Force on a Conductor

8 Key excerpts on "Force on a Conductor"

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  eBook - ePub

  Electromagnetism

  • I. S. Grant, W. R. Phillips(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 4

  Steady currents and magnetic fields

  The previous chapters were concerned with the forces between stationary charges and their description in terms of the electrostatic field and potential. We now consider what happens when charges are in motion. As well as the force represented by the electric field, a moving charge may experience an additional force, called a magnetic force, when it is in the presence of other moving charges or permanent magnets. Magnetic forces are also conveniently described in terms of a vector field—the magnetic field—from which we can derive the force on a moving charge at any point.

  A steady electric current in a conducting wire arises from the continuous motion of charge along the wire, and a moving charge near a current-carrying wire may therefore experience a magnetic force. In this chapter we shall first explain the rules governing the flow of current in conductors, and then go on to discuss the magnetic fields caused by steady currents in conductors.

  4.1 ELECTROMOTIVE FORCE AND CONDUCTION

  4.1.1 Current and resistance

  Some of the electrons inside conductors are not bound to particular atoms but are free to move from atom to atom. In a metal we may picture these free electrons, the conduction electrons, as behaving more or less like gas molecules in a box, moving about at random with large speeds, and colliding with atoms in the metallic lattice. In the absence of an applied electric field the mean electron velocity, averaged over any volume which is very large on a microscopic scale, will be zero. If an electric field E is maintained in the conductor the electrons acquire an average velocity v in the direction opposite to the field. This is because, between collisions with atoms in the conductor, the electrons are acted upon by a force – e E. The mean drift velocity v

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  Advanced Electromagnetic Waves

  • Saad Osman Bashir(Author)
  • 2015(Publication Date)
  • IntechOpen

    (Publisher)

  Having done that analysis, it remains to take into account to the effects of the Special Relativity theory, especially the Lorentz transformation of lengths. Since that effect is related only to the relative movements of the two coordinate axes, and has nothing to do with the propagation delay an observer faces, it may be multiplied straightforwardly to the effect of propagation delay. An electric current carried by a conductor implies that both the immobile lattice ions and the moving electrons contribute to the force exerted on other charges. In the case those are embedded in a neighboring electric conductor that is also carrying an electric current, they interact with both positive lattice ions and moving conductor electrons. This implies that altogether four kinds of interaction will take place, each demanding their own mathematical treatment respectively: from the positive ions of the first conductor to both kinds of charges of the second conductor and from the electrons of the first conductor to both kinds of charges The Electromagnetic Force between Two Parallel Current Conductors Explained Using Coulomb’s Law http://dx.doi.org/10.5772/61221 63 of the second conductor. In the case of quadrangle circuits, the two respective currents appear both parallel and perpendicular to each other. 2.4. Coulomb’s law, basic formulation In order to integrate the contribution to the total force between two currents, carried by conductors, it is most suitable to use the differential force that an incremental segment gives rise to.

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  Magnetic Fields

  A Comprehensive Theoretical Treatise for Practical Use

  • Heinz E. Knoepfel(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  Theelectrons flowing with drift velocity vd along the conductor (figure 6.2-14) experience the Lorentz force vd xB, thereby accumulating on oneside of the conductor; this establishes anelectric Hall field EH, which inturn acts on the electrons. For completeness' sake, we may even addanadditional transverse force FL deriving from various interaction processes ofthe electrons with the lattice' 2'. Insteady state, the mean value of these transverse forces just balances: -eEH-e(vd xB)+FL =O . (6.2-87) The electrons drift longitudinally, driven by theprimary electric field, butestablish a transverse concentration gradient determined by this equation. On the other hand, the Hall field pulls on the (positively charged) lattice towhich, by reaction, the force FL also applies, with the result that ameantransverse force density f=neEH-nFL=-ne(vdXB)=jXB (6.2-88) acts on the solid-state (lattice) conductor; this is just the Amp&reforce (6.1-10). [Here we have used (6.1-1 l), and n is the number of electrons per unit volume.] To illustrate a case of practical interest, we consider a conductor whosesurface is subjected to a (monotonically increasing or constant) magnetic field pulse H(t). For 6.2 MAGNETIC FORCE ON RIGIDCONDUCTORS 341 simplicity we take the plane case, where an outer magnetic field is applied parallel to an infinitely thick conductor (figure 6.2- 15) and the (monotonically decreasing) current density (6.2-89) JHz (XJ) jy(x,t) = - dX is induced. Inthe infinitesimal layer between x and x + Ax beneath the surface, there is the current density j,(x,f) carried by electrons (and kept to this position by the diffusion process), which drift at velocity and are subject to the transverse Hall force 1 dH F,(x,r) = -evyBz = - Bz 2 n d x or force density dH dx f,(x,r) = -nFx = -B 2 (6.2-90) (6.2-91 a) (6.2-9 1 b) Each layer contributes with a force component per unit surface (or pressure)f,dx.

