Biot Savart Law

7 Key excerpts on "Biot Savart Law"

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  eBook - ePub

  Pocket Book of Electrical Engineering Formulas

  • Richard C. Dorf, Ronald J. Tallarida(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  27 Static Magnetic Fields 1.  Biot–Savart Law

  A current I flowing in a differential vector length dL results in a magnetic field intensity H as

  d H =

  I d L ×

  a R

  4 π

  R 2

        (

  A / m

  )

  Expressed in terms of current density J, we have

  H =

  ∫ volume

  J ×

  a R

  4 π

  R 2

  d υ

  2.  Ampere’s Law The line integral of H about any closed path is equal to the direct current enclosed by that path:

  ∮

  H · d L = I

  3.  Maxwell’s Equations for Static Fields

  ∇ × H = J

  and

  ∇ × E = 0

  4.  Stokes’ Theorem

  ∮

  H · d L =

  ∫

  surface     S

  ( ∇ × H ) · d S

  5.  Magnetic Flux Density

  Magnetic flux density B in free space is

  B =

  μ 0

  H         ( T )

  where T is teslas and μ0 = 4π × 10−7 H/m.

  Then, the divergence theorem provides

  ∇ · B = 0

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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)

  Chapter 2 Static Magnetic Fields 2.1    Introduction The electro static field is characterized by electric field intensity Ē and electric flux density D ¯ and they are related as D ¯ = ∊ E ¯. Similarly static magnetic fields are characterized by magnetic field intensity H ¯ and magnetic flux density B ¯ and they are related as B ¯ = μ H ¯. Static electric fields are obtained by static charges. Static magnetic fields are obtained when the static charges are moving with constant velocity or static magnetic fields are generated by constant current flow. Electrostatic fields are described by Coulomb’s law and Gauss’s law. Similarly magneto static fields are described by Biot-Savart’s law and Ampere’s circuit law. In general case we use Biot-Savart’s law to find either H ¯ or B ¯. If the distribution is symmetry we use Ampere’s circuit law to find either B ¯ or H ¯. 2.2    Biot-Savart’s Law It states that the elemental magnetic field intensity d H ¯ produced at point ‘p’ as shown in Fig.2.1, by a differential current element Idl is proportional to the product Idl and sine of the angle ‘α’ between the Idl and line joining the element to point ‘p’ and is inversely proportional to square of the distance between element and point ‘p’. Fig. 2.1 Magnetic field due to current element d H ¯ ∝ I d l sin α R 2 d H ¯ = K I d l sin α R 2 K = proportional constant = 1 4 π ∴ d H ¯ = I d l sin α 4 π R 2 The above equation can be written. as d H ¯ = I d l ¯ × a ¯ R 4 π R 2 where a ¯ R = R ¯ | R ¯ | d H ¯ = I d l ¯ × R ¯ 4 π R 3 (2.2.1) To indicate the direction of magnetic field intensity we use right hand thumb rule. In this thumb indicates direction of current and the fingers encircling the wire indicates the direction of magnetic field intensity. We can also use right hand screw rule to find the direction of magnetic field intensity

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  eBook - ePub

  Energy Medicine - E-Book

  The Scientific Basis

  • James L. Oschman(Author)
  • 2015(Publication Date)
  • Churchill Livingstone

    (Publisher)

  Figure 5.17 , the various organs in the body generate electrical currents that flow through the tissues and therefore generate magnetic fields both within and around the body.

  Figure 2.5 André-Marie Ampère (1775–1836).

  The strongest electrical field is produced by the heart and generates a current that flows through the circulatory system, which is a good conductor. In accord with Ampère’s law, this current produces the strongest biomagnetic field of the body (Figure 2.6 ). This point is important because of the skepticism about energy fields around the body that are described and used by various complementary and alternative medicine (CAM) practitioners. The laws of physics require the production of a biomagnetic field around the body as a result of the electrical activity of the heart and other organs. In Chapter 8 we will see the methods that have been used to measure these biomagnetic fields.

