Faraday paradox

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The Faraday Paradox or Faraday's paradox is any experiment in which Michael Faraday's electromagnetic induction law appears to predict the wrong result. [B] Faraday paradox or Faraday's paradox Paradox falls into two classes:

• Faraday's law appears to predict that there will be zero EMF but there is a non-zero EMF.
• Faraday's law appears to predict that there will be a non-zero EMF but no EMF.

Faraday concluded his inductive law in 1831, after creating the first electromagnetic generator or dynamo, but was never satisfied with his own explanation of the paradox.

Video Faraday paradox

Hukum Faraday dibandingkan dengan persamaan Maxwell-Faraday

Hukum Faraday (juga dikenal sebagai Hukum Faraday-Lenz ) menyatakan bahwa gaya elektromotif (EMF) diberikan oleh turunan total fluks magnetik sehubungan dengan waktu t :

${\ displaystyle {\ mathcal {E}} = - {d \ Phi _ {B} \ over dt}, \}  ÂÂ$

di mana ${\ displaystyle {\ mathcal {E}}}  ÂÂ$ adalah EMF dan? _B adalah fluks magnetik. Arah gaya gerak listrik diberikan oleh hukum Lenz. Fakta yang sering diabaikan adalah bahwa hukum Faraday didasarkan pada turunan total, bukan turunan parsial, dari fluks magnetik. Ini berarti bahwa EMF dapat dihasilkan bahkan jika total fluks melalui permukaan konstan. Untuk mengatasi masalah ini, teknik khusus dapat digunakan. Lihat di bawah untuk bagian tentang Penggunaan teknik khusus dengan hukum Faraday. Namun, interpretasi yang paling umum dari hukum Faraday adalah bahwa:

The induced electromotive force in a closed circuit is equal to the negative time rate of the magnetic flux changes enclosed by the circuit.

Faraday's legal version is strictly applicable only when the closed circuit is an infinite loop of thin wire, and is invalid in other circumstances. This ignores the fact that Faraday's law is defined by total, not partial, magnetic flux derivatives and also the fact that EMF is not always limited to closed paths but may also have radial components as discussed below. A different version, the Maxwell-Faraday equation (discussed below), applies in all circumstances, and when used in conjunction with Lorentz's law of force is consistent with the correct application of Faraday's law.

The Maxwell-Faraday equation is a generalization of Faraday's law which states that a time varying magnetic field is always accompanied by a spatially varying non-conservative electric field, and vice versa. Maxwell-Faraday's equations are:

(dalam satuan SI) di mana ${\ displaystyle \ parsial}  ÂÂ$ adalah operator turunan parsial, ${\ displaystyle \ nabla \ times}  ÂÂ$ adalah operator kurung dan lagi E ( r , t ) adalah medan listrik dan B ( r , t ) adalah medan magnet. Bidang-bidang ini umumnya dapat berfungsi dari posisi r dan waktu t .

The Maxwell-Faraday equation is one of four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in integral form by the Kelvin-Stokes theorem.

The paradox where Faraday's law of induction seems to predict the EMF zero but actually predict non-zero EMF

This paradox is generally solved by the fact that EMF can be made by flux changes in the circuit as described in Faraday's law or by the conductor movement in the magnetic field. This is explained by Feynman as written below. See also A. Sommerfeld, Vol III Electrodynamics Academic Press, page 362.

Equipment

This experiment requires several simple components (see Figure 1): cylindrical magnets, conduction discs with rims, conduction axes, multiple cables, and galvanometers. Disks and magnets are installed apart short distances on the shaft, where they are free to rotate about the axes of their own symmetry. Electrical circuits are formed by connecting the shear contacts: one to the disk shaft, the other to the rim. A galvanometer can be incorporated into the circuit to measure the current.

Procedures

The experiment takes place in three steps:

1. The magnet is held to prevent it from spinning, while the disc is rotated on its axis. The result is that the galvanometer records a direct current. Therefore the apparatus acts as a generator, called a Faraday generator, a Faraday disc, or a homopolar (or unipolar) generator.
2. The disk is held while the magnet rotates on its axis. The result is that the galvanometer does not record the current.
3. Disk and magnet are played together. The Galvanometer records the current, as happens in step 1.

Why is this a paradox?

This experiment is described by some as a "paradox" as it seems, at first sight, to violate Faraday's law of electromagnetic induction, since the flux through the disk looks the same no matter what turns. Therefore, EMF is predicted to be zero in all three cases of rotation. The discussion below shows this point of view coming from the wrong surface choice to calculate the flux.

