Faraday paradox
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The Faraday paradox or Faraday's paradox is any experiment in which
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
's law of
electromagnetic induction Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field. Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk ...
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 Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...
or
dynamo file:DynamoElectricMachinesEndViewPartlySection USP284110.png, "Dynamo Electric Machine" (end view, partly section, ) A dynamo is an electrical generator that creates direct current using a commutator (electric), commutator. Dynamos were the f ...
, 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 In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
(EMF) is given by the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
of the magnetic flux with respect to time ''t'': :\mathcal = -\frac, where \mathcal is the EMF and Φ''B'' is the
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...
through a loop of wire. The direction of the electromotive force is given by
Lenz's law Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
. 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 Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
(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 Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
electric field, and vice versa. The Maxwell–Faraday equation is: (in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
) where \partial is the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
operator, \nabla\times is the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
operator and again E(r, ''t'') is the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
and B(r, ''t'') is the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. These fields can generally be functions of position r and time ''t''. The Maxwell–Faraday equation is one of the four
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, and therefore plays a fundamental role in the theory of
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
. It can also be written in an integral form by the
Kelvin–Stokes theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
.


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 magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, ...
, a conducting disc with a conducting rim, a conducting axle, some wiring, and a
galvanometer A galvanometer is an electromechanical measuring instrument for electric current. Early galvanometers were uncalibrated, but improved versions, called ammeters, were calibrated and could measure the flow of current more precisely. A galvanom ...
. 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 Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
. The apparatus therefore acts as a
generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...
, variously called the Faraday generator, the
Faraday disc A homopolar generator is a DC electrical generator comprising an electrically conductive disc or cylinder rotating in a plane perpendicular to a uniform static magnetic field. A potential difference is created between the center of the disc and th ...
, 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 A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
consisted of imaginary
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
of
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...
, 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 we shall see in the next section, modern physics (since the discovery of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
) 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 USMLE Step 1 (more commonly just Step 1 or colloquially, The Boards) is the first part of the United States Medical Licensing Examination. It aims to assess whether medical school students or graduates can apply important concepts of the found ...
, 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 Step(s) or STEP may refer to: Common meanings * Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * ''Steps'' (TV series), Hong Kong * ''Step'' (film), US, 2017 Literature * '' ...
, 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, 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: : \mathbf = \frac \frac\frac where \mathbf \hat r' 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 \mathbf v and \mathbf r'. The magnetic field B of a charge is: : \mathbf = \frac \mathbf \times \mathbf 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 The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
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 In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
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 The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
), or consider the effective/relative definition of motion/rotation of a field (see below). This note helps resolve the paradox. The Lorentz force is perpendicular to both the velocity of the electrons, which is in the plane of the disc, and to the magnetic field, which is normal (
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
) to the disc. An electron at rest in the frame of the disc moves circularly with the disc relative to the B-field (i.e. the rotational axis or the laboratory frame, remember the note above), and so experiences a radial Lorentz force. In Figure 1 this force (on a ''positive'' charge, not an electron) is outward toward the rim according to the right-hand rule. Of course, this radial force, which is the cause of the current, creates a radial component of electron velocity, generating in turn its own Lorentz force component that opposes the circular motion of the electrons, tending to slow the disc's rotation, but the electrons retain a component of circular motion that continues to drive the current via the radial Lorentz force.