  Figure 2.6 The strongest electrical field is produced by the heart and generates an electrical field that is conducted throughout the body via the circulatory system (A), which is a good conductor. The electrical field of the heart is recorded in the electrocardiogram (B). In accord with Ampère’s law, this current produces the strongest biomagnetic field of any organ, and the field is radiated into the space surrounding the body (C). Modern devices can record the biomagnetic field of the heart, which is called a magnetocardiogram (D). Biomagnetic fields are discussed in detail in Chapter 8 .

  Electricity from Magnetism: Faraday’s Law of Induction

  About 11 years after Ørsted’s important discovery in Denmark, another important finding took place simultaneously in England and America. Electromagnetic induction is the reverse of Ampère’s law, i.e., magnetic fields can cause currents to flow through conductors. Electromagnetic induction was discovered by the English chemist and physicist, Michael Faraday in 1831, and, independently and at the same time, by an American scientist, Joseph Henry (Figure 2.7 ). The resulting law of physics is known as Faraday’s law of induction because Faraday published his results before Henry. Note from Figure 2.7C

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  eBook - ePub

  Liquid Crystal Displays

  Fundamental Physics and Technology

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

    (Publisher)

  But Faraday observed that the effect that a magnet has on a current-carrying wire is to move it, as shown in the schematic drawing in Figure 2.2 as the dashed line. That is, when the current is turned on, the wire near the magnet will move horizontally in a direction perpendicular to the direction of the South to North poles of the magnet, so mathematically the magnetic force emanating from the magnet produces a mechanical force F that can be described by the vector equation. The equation says that a point charge traveling in the wire at a velocity (v) will be subject to a force (F) that is proportional and perpendicular to both v and B (the cross product in the vector calculus). The electric and magnetic forces combined in a single equation is the well-known Lorentz force, Figure 2.2 Magnetic force acting on a current in a wire moves the wire. From the above equation, it is clear that while there is a force (E) associated directly with an electric charge (q), a magnetic force requires motion (v) to act. Other observations were not so simply describable, however; for example, the subsequent mutual interaction of the current, the generated magnetic force, and the magnet’s magnetic force. To help matters along, Faraday here visualized the force effect of the magnet as a pattern of lines of force, the grouping together of which constituted a flux of force lines, the number and closeness of the lines representing the density of the flux, and that flux density indicating the strength or intensity of the magnetic force. Faraday’s own drawing of the lines of force emanating from a bar magnet is shown in Figure 2.3, where he described the flux lines as a field

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  eBook - ePub

  Essentials of MRI Safety

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

    (Publisher)

  Hall effect.

  Lorentz force

  The magnitude of the Lorentz force on a charge Q possessing velocity v is given as

  (2.20)

  The direction of the force can be determined by Fleming’s left‐hand rule.

  Magneto‐hydrodynamic effect

  A similar effect is the generation of an electric field E by the flow of charge within an external magnetic field (Figure 2.26 ). This is analogous to the Hall effect observed in semiconductors.

  (2.21)

  Figure 2.26

  Magneto‐hydrodynamic and Hall effect.

  In terms of induced voltage or electrical potential, V, where

  (2.22)

  and d is the distance between charged surfaces (as in a capacitor), we have an induced voltage

  (2.23)

  The effect is most commonly encountered in MRI as an artefact in ECG traces.

  LAWS OF INDUCTION

  The laws of induction follow from Maxwell’s third equation or Faraday’s law. If we consider a wire loop within a time‐varying B‐field the magnitude of the induced E‐field is [3 ]

  (2.24)

  This applies for both the electric field induced by the imaging gradients responsible for peripheral nerve stimulation (PNS), and the electric field induced by the RF B1 ‐field responsible for SAR and tissue (and implant) heating. The direction of E follows a left‐hand rule, as any magnetic field produced by the induced current in the wire opposes the rate of change of flux that induced it.

  Faraday induction from the gradients

  Biological tissues conduct electricity by means of water and electrolytes. Rather than considering electrical current in tissue (as in wires), we consider the current densityJ, a vector (Figure 2.27 )

  (2.25)

  Figure 2.27

  Ohm’s law in a circuit and a volume conductor.

  σ is the tissue conductivity in siemens per meter (S m−1 ). Some representative values are shown in Table 2.3 .