The paradox looks slightly different from the flux's point of view: in Faraday's electromagnetic induction model, the magnetic field consists of an imaginary line of magnetic flux, similar to the line that appears when iron filings are sown on paper and held near the magnet. The EMF is proposed to be proportional to the rate of flux line cutting. If the lines of flux are imagined to start with a magnet, then they will be stationary in the magnetic frame, and rotating the disk relative to the magnet, whether by rotating magnets or discs, must produce EMF, but spinning them both should not.

Maps Faraday paradox

Faraday's explanation

In Faraday's model of electromagnetic induction, the circuit receives an induced current when it cuts a magnetic flux line. According to this model, the Faraday disk should work when the disc or magnet is rotated, but not both. Faraday attempts to explain disagreement with observation by assuming that the magnetic field, complete with its flux line, remains silent when the magnet is rotated (a perfectly accurate picture, but may not be intuitive in the flux-line model). In other words, the flux line has its own frame of reference. As we will see in the next section, modern physics (since the invention of electrons) does not require a flux-line drawing and eliminates paradox.

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Modern explanation

Retrieve path back to account

In step 2, since no current is observed, one can conclude that the magnetic field does not rotate with a rotating magnet. (Whether it is not or is ineffective or relative, the Lorentz style is zero because v is zero relative to the laboratory frame.So there is no current measurement from the laboratory frame.) The use of the Lorentz equation to explain this paradox has led to debate in the literature, whether the magnetic field is spinning or not with a magnet. Since the force on the charge expressed by the Lorentz equation depends on the relative motion of the magnetic field (ie the laboratory frame) to the conductor where the EMF is located it speculates that in the case when the magnet rotates with the disk but the voltage is still developing, the magnetic field (ie the laboratory frame) does not rotate with magnetic material (of course because it is a laboratory frame), whereas the effective definition of the frame magnetic field or "effective/relative rotation of the field" changes without motion relative to the conductive disk.

Careful thinking shows that, if the magnetic field is assumed to be rotating with magnets and magnets rotated by disk, the current must still be produced, not by the EMF on the disk (no relative motion between disk and magnet) but in the external circuit connecting the brush, in motion relative to the rotating magnet. (Brush is in the lab frame.)

This mechanism agrees with observations involving the return path: the EMF is generated whenever the disk moves relative to the return path, regardless of the magnetic rotation. In fact it shows that as long as the current loop is used to measure EMFs induced from the movement of disks and magnets it is not possible to know whether the magnetic field is not or does not rotate with the magnet. (This depends on the definition, the motion of a field can only be defined effectively/relative.If you hold the view that the field flux is a physical entity, it rotates or depends on how it is produced.But this does not change what is used in the Lorentz formula, b> v , the carrier velocity relative to the frame at which the measurement occurs and the field strength varies according to the relativity at the point of time.)

Several experiments have been proposed using electrostatic measurements or electron beams to solve the problem, but nothing seems to be done successfully to date.

Using Lorentz style

The F force acting on electric charge particles q with an instantaneous velocity v , due to the external electric field E and the magnetic field B , given by Lorentz forces:

di mana ÃÆ' - adalah produk vektor silang. Semua kuantitas tebal adalah vektor. The relativistically-correct medan listrik dari muatan titik bervariasi dengan kecepatan seperti:

${\ displaystyle \ mathbf {E} = {\ frac {q} {4 \ pi \ epsilon _ {0}}} {\ frac {1-v ^ {2}/c ^ {2}} {(1-v ^ {2} \ sin ^ {2} \ theta/c ^ {2}) ^ {3/2}}} {\ frac {\ mathbf {{\ hat { r}} '}} {| \ mathbf {r}' | ^ {2}}}}  ÂÂ$

di mana ${\ displaystyle \ mathbf {\ hat {r}} '} Â$ adalah vektor satuan yang menunjuk dari posisi saat ini (non-terbelakang) dari partikel ke titik di mana lapangan sedang diukur, dan? adalah sudut antara ${\ displaystyle \ mathbf {v}} Â$ dan ${\ displaystyle \ mathbf {r} '} Â Â$ . Medan magnet B muatan adalah:

${\ displaystyle \ mathbf {B} = {\ frac {1} {c2}} \ mathbf {v} \ times \ mathbf {E }} Â Â$

At the most basic level, the total Lorentz force is the cumulative result of the electric field E and the magnetic field B of each charge acting on any other load.

When the magnet rotates, but the flux lines of stationary and stationary conductor

Consider the special case in which the stationary cylindrical shaping disks but the cylindrical magnet disks are rotating. In such situations, the average velocity v of charge in the grounding disk is initially zero, and therefore the magnetic force F = q v ÃÆ'â € " B is 0, where v is the average speed of charge q of the circuit relative to the frame in which the measurement is taken, and q is the charge on the electron.

