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: Faraday's law appears to predict that there will be zero electromotive force (EMF) but there is a non-zero EMF. Faraday's law appears to predict that there will be a non-zero EMF but there is zero EMF.
Faraday deduced his law of induction in 1831, after inventing the first electromagnetic generator or dynamo, but was never satisfied with his own explanation of the paradox.
Faraday's law compared to the Maxwell–Faraday equation
Faraday's law (also known as the Faraday–Lenz law) states that the electromotive force (EMF) is given by the total derivative of the magnetic flux with respect to time t:
where is the EMF and ΦB is the magnetic flux through a loop of wire. The direction of the electromotive force is given by Lenz's law. An often overlooked fact is that Faraday's law is based on the total derivative, not the partial derivative, of the magnetic flux. This means that an EMF may be generated even if total flux through the surface is constant. To overcome this issue, special techniques may be used. See below for the section on Use of special techniques with Faraday's law.
However, the most common interpretation of Faraday's law is that:
This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire,"The flux rule" is the terminology that Feynman uses to refer to the law relating magnetic flux to EMF. and is invalid in other circumstances. It ignores the fact that Faraday's law is defined by the total, not partial, derivative of magnetic flux and also the fact that EMF is not necessarily confined to a closed path but may also have radial components as discussed below. A different version, the Maxwell–Faraday equation (discussed below), is valid in all circumstances, and when used in conjunction with the Lorentz force law it is consistent with correct application of Faraday's law.
The Maxwell–Faraday equation is a generalization of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is:
(in SI units) where is the partial derivative operator, is the curl operator and again E(r, t) is the electric field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t.
The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin–Stokes theorem.
Paradoxes in which Faraday's law of induction seems to predict zero EMF but actually predicts 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 Feynman as noted below. See also A. Sommerfeld, Vol III Electrodynamics Academic Press, page 362.
The equipment
The experiment requires a few simple components (see Figure 1): a cylindrical magnet, a conducting disc with a conducting rim, a conducting axle, some wiring, and a galvanometer. The disc and the magnet are fitted a short distance apart on the axle, on which they are free to rotate about their own axes of symmetry. An electrical circuit is formed by connecting sliding contacts: one to the axle of the disc, the other to its rim. A galvanometer can be inserted in the circuit to measure the current.
The procedure The experiment proceeds in three steps: The magnet is held to prevent it from rotating, while the disc is spun on its axis. The result is that the galvanometer registers a direct current. The apparatus therefore acts as a generator, variously called the Faraday generator, the Faraday disc, or the homopolar (or unipolar) generator. The disc is held stationary while the magnet is spun on its axis. The result is that the galvanometer registers no current. The disc and magnet are spun together. The galvanometer registers a current, as it did in step 1.
Why is this paradoxical? The experiment is described by some as a "paradox" as it seems, at first sight, to violate Faraday's law of electromagnetic induction, because the flux through the disc appears to be the same no matter what is rotating. Hence, the EMF is predicted to be zero in all three cases of rotation. The discussion below shows this viewpoint stems from an incorrect choice of surface over which to calculate the flux.
The paradox appears a bit different from the lines of flux viewpoint: in Faraday's model of electromagnetic induction, a magnetic field consisted of imaginary lines of magnetic flux, similar to the lines that appear when iron filings are sprinkled on paper and held near a magnet. The EMF is proposed to be proportional to the rate of cutting lines of flux. If the lines of flux are imagined to originate in the magnet, then they would be stationary in the frame of the magnet, and rotating the disc relative to the magnet, whether by rotating the magnet or the disc, should produce an EMF, but rotating both of them together should not.
Faraday's explanation In Faraday's model of electromagnetic induction, a circuit received an induced current when it cut lines of magnetic flux. According to this model, the Faraday disc should have worked when either the disc or the magnet was rotated, but not both. Faraday attempted to explain the disagreement with observation by assuming that the magnet's field, complete with its lines of flux, remained stationary as the magnet rotated (a completely accurate picture, but maybe not intuitive in the lines-of-flux model). In other words, the lines of flux have their own frame of reference. As is shown in the next section, modern physics (since the discovery of the electron) does not need the lines-of-flux picture and dispels the paradox.
