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Faraday Paradox Faraday paradox This article describes the Faraday paradox in 1 Paradoxes in which Faraday’s electromagnetism. There is a different Faraday law of induction predicts zero paradox in electrochemistry: see Faraday para- dox (electrochemistry). EMF but there is a non-zero EMF These paradoxes are generally resolved by the fact that an EMF may be created by a changing flux in a circuit as explained in Faraday’s law or by the movement of a conductor in a magnetic field. This is explained by Feyn- man as noted below. See also A. Sommerfeld, Vol III 'Electrodynamics Academic Press, page 362. 1.1 The equipment See also: electrical generator The experiment requires a few simple components (see Michael Faraday The Faraday paradox (or Faraday’s paradox) is any experiment in which Michael Faraday's law of electromagnetic induction appears to predict an incorrect result. The paradoxes fall into two classes: 1. Faraday’s law predicts that there will be zero EMF but there is a non-zero EMF. 2. Faraday’s law predicts that there will be a non-zero EMF but there is a zero EMF. Figure 1: Faraday’s disc electric generator. The disc rotates with angular rate ω, sweeping the conducting disc circularly in the static magnetic field B due to a permanent magnet. The mag- netic Lorentz force v × B drives the current radially across the Faraday deduced this law in 1831, after inventing the first conducting disc to the conducting rim, and from there the circuit electromagnetic generator or dynamo, but was never sat- path completes through the lower brush and the axle supporting isfied with his own explanation of the paradox. the disc. Thus, current is generated from mechanical motion. 1 21 PARADOXES IN WHICH FARADAY’SLAWOF INDUCTION PREDICTS ZERO EMF BUT THERE IS A NON-ZERO EMF Figure 1): a cylindrical magnet, a conducting disc with have worked when either the disc or the magnet was ro- a conducting rim, a conducting axle, some wiring, and a tated, but not both. Faraday attempted to explain the dis- galvanometer. The disc and the magnet are fitted a short agreement with observation by assuming that the mag- distance apart on the axle, on which they are free to rotate net’s field, complete with its lines of flux, remained sta- about their own axes of symmetry. An electrical circuit tionary as the magnet rotated (a completely accurate pic- is formed by connecting sliding contacts: one to the axle ture, but maybe not intuitive in the lines-of-flux model). of the disc, the other to its rim. A galvanometer can be In other words, the lines of flux have their own frame of inserted in the circuit to measure the current. reference. As we shall see in the next section, modern physics (since the discovery of the electron) does not need the lines-of-flux picture and dispels the paradox. 1.2 The procedure The experiment proceeds in three steps: 1.5 Modern explanations 1. The magnet is held to prevent it from rotating, while 1.5.1 Using the Lorentz force the disc is spun on its axis. The result is that the galvanometer registers a direct current. The appa- See also: Lorentz force law ratus therefore acts as a generator, variously called the Faraday generator, the Faraday disc, or the After the discovery of the electron and the forces that af- homopolar (or unipolar) generator. fect it, a microscopic resolution of the paradox became 2. The disc is held stationary while the magnet is spun possible. See Figure 1. The metal portions of the appara- on its axis. The result is that the galvanometer reg- tus are conducting, and confine a current due to electronic isters no current. motion to within the metal boundaries. All electrons that move in a magnetic field experience a Lorentz force of 3. The disc and magnet are spun together. The gal- F = qv × B, where v is the velocity of the electrons and vanometer registers a current, as it did in step 1. q is the charge on an electron. This force is perpendic- ular to both the velocity of the electrons, which is in the plane of the disc, and to the magnetic field, which is nor- 1.3 Why is this paradoxical? mal (surface normal) to the disc. An electron at rest in the frame of the disc moves circularly with the disc rel- The experiment is described by some as a “paradox” as ative to the B-field, and so experiences a radial Lorentz it seems, at first sight, to violate Faraday’s law of elec- force. In Figure 1 this force (on a positive charge, not tromagnetic induction, because the flux through the disc an electron) is outward toward the rim according to the appears to be the same no matter what is rotating. Hence, right-hand rule. the EMF is predicted to be zero in all three cases of ro- Of course, this radial force, which is the cause of the cur- tation. The discussion below shows this viewpoint stems rent, creates a radial component of electron velocity, gen- from an incorrect choice of surface over which to calcu- erating in turn its own Lorentz force component that op- late the flux. poses the circular motion of the electrons, tending to slow The paradox appears a bit different from the lines of the disc’s rotation, but the electrons retain a component flux viewpoint: in Faraday’s model of electromagnetic in- of circular motion that continues to drive the current via duction, a magnetic field consisted of imaginary lines of the radial Lorentz force. magnetic flux, similar to the lines that appear when iron This mechanism agrees with the observations: an EMF is filings are sprinkled on paper and held near a magnet. The generated whenever the disc moves relative to the mag- EMF is proposed to be proportional to the rate of cutting netic field, regardless of how that field is generated. lines of flux. If the lines of flux are imagined to originate in the magnet, then they would be stationary in the frame The use of the Lorentz equation to explain the Faraday of the magnet, and rotating the disc relative to the magnet, Paradox has led to a debate in the literature as to whether whether by rotating the magnet or the disc, should pro- or not a magnetic field rotates with a magnet. Since the duce an EMF, but rotating both of them together should force on charges expressed by the Lorentz equation de- not. pends upon the relative motion of the magnetic field to the conductor where the EMF is located it was speculated that in the case when the magnet rotates with the disk but 1.4 Faraday’s explanation a voltage still develops, that the magnetic field must there- fore not rotate with the magnetic material as it turns with In Faraday’s model of electromagnetic induction, a circuit no relative motion with respect to the conductive disk. received an induced current when it cut lines of magnetic However, careful thought showed if the magnetic field flux. According to this model, the Faraday disc should was assumed to rotate with the magnet and the magnet 1.5 Modern explanations 3 rotated with the disk that a current should still be pro- duced, not by EMF in the disk (there is no relative motion between the disk and magnet) but in the external circuit linking the brushes[1] which is in fact in relative motion with respect to the rotating magnet. In fact it was shown that so long as a current loop was used to measure in- duced EMFs from the motion of the disk and magnet it is not possible to tell if the magnetic field does or does not rotate with the magnet. Several experiments have been proposed using electro- static measurements or electron beams to resolve the issue, but apparently none have been successfully per- formed to date. However, In case 2, since there is no current observed— the magnetic field did not rotate with the rotating magnet. 1.5.2 Relation to Faraday’s law of induction See also: Faraday’s law of induction The flux through the portion of the path from the brush at the rim, through the outside loop and the axle to the center Figure 2: Two possible loops for finding EMF: the geometrically of the disc is always zero because the magnetic field is in simple path is easy to use, but the other provides the same EMF. the plane of this path (not perpendicular to it), no matter Neither is intended to imitate any line of physical current flow. what is rotating, so the integrated emf around this part of the path is always zero. Therefore, attention is focused on the portion of the path from the axle across the disc to so the rate that flux sweeps past the imaginary line is the brush at the rim. Faraday’s law of induction can be stated in words as:[2] dΦ dA R2 dθ R2 E = − B = B = B = B !; The induced electromotive force or EMF dt dt 2 dt 2 in any closed circuit is equal to the time rate of with ω = d θ / dt the angular rate of rotation. The sign change of the magnetic flux through the circuit. is chosen based upon Lenz’s law: the field generated by the motion must oppose the change in flux caused by the Mathematically, the law is stated: rotation. For example, the circuit with the radial segment in Figure 2 according to the right-hand rule adds to the ZZ applied B-field, tending to increase the flux linkage.
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