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An error in The Lectures on Physics by Feyman, Leighton and Sands is discussed and corrected. It is caused by the improper application of Faraday’s in-duced voltage law.

There exists a great amount of confusion about the calculation of electromotive force commonly denoted as EMF. The introductory treatment for the calculation of EMF as presented in most elementary texts is inadequate both because the quantities often are not carefully defined and because the restrictions upon the validity of the formulas given are not emphasized. A fallacious understanding of Faraday’s law is given in many texts and in various internet sources. This short paper considers the EMF generated by plates rocking in a uniform magnetic field. This problem is taken from Feynman et al. [

In

making a complete circuit including the galvanometer. As the plates are rocked the place of contact moves continuously from point P to point P'. If we imagine the circuit to be completed through the plates on the dotted line shown in the figure, the magnetic flux through this circuit changes by a large amount as the plates are rocked. Feynman, Leighton and Sands [

EMF = − ∫ S ∂ B / ∂ t ⋅ d S + ∮ v × B ⋅ d l (1)

along a path and embraced area including the brushes and voltmeter leads. This equation is derived by Young [

We develop a definition for the electromotive force about any closed metallic circuit whose various parts may be moving relative to each other and which may or may not contain sliding contacts. Once again the desired definition of EMF must be consistent with Kirchhoff’s Law. When there is relative motion between parts of the circuit the resistance of a segment must be defined in such a way that it does not change if the segment is moving. That is, if one wishes to retain the formula for the dissipation in an element as I 2 R and because both the dissipation and the current are independent of the observer, it follows that the resistance to be used in this formula should be the same as the resistance calculated by an observer stationary with respect to that element. In that case the resistance of any element is defined as

R i = ( ∫ i E ′ ⋅ d l ) / ( ∫ i J ′ ⋅ d S ) (2)

where the primes denote that the fields are measured by an observer at rest with respect to the conductor at the point d l . (if the conductor is flexible it can be considered as comprising an infinite number of increments each of which has a resistance that is evaluated by an observer on that particular increment.) Because the current flowing through a conductor is independent of the observer, the I R drops about the circuit can be written as

∑ i I i R i = { ∑ i ( ∫ i J ′ ⋅ d S ) / [ ( ∫ i E ′ ⋅ d l ) / ( ∫ i J ′ ⋅ d S ) ] } = ∑ i ∫ i E ′ ⋅ d l = ∮ E ′ ⋅ d l (3)

It thus appears that a reasonable definition of the electromotive force about any prescribed path in a non-rigid, conducting circuit is given by

e = ∮ E ′ ⋅ d l (4)

where E ′ is defined as the electric field intensity evaluated at the element d l by an observer stationary with respect to the conductor at that point. To avoid having to use several reference frames in evaluating Equation (4),

E ′ = E + v × B (5)

Equation (5) from the special theory of relativity can be used to transform each element of Equation (4) to single given reference system. The fields measured by an observer in this fixed reference frame will be denoted by the unprimed quantities. By solving Equation (5) for E and substituting it into the Maxwell equation ∇ × E = − ∂ B / ∂ t , there results

∇ × E ′ = − ∂ B / ∂ t + ∇ × ( v × B ) (6)

Integrating both sides of Equation (6) over the surface enclosed by the path about which the EMF is to be calculated and applying Stokes Theorem there results

e = − ∫ S ∂ B / ∂ t ⋅ d S + ∮ v × B ⋅ d l (7)

where B is the total magnetic induction field due to all causes and has been assumed to be independent of the observer.

v is the velocity of the segment of the conductor at the point corresponding to d l relative to some given observer. (Note carefully that v is the velocity of the conductor and not the velocity of the path element d l in case the two differ as will be shown in later examples.)

∂ B / ∂ t is the vector time rate of change of B relative to the same observer.

Warning!! This formula has been derived from the results of the Special Theory of Relativity and hence is strictly justified only for the case of uniform motion. However, it is found experimentally to hold with negligible error for all accelerations that are physically attainable in the terrestrial laboratory. The resulting error only can be found from The General Theory of Relativity.

Once again the expression for the electromotive force (EMF) as exhibited in Equation (7) contains a surface integral and hence can be evaluated only for a closed circuit. It is nevertheless possible to define a unique potential difference between the terminals of a generator, just as in the case of rigid circuits. These terminals and the external circuit must lie within a region that to some observer is irrotational and that observer must see no part of the external circuit moving through a magnetic field. In other words, if it is possible to connect these terminals with a rigid line so that there is no EMF in any closed circuit formed by this line and any permitted external circuit, then it is possible to uniquely define a terminal voltage which is equal to the EMF (computed about the internal circuit and this rigid line) minus the IR drop in the generator: e.g.

V A B = − ∫ A B E ⋅ d l . As before the sign convention is such that the line integral of

Equation (4) is taken as positive in the direction going from the negative

terminal, A, through the generator to the positive terminal, B, and ∫ A B E ′ ⋅ d l ′ is

taken from A to B through the generator.

This section is restricted to cases where any segment of the circuit moving relative to any observer through a magnetic field is a conductor. The case of a dielectric moving through a magnetic field has been presented by Young [

It is best not to mix special relativistic equations such as Equation (5) with Faraday’s law that was established more than a century earlier. That was the mistake made by Feynman et al. [

Young, F.J. (2018) A Corrected EMF Problem. Open Access Library Journal, 5: e4270. https://doi.org/10.4236/oalib.1104270