Use of special techniques with Faraday's law

The flux through the portion of the path from the brush at the rim, through the outside loop and the axle to the center of the disc is always zero because the magnetic field is in the plane of this path (not perpendicular to it), no matter 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 the brush at the rim. Faraday's law of induction can be stated in words as: Mathematically, the law is stated: :\mathcal = - \frac = -\frac \iint_ d \mathbf \cdot \mathbf (\mathbf,\ t) \ , where ΦB is the flux, and ''d''A is a vector element of area of a moving surface Σ(''t'') bounded by the loop around which the EMF is to be found. How can this law be connected to the Faraday disc generator, where the flux linkage appears to be just the B-field multiplied by the area of the disc? One approach is to define the notion of "rate of change of flux linkage" by drawing a hypothetical line across the disc from the brush to the axle and asking how much flux linkage is swept past this line per unit time. See Figure 2. Assuming a radius ''R'' for the disc, a sector of disc with central angle ''θ'' has an area: : A = \frac \pi R^2 \ , so the rate that flux sweeps past the imaginary line is :\mathcal = - \frac = B \frac = B\ \frac \ \frac =B\ \frac \omega \ , with ''ω'' = ''dθ'' / ''dt'' the angular rate of rotation. The sign is chosen based upon
Lenz's law Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
: the field generated by the motion must oppose the change in flux caused by the rotation. For example, the circuit with the radial segment in Figure 2 according to the right-hand rule ''adds'' to the applied B-field, tending to increase the flux linkage. That suggests that the flux through this path is decreasing due to the rotation, so ''dθ'' / ''dt '' is negative. This flux-cutting result for EMF can be compared to calculating the work done per unit charge making an infinitesimal test charge traverse the hypothetical line using the Lorentz force / unit charge at radius ''r'', namely = ''Bv'' = ''Brω'': : \mathcal = \int_0^R dr Br \omega = \frac B \omega \ , which is the same result. The above methodology for finding the flux cut by the circuit is formalized in the flux law by properly treating the time derivative of the bounding surface Σ(''t''). Of course, the time derivative of an integral with time dependent limits is ''not'' simply the time derivative of the integrand alone, a point often forgotten; see
Leibniz integral rule In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are
and
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
. In choosing the surface Σ(''t''), the restrictions are that (i) it has to be bounded by a closed curve around which the EMF is to be found, and (ii) it has to capture the relative motion of all moving parts of the circuit. It is emphatically ''not'' required that the bounding curve corresponds to a physical line of flow of the current. On the other hand, induction is all about relative motion, and the path emphatically ''must'' capture any relative motion. In a case like Figure 1 where a portion of the current path is distributed over a region in space, the EMF driving the current can be found using a variety of paths. Figure 2 shows two possibilities. All paths include the obvious return loop, but in the disc two paths are shown: one is a geometrically simple path, the other a tortuous one. We are free to choose whatever path we like, but a portion of any acceptable path is ''fixed in the disc itself'' and turns with the disc. The flux is calculated though the entire path, return loop ''plus'' disc segment, and its rate-of change found. In this example, all these paths lead to the same rate of change of flux, and hence the same EMF. To provide some intuition about this path independence, in Figure 3 the Faraday disc is unwrapped onto a strip, making it resemble a sliding rectangle problem. In the sliding rectangle case, it becomes obvious that the pattern of current flow inside the rectangle is time-independent and therefore irrelevant to the rate of change of flux linking the circuit. There is no need to consider exactly how the current traverses the rectangle (or the disc). Any choice of path connecting the top and bottom of the rectangle (axle-to-brush in the disc) and moving with the rectangle (rotating with the disc) sweeps out the same rate-of-change of flux, and predicts the same EMF. For the disc, this rate-of-change of flux estimation is the same as that done above based upon rotation of the disc past a line joining the brush to the axle.


Configuration with a return path

Whether the magnet is "moving" is irrelevant in this analysis, due to the flux induced in the return path. The crucial relative motion is that of the disk and the return path, not of the disk and the magnet. This becomes clearer if a modified Faraday disk is used in which the return path is not a wire but another disk. That is, mount two conducting disks just next to each other on the same axle and let them have sliding electrical contact at the center and at the circumference. The current will be proportional to the relative rotation of the two disks and independent of any rotation of the magnet.


Configuration without a return path

A Faraday disk can also be operated with neither a galvanometer nor a return path. When the disk spins, the electrons collect along the rim and leave a deficit near the axis (or the other way around). It is possible in principle to measure the distribution of charge, for example, through the
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
generated between the rim and the axle (though not necessarily easy). This charge separation will be proportional to the relative rotational velocity between the disk and the magnet.


Paradoxes in which Faraday's law of induction seems to predict non-zero EMF but actually predicts zero EMF

These paradoxes are generally resolved by determining that the apparent motion of the circuit is actually deconstruction of the circuit followed by reconstruction of the circuit on a different path.


