Hering's Paradox
Hering's paradox describes a physical experiment in the field of electromagnetism that seems to contradict Maxwell's equation in general, and Faraday's Law of Induction and the flux rule in particular.
In his study on the subject, Carl Hering concluded in 1908 that the usual statement of Faraday's Law (at the turn of the century) was imperfect and that it required to be modified in order to become universal.
[1] Since then, Hering's paradox has been used repeatedly in physics didactics to demonstrate the application of Faraday's Law of Induction,[2][3][4][5] and it can be considered to be completely understood within the theory of classical electrodynamics.
Grabinski criticizes, however, that most of the presentations in introductory textbooks were problematical.
Either, Faraday's Law was misinterpreted in a way that leads to confusion, or solely such frames of reference were chosen that avoid the need of an explanation.
[6] In the following, Hering's paradox is first shown experimentally in a video and -- in a similar way as suggested by Grabinski -- it is shown, that when carefully treated with full mathematical consistency, the experiment does not contradict Faraday's Law of Induction.
Finally, the typical pitfalls of applying Faraday's Law are mentioned.
In the experiment, a slotted iron core is used, where a coil fed with a direct current generates a constant magnetic field in the core and in its slot.
The easiest way to understand the outcome of the experiment is to view it from the rest frame of the magnet, i. e. the magnet is at rest, and the oscilloscope and the wires are at motion.
To conclude, there is While the perspective from the rest frame of the magnet causes no difficulties in understanding, this is not the case when viewed from a frame of reference in which the oscilloscope[11] and the cables are at rest and an electrically conductive permanent magnet moves into a conductor loop at a speed of
Under these circumstances, there is rest induction due to the movement of the magnet (
The correct answer to this question is "Yes, it does", and it is one of the pitfalls concerning the application of Faraday's Law.
For some people it is contraintuitive to assume that a Lorentz force is exerted to a charge although there is no relative motion between the magnet and the charge.
[12] An essential step of solving the paradox is the realization that the inside of the conductive moving magnet is not field-free, but that a non-zero electric field strength
However, the induced voltage is not localized in the oscilloscope, but in the magnet.
can be derived from the consideration that there is obviously no current-driving force acting on any section of the circuit.
Finally, the following electric field strengths result for the various sections of the conductor loop: To check whether the outcome of the experiment is compatible with Maxwell's equations, we first write down the Maxwell Faraday equation in integral notation: Here
is its boundary curve, which is assumed to be composed of the (stationary) sections
The direction of integration (clockwise) and the surface orientation (pointing into the screen) are right-handed to each other as assumed in the Maxwell Faraday equation.
Considering the electrical field strengths shown in the table, the left side of the Maxwell Faraday equation can be written as: The minus sign is due to the fact that the direction of integration is opposite to the direction of the electric field strength (
To calculate the right-hand side of the equation, we state that within the time
the magnetic field of the induction surface increases from
This shows that Hering's paradox is in perfect agreement with the Maxwell Faraday equation.
of the Maxwell-Faraday equation where neither the induction area nor its boundary occurs.
From a mathematical point of view, the boundary curve is just an imaginary line that had to be introduced to convert the Maxwell-Faraday equation to its integral notation such as to establish a relationship to electical voltages.
Because the boundary curve is physically of no importance, the outcome of an experiment does not depends on the speed of this curve and it is not affected by whether or not the speed of the boundary curve corresponds to the speed of a conductor wire being located at the same place.
For reasons of simplicity,[13] the speed of the boundary curve is assumed to be zero in this article.
It affects the value of the electric field strength inside the magnet and is thus accounted for in the Maxwell-Faraday equation via the numerical value of the vector field
The difficulties in understanding Hering's paradox and similar problems are usually based on three misunderstandings: If these points are consistently considered, Hering's paradox turns out to be in perfect agreement to Faraday's law of induction (given by the Maxwell Faraday equation) viewed from any frame of reference whatsoever.
Furthermore, the difficulties in understanding the (thought) experiments described in the chapter "Exceptions to the flow rule" in the "Feynman Lectures" are due to the same misunderstandings.
