig. 1.21 Flux lines Physically divergence can be interpreted as the measure of how much field diverges or emanates from a point. Let us consider the Fig.1.21(a) in which field is reaching to the point. Divergence at that point is −Ve or it is also called as convergence. In Fig.1.21(b) the field is going away from the point, therefore divergence is +Ve. In Fig.1.21(c) some of the flux lines or field lines are reaching to the point and same number of field lines are leaving from the point hence the divergence is zero. To determine ∇. Ā let us consider the volume in Cartesian co-ordinate systems as shown in the Fig. 1.22. In Cartesian co-ordinate system, the vector Ā with it’s unit vectors and components along X, Y, Z is A ¯ = A x a ¯ x + A y a ¯ y + A z a ¯ z Fig. 1.22 Evaluation of ∇. Ā Assume the elemental volume ∆V = ∆x∆y∆z

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  Engineering Electrodynamics

  Electric Machine, Transformer, and Power Equipment Design

  • Janusz Turowski, Marek Turowski(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Using the so-called surface divergence, this condition can be presented as D i v B = B 2 n - B 1 n = 0 ⁢ (2.191a) Equality of the tangential components of electric field intensity E 2 t = E 1 t ⁢ (2.192) This condition follows from Equation 2.17. Using the so-called surface curl, it can be presented as Curl E = n (E 2 t - E 1 t) = 0 ⁢ (2.192a) Equality of the tangential components of magnetic field intensity H 2 t = H 1 t ⁢ (2.193) which follows from Equation 2.1 and is valid in the case of continuous current distribution nearby boundary surface, that is, for a finite current density (J ≠ ∞). Only in the case of an ideal theoretical superconductor we have H 2 t - H 1 t = J s u r f ⁢ (2.193a) where J surf is the linear density of the surface current (A/m) flowing in a dimensionless. boundary layer. Applying the surface curl, we get Curl H = n (H 2t − H 1t) = J surf Condition for the normal components of electric field intensity ε 2 E 2 n - ε 1 E ln = ρ s u r f ⁢ (2.194) corresponding to the surface divergence: Div D = ρ surf, where ρ surf is the surface density of electric charge (C/m 2) placed in a dimensionless boundary layer. At an absence of such charge, there exists the equality E 2 n E 1 n = ε 1 ε 2 ⁢ (2.194a) Equations 2.191a, 2.192a, and 2.193 create the so-called system of “Maxwell’s surface equations.” Boundary conditions expressed by the magnetic vector potential A, according to its definition 2.50 B = curl A, for a plane wave (A y = A z = 0), take the form B 2 n = B 1 n which corresponds to ∂ A 2 ∂ t = ∂ A 1 ∂ t ⁢ (2.191b) H 2 t = H 1 t which corresponds

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