Modern explanations
A circuit is not necessarily a loop
In step 1, the paradox can be readily solved: the circuit does not constitute a simple loop of wire, as postulated by Faraday's law of induction; it is rather the union of two loops, because the current can flow through the two halves of the rim (see figure 2). If, on the other hand, one keep only one part of the rim from the radius junction to the brush, then the whole circuit is now a true loop whose shape varies with the time; then Faraday's law applies and leads to correct results.
Taking the return path into account
In step 2, since there is no current observed, one might conclude that the magnetic field did not rotate with the rotating magnet. (Whether it does or does not effectively or relatively, the Lorentz force is zero since v is zero relative to the laboratory frame. So there is no current measuring from laboratory frame.) The use of the Lorentz equation to explain this paradox has led to a debate in the literature as to whether or not a magnetic field rotates with a magnet. Since the force on charges expressed by the Lorentz equation depends upon the relative motion of the magnetic field (i.e. the laboratory frame) to the conductor where the EMF is located it was speculated that in the case when the magnet rotates with the disk but a voltage still develops, the magnetic field (i.e. the laboratory frame) must therefore not rotate with the magnetic material (of course since it is the laboratory frame), while the effective definition of magnetic field frame or the "effective/relative rotation of the field" turns with no relative motion with respect to the conductive disk.
Careful thought showed that, if the magnetic field was assumed to rotate with the magnet and the magnet rotated with the disk, a current should still be produced, not by EMF in the disk (there is no relative motion between the disk and the magnet) but in the external circuit linking the brushes,A. G. Kelly, Monographs 5 & 6 of the Institution of Engineers of Ireland, 1998, and ] which is in fact in relative motion with respect to the rotating magnet. (The brushes are in the laboratory frame.)
This mechanism agrees with the observations involving return paths: an EMF is generated whenever the disc moves relative to the return path, regardless of the rotation of the magnet. In fact it was shown that so long as a current loop is used to measure induced 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. (This depends on the definition, the motion of a field can be only defined effectively/relatively. If you hold the view that the field flux is a physical entity, it does rotate or depends on how it is generated. But this does not alter what is used in the Lorentz formula, especially the v, the velocity of the charge carrier relative to the frame where measurement takes place and field strength varies according to relativity at any spacetime point.)
Several experiments have been proposed using electrostatic measurements or electron beams to resolve the issue, but apparently none have been successfully performed to date.
Using the Lorentz force
The force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by the Lorentz force:See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation, F = q ( E+ v × B ), which gives the force acting on a point charge q in the presence of electromagnetic fields."
where × is the vector cross product. All boldface quantities are vectors. The relativistically-correct electric field of a point charge varies with velocity as:
where is the unit vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured, and θ is the angle between and . The magnetic field B of a charge is:
At the most underlying level, the total Lorentz force is the cumulative result of the electric fields E and magnetic fields B of every charge acting on every other charge.
When the magnet is rotating, but flux lines are stationary, and the conductor is stationary
Consider the special case where the cylindrical conducting disk is stationary but the cylindrical magnetic disk is rotating. In such a situation, the mean velocity v of charges in the conducting disk is initially zero, and therefore the magnetic force is 0, where v is the mean velocity of a charge q of the circuit relative to the frame where measurements are taken, and q is the charge on an electron.
When the magnet and the flux lines are stationary and the conductor is rotating
After the discovery of the electron and the forces that affect it, a microscopic resolution of the paradox became possible. See Figure 1. The metal portions of the apparatus are conducting, and confine a current due to electronic motion to within the metal boundaries. All electrons that move in a magnetic field experience a Lorentz force of , where v is the velocity of the electrons relative to the frame where measurements are taken, and q is the charge on an electron. Remember, there is no such frame as "frame of the electromagnetic field". A frame is set on a specific spacetime point, not an extending field or a flux line as a mathematical object. It is a different issue if you consider flux as a physical entity (see Magnetic flux quantum), or consider the effective/relative definition of motion/rotation of a field (see below). This note helps resolve the paradox.