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  Steady Electric Fields and Currents

  Elementary Electromagnetic Theory

  • B. H. Chirgwin, C. Plumpton, C. W. Kilmister(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  3.4 Forces and energy We turn now to an aspect of the theory which has not so far been consi-dered, namely that of the energy required to establish an electrostatic field. In the picture of a stretched string, used as an illustration in Chapter 1, when energy has been stored in the string a force is exerted by the string at either end. For the special case of a string obeying Hooke's law the energy is given by W — λχ 2 /(2ά) when the displacement at the end is x. Furthermore, the force exerted by the string which tends to increase x is the tension, given by a ax (This result is characteristic of the general relation between force and poten-tial energy in mechanics.) We have already established two fundamental results concerning the forces experienced by charges in an electrostatic field. They are: (1) a point charge Q, at a place where the field caused by all charges excluding Q itself is E, experiences a force QE § 3.4 CONDUCTORS IN THE ELECTROSTATIC FIELD 91 (2) a small area a of surface carrying charge density σ, where the fields on the sides of a are E + , E_, experiences a force <χ.σ(Ε+ +E„); this force arises from the field caused by all charges except the immediate local surface charge on a, i.e. the force per unit area on such a surface distribution is In the case of a surface charge on a conductor this force is always equivalent to an outward force per unit area of cr 2 /(2e 0 ). Also we can expect a force to act on a small volume v containing a charge density ρ. Since the contribution to the field from v itself becomes vanishingly small as v is reduced we expect that a force ρΕ per unit volume acts at a point of a volume charge distribution where the electric field is E. The action on any finite distribution of charge could be obtained by combining these forces using integrations where necessary to give a total resultant force at any point together with a couple.

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  Fields of Force

  The Development of a World View from Faraday to Einstein.

  • William Berkson(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  He began by giving a precise definition of a line of magnetic force. It is a line whose tangent is always in the direction of the magnetic force at that point (that is in the direction a small magnetic needle would take at that point), or it can be considered as a line along which no electromagnetic induction takes place if a wire is moved along it. Further, the intensity of force in a region can be represented by the density of lines of force drawn through it. One could fix a ‘unit line’ of force to draw these lines, which would be varying distances apart, depending on the amount of force between them. Such lines may possibly be lines of flow of ether, lines of action at a distance or lines of vibrations. Faraday did not commit himself to any one of these views, but he did state, ‘I am more inclined to the notion that in the transmission of the force there is such an action, external to the magnet, than that the effects are merely attraction and repulsion at a distance.’ 2 In his argument, Faraday then returned to the experiments which originally convinced him of the value of the lines of force view—the unipolar induction experiments. 3 What Faraday wished to show was that the lines of force clearly and exactly represent the phenomena which occur in such experiments. First, he showed that the one rule of electromagnetic induction is that the [total] induced current is directly proportional solely to the number of lines of force cut (for a wire of given conductivity). Second, he showed that the magnetic lines of force go through the magnet itself, and so are always closed; there are no magnetic ‘poles’, but only places where the lines of force happen to enter and leave the magnet