An additional rule

In the case when the disk alone spins there is no change in flux through the circuit, however, there is an electromotive force induced contrary to Faraday's law. We can also show an example when there is a change in flux, but no induced voltage. Figure 5 (near right) shows the setup used in Tilley's experiment.Tilley, D. E., Am. J. Phys. 36, 458 (1968) It is a circuit with two loops or meshes. There is a galvanometer connected in the right-hand loop, a magnet in the center of the left-hand loop, a switch in the left-hand loop, and a switch between the loops. We start with the switch on the left open and that on the right closed. When the switch on the left is closed and the switch on the right is open there is no change in the field of the magnet, but there is a change in the area of the galvanometer circuit. This means that there is a change in flux. However the galvanometer did not deflect meaning there was no induced voltage, and Faraday's law does not work in this case. According to A. G. Kelly this suggests that an induced voltage in Faraday's experiment is due to the "cutting" of the circuit by the flux lines, and not by "flux linking" or the actual change in flux. This follows from the Tilley experiment because there is no movement of the lines of force across the circuit and therefore no current induced although there is a change in flux through the circuit. Nussbaum suggests that for Faraday's law to be valid, work must be done in producing the change in flux. Nussbaum, A., "Faraday's Law Paradoxes", http://www.iop.org/EJ/article/0031-9120/7/4/006/pev7i4p231.pdf?request-id=49fbce3f-dbc4-4d6c-98e9-8258814e6c30
To understand this idea, we will step through the argument given by Nussbaum. We start by calculating the force between two current carrying wires. The force on wire 1 due to wire 2 is given by: : \mathbf_ = \frac I_1 I_2 \oint_ \oint_ \frac The magnetic field from the second wire is given by: : \mathbf_ = \frac I_2 \oint_ \frac So we can rewrite the force on wire 1 as: : \mathbf_ = I_1 \oint_ d \mathbf\ \mathbf \mathbf_ Now consider a segment d \mathbf of a conductor displaced d \mathbf in a constant magnetic field. The work done is found from: :d W = d \mathbf \cdot d \mathbf If we plug in what we previously found for d \mathbf we get: :d W = (I d \mathbf \mathbf \mathbf) \cdot d \mathbf The area covered by the displacement of the conductor is: :d \mathbf= d\mathbf \mathbf d\mathbf Therefore: :d W = I \mathbf \cdot d\mathbf = I d \Phi_B The differential work can also be given in terms of charge dq and potential difference V: :d W = V dq = V I dt By setting the two equations for differential work equal to each other we arrive at Faraday's Law. :d \Phi_B = V dt Furthermore, we now see that this is only true if d W is non-vanishing. Meaning, Faraday's Law is only valid if work is performed in bringing about the change in flux. A mathematical way to validate Faraday's Law in these kind of situations is to generalize the definition of EMF as in the proof of
Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
: :\mathrm = \oint \left(\mathbf + \mathbf\times\mathbf\right) \cdot d\boldsymbol The galvanometer usually only measures the first term in the EMF which contributes the current in circuit, although sometimes it can measure the incorporation of the second term such as when the second term contributes part of the current which the galvanometer measures as motional EMF, e.g. in the Faraday's disk experiment. In the situation above, the first term is zero and only the first term leads a current that the galvanometer measures, so there is no induced voltage. However, Faraday's Law still holds since the apparent change of the magnetic flux goes to the second term in the above generalization of EMF. But it is not measured by the galvanometer. Remember \mathbf is the local velocity of a point on the circuit, not a charge carrier. After all, both/all these situations are consistent with the concern of relativity and microstructure of matter, and/or the completeness of Maxwell equation and Lorentz formula, or the combination of them,
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
.


See also

*
Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
*
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
*
Moving magnet and conductor problem The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
* Corotation electric field


References


Further reading

*Michael Faraday, Experimental Researches in Electricity, Vol I, First Series, 1831 in Great Books of the Western World, Vol 45, R. M. Hutchins, ed., Encyclopædia Britannica, Inc., The University of Chicago, 1952
"Electromagnetic induction: physics and flashbacks" (PDF)
by Giuseppe Giuliani – details of the Lorentz force in Faraday's disc

– contains derivation of equation for EMF of a Faraday disc
Don Lancaster's "Tech Musings" column, Feb 1998
– on practical inefficiencies of Faraday disc
"Faraday's Final Riddle; Does the Field Rotate with a Magnet?" (PDF)
– contrarian theory, but contains useful references to Faraday's experiments *P. J. Scanlon, R. N. Henriksen, and J. R. Allen, "Approaches to electromagnetic induction," Am. J. Phys. 37, 698–708 (1969). – describes how to apply Faraday's law to Faraday's disc *Jorge Guala-Valverde, Pedro Mazzoni, Ricardo Achilles "The homopolar motor: A true relativistic engine," Am. J. Phys. 70 (10), 1052–1055 (Oct. 2002). – argues that only the Lorentz force can explain Faraday's disc and describes some experimental evidence for this *Frank Munley, Challenges to Faraday's flux rule, Am. J. Phys. 72, 1478 (2004). – an updated discussion of concepts in the Scanlon reference above. *Richard Feynman, Robert Leighton, Matthew Sands, "The Feynman Lectures on Physics Volume II", Chapter 17 – In addition to the Faraday "paradox" (where linked flux does not change but an emf is induced), he describes the "rocking plates" experiment where linked flux changes but no emf is induced. He shows that the correct physics is always given by the combination of the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
with the Maxwell–Faraday equation (see quotation box) and poses these two "paradoxes" of his own.
The rotation of magnetic field
by Vanja Janezic – describes a simple experiment that anyone can do. Because it only involves two bodies, its result is less ambiguous than the three-body Faraday, Kelly and Guala-Valverde experiments. *W. F. Hughes and F. J. Young, The Electromagnetodynamics of Fluids, John Wiley & Sons (1965) LCCC #66-17631. Chapters 1. Principles of Special Relativity and 2. The Electrodynamics of Moving Media. From these chapters it is possible to work all induced emf problems and explain all the associated paradoxes found in the literature. {{Michael Faraday Electrodynamics Michael Faraday