  Table 2.3

  Tissue conductivity at various frequencies. Electrical properties from https://itis.swiss/virtual‐population/tissue‐properties/database

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  eBook - ePub

  Engineering Electrodynamics

  Electric Machine, Transformer, and Power Equipment Design

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

    (Publisher)

  Figure 2.2 ).

  If we apply Stokes’ theorem to the second Maxwell’s Equation 2.2

  ∬ A

  curl E ⋅ d A =

  ∮ l

  E ⋅ d l = -

  ∬ A

  ∂ B

  ∂ t

  ⋅ d A

  ⁢ ( 2.17 )

  we obtain Faraday’s law of electromagnetic induction in the mathematical Maxwell’s formulation ([2.8 ], p. 234)

  e =

  ∮ l

  E ⋅ d l = - N

  d Φ

  d t

  ⁢ ( 2.18 )

  and the experimental Faraday’s formulation

  d Q = -

  d Φ

  R

  or

  - N

  d Φ

  d t

  = R i

  ⁢ ( 2.18a )

  Figure 2.2 Illustration of the Biot–Savart Law.

  In Equations 2.18 , N is the number of turns enclosing the flux Φ = ∫∫A B ⋅ dA ; dQ = i dt is the electric charge. When the turns are so distributed that every of them encircles only a part of the total flux, then the electromotive force e is calculated from the equivalent number of turns N ⋅ k w , where k w < 1 is the so-called winding factor (Turowski [1.18]). The minus sign means the well-known electromagnetic inertia rule of Lenz (1833), as follows:

  In electric circuits and conducting bodies there exists a tendency to preserve in unchanged state the magnetic flux coupled with these circuits.

  Any effort to change the flux in a circuit induces electromotive forces acting in direction opposite to these changes. This phenomenon appears most distinctively in superconductors [Meissner Effect (2.117), (2.118)].

  Although Equations 2.18 and 2.18 a are equivalent to each other, (2.18a) clearly explains that in a circuit with R ≈ 0, the alternating flux dΦ/dt is completely displaced outside the circuit. Equation 2.18 concerns generally cutting conductor sections by the flux Φ. It is, therefore, more broad than Equation 2.18

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  eBook - ePub

  Fundamental Electrical and Electronic Principles

  • C R Robertson(Author)
  • 2008(Publication Date)
  • Routledge

    (Publisher)

  The experimental procedure described above is purely qualitative. However, if it was refined and performed under controlled conditions, then it would yield the following results:

  The magnitude of the induced emf is directly proportional to the value of magnetic flux, the rate at which this flux links with the coil, and the number of turns on the coil. Expressed as an equation we have:

  Notes:

  1    The symbol for the induced emf is shown as a lower-case letter e. This is because it is only present for the short interval of time during which there is relative movement taking place, and so has only a momentary value.

  2    The term dΦ/dt is simply a mathematical means of stating ‘the rate of change of flux with time’. The combination NΦ/dt is often referred to as the ‘rate of change of flux linkages’.

  3    The minus sign is a reminder that Lenz’s law applies. This law is described in the next section.

  4    Equation (5.1) forms the basis for the definition of the unit of magnetic flux, the weber, thus:

  The weber is that magnetic flux which, linking a circuit of one turn, induces in it an emf of one volt when the flux is reduced to zero at a uniform rate in one second. In other words, 1 volt = 1 weber/second or 1 weber = 1 volt second. 5.2 Lenz’s Law

  This law states that the polarity of an induced emf is always such that it opposes the change which produced it. This is similar to the statement in mechanics, that for every force there is an opposite reaction.

  5.3 Fleming’s Righthand Rule

  This is a convenient means of determining the polarity of an induced emf in a conductor. Also, provided that the conductor forms part of a complete circuit, it will indicate the direction of the resulting current flow.

  The first finger, the second finger and the thumb of the right hand are held out mutually at right angles to each other (like the three edges of a cube as shown in Fig. 5.3 ). The First finger indicates the direction of the Flux, the thuMb the direction of Motion of the conductor relative to the flux, and the sECond finger indicates the polarity of the induced Emf, and direction of Current flow. This process is illustrated in Fig. 5.4

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