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  General Physics. Electromagnetism Optics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2023(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Hence, the work done by the magnetic force matches the product of the variation of the magnetic flux linked to the circuit times the current which flows in it dW m = i  BidA  u n = i dΦ . The energy needed by the magnetic force to do this work is supplied by the voltage generator E which, likewise, supplies the energy wasted, due to Joule effect, by the total resistance R of the circuit. In an infinitesimal time dt the generator moves an amount of charge dq = i dt between its two poles and therefore it does the work dW = Edq = Ei dt which is then divided as work done by the magnetic force and energy dissipated by Joule effect dW = dW m + dW R ⇒ Ei dt = Ri 2 dt + i dΦ B , so the flowing current is given by i = 1 R E − dΦ B dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , which is indeed Faraday’s law in presence of a voltage generator. The evaluation can also be done by considering the action of Lorentz force. In fact, con- sider a b long, thin, conductive rod translating with velocity  v = v  u x in a uniform magnetic field  B = B  u z . The rod is represented by the vector  b = b  u which forms an angle θ with vector  B. The free electrons of the metal are subject to the action of the Lorentz force  F = −e  v ×  B = evB  u y and therefore they displace towards the end Q of the rod, leaving an opposite net charge at R E B b P Q P ′ Q ′ i i dx v Chapter 12 Electromagnetic Induction 191 the other end. Then an electromotive force E i appears at its extremities. The force that causes the displacement of the free electrons along the rod is the compo- nent parallel to the rod of the magnetic force F  =  Fi  u = −e  v ×  Bi  u = −e  v ×  Bi  b b , because the normal component produces a negligible effect, be- ing the rod very thin.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  541 CHAPTER 18 LEARNING OBJECTIVES After reading this module, you should be able to... 18.1 Define electric charge. 18.2 Describe the electric force between charged particles. 18.3 Distinguish between conductors and insulators. 18.4 Explain charging by contact and charging by induction. 18.5 Use Coulomb’s law to calculate the force on a point charge due to other point charges. 18.6 Calculate the net electric field due to a configuration of point charges. 18.7 Draw electric field lines. 18.8 Describe the electric field inside a conductor. 18.9 Use Gauss’ law to obtain the value of the electric field due to charge distributions. We have all experienced static electricity in our hair and on our clothes, and have been zapped on occasion when touching a doorknob after walking on carpet. These phenomena occur when electric charges, one of the fundamental building blocks of atoms, separate, and one type (either positive or negative) becomes more abundant than the other. As we will see in this chapter, like charges repel, which is why the toddler’s hair is standing on end. Rachel Hopper/Dreamstime.com *The definition of the coulomb depends on electric currents and magnetic fields, concepts that will be discussed later. Therefore, we postpone its definition until Section 21.7. Electric Forces and Electric Fields 18.1 The Origin of Electricity The electrical nature of matter is inherent in atomic structure. An atom consists of a small, relatively massive nucleus that contains particles called protons and neutrons. A proton has a mass of 1.673 × 10 −27 kg, and a neutron has a slightly greater mass of 1.675 × 10 −27 kg. Surround- ing the nucleus is a diffuse cloud of orbiting particles called electrons, as Figure 18.1 suggests. An electron has a mass of 9.11 × 10 −31 kg. Like mass, electric charge is an intrinsic property of protons and electrons, and only two types of charge have been discovered, positive and negative.

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  Special Relativity

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  2 . Hence the magnitude of the force on this length of wire is given by

  F =

  λ 2

  l

  v 2

  (

  2 k

  I 1

  c 2

  b

  )

  i.e.,

  F =

  2 k

  c 2

  I 1

  I 2

  b

  l =

  B 2

  I 2

  l

  ⁢

  ( 8-20 )

  This force, as we have seen, is in the plane of the wires. If the currents are in the same direction, the force is attractive. If the currents are oppositely directed, the force is repulsive.

  The constant k , remember, is the constant of proportionality in Coulomb’s law. We see now that it appears in the expression for the force between two current-carrying wires and can therefore be used to calculate that force when known currents are flowing. In the MKS system, as we mentioned at the beginning of the chapter, its value is about 9 × 109 newton • m2 /coulomb2 . Using Eq. (8-20) , one can calculate from this that the force per unit length between two parallel wires, 1 cm apart, each carrying a current of 10 amp, is 0.002 newton/m. If there were no positive ions to neutralize the electric field of the electrons, the electric force between the wires would be a factor of c 2 /v 2 greater than this, or about 5 × 1020 newton/m!

  Units And Systems of Electric and Magnetic Measurement

  To wrap up this discussion, as it were, let us return to the quantitative statements of the electric and magnetic force laws with which we began the chapter. The subject of basic units, etc., may often seem rather dull, but the unification of electricity and magnetism through relativity provides a special interest in this case.

  Equation (8-1) states the basic law of force between electric charges (Coulomb’s law). The last equation [Eq. (8-20) ] states a law of force between parallel currents. The CGS unit of charge was defined by setting the constant k equal to unity in the Coulomb law. But what if we let Eq. (8-20) define our unit of charge? We can do this by asserting that two parallel wires, placed a unit distance apart and each carrying a current of unit strength, repel or attract one another with exactly 2 units of force per unit length. Why 2 units of force and not 1? To correspond with the factor 2 on the right side of Eq. (8-20) . Historically, this was in effect done to define an absolute unit of current, and the units of mechanical measurement were CGS units.1

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