Note that v does not represent the speed that the magnetic field lines run through the conductor. The observed magnetic field pattern depends on the frame of reference. They do not have their own speed. To illustrate, imagine that someone took a magnet and turned it over 180 degrees. The assumption that these field lines actually have their own velocities will be magnetic field lines that will far swing to the other side, potentially faster than the speed of light, but that's not what happens.

Instead, what happens is that rotating the magnet causes the subatomic particles in the magnet to obtain a change in velocity. However, in the rotated magnet, the velocity of electrons will vary in front of or behind other particles, due to the lower mass compared to the nuclei. There is a long, diffuse field contraction of the moving charge, and the electric field length contraction E of the electron will be greater or less than the contraction of the E field of the positive nucleus depending on whether magnetic rotation is aligned with, or opposite to, the electron spin that generates magnets.

In the case of a symmetrical spinning magnet at a constant velocity, the magnetic field intensity distribution B magnet is constant with time, even after taking into account the relativistic correction for B , and therefore by equation Maxwell-Faraday electric field caused by curl-free magnetic rotation, caused purely by the contraction of electric field length which is a subatomic particle. This means that in this particular example with a rotating magnetic disk and a stationary drive disk, the induced electric field can not be explained by the Maxwell-Faraday equation, which describes the electric field curl induced by the change in magnetic flux density.

Therefore, in this view, the magnetic field does not rotate with its magnetic source, and they exist independently of them. However, Lorentz forces generated by magnetic rotation are as if these lines are rotated simultaneously with it, but this is actually due to the effects of relativity on the electric field.

When magnets and stationary flux lines and conductors spin

After the discovery of electrons and forces that affect it, microscopic resolution of the paradox becomes possible. See Figure 1. The metal parts of the equipment are performing, and limit the currents caused by electronic movement into metal boundaries. All electrons moving in the magnetic field experience the Lorentz force F = q v ÃÆ'â € " B , where v is the velocity of the electron relative to the frame in which the measurement is taken, and q is the charge on the electron. Remember, there is no frame like "electromagnetic field frame". The frame is fixed at a particular point of time, not an expanded field or a flux line as a mathematical object. This is a different matter if you think of flux as a physical entity (see quantum magnetic flux), or consider the effective/relative definition of the field motion/rotation (see below). This note helps resolve the paradox.

The Lorentz force is perpendicular to the velocity of the electrons, which are in the plane of the disk, and the magnetic field, which is normal (normal surface) to the disc. An electron at rest in a disc frame moves circularly with a disc relative to the B-field (ie the rotational axis or laboratory frame, remember the note above), and so experiences the Lorentz radial force. In Figure 1 this force (on a positive charge, not an electron) comes out to the edge according to the right hand rule.

Of course, this radial force, which is the cause of the current, creates the radial component of the electron velocity, producing its own Lorentz force component opposed to the circular motion of electrons, tending to slow the rotation of the disk, but the electrons maintain a circular motion component that continues to drive current through the Lorentz radial force.

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Use of special techniques with Faraday's law

Flux through the road part of the brush at the edge, through the outer loop and the shaft to the center of the disk is always zero because of the magnetic field in the field of this path (no perpendicular to it), no matter what is spinning, so emf is integrated around this part of the road always zero. Therefore, attention is focused on the portion of the road from the axis in the disc to the brush at the edge.

Faraday's inductive law can be expressed in words as:

The induced electromotive force or EMF in a closed circuit is equal to the rate of change of magnetic flux through the circuit.

Secara matematis, hukum dinyatakan:

${\ displaystyle {\ mathcal {E}} = - {\ frac {d \ Phi B}} {dt}} = - {\ frac { d} {dt}} \ iint_ {\ Sigma (t)} d {\ boldsymbol {A}} \ cdot \ mathbf {B} (\ mathbf {r}, \ t) \,} Â Â$

Where? _B is the flux, and d A is the vector element of the moving surface area ? ( t ) is limited by the loop where EMF can be found.

How can this law be connected to a Faraday disk generator, where the flux relationship seems to be just a B-field multiplied by the disk area?

One approach is to define the idea of ​​"the rate of change of the flux relationship" by drawing a hypothetical line across the disc from the brush to the shaft and asking how many flux relations are swept across this line per unit of time. See Figure 2. Assuming the radius of R for disk, disk sector with the central angle ? has an area:

${\ displaystyle A = {\ frac {\ theta} {2 \ pi}} \ pi R ^ {2} \,}$

Sehingga laju fluks yang melewati garis imajiner adalah

${\ displaystyle {\ mathcal {E}} = - {\ frac {d \ Phi B}} {dt}} = B {\ frac { dA} {dt}} = B \ {\ frac {R2} {2}} \ {\ frac {d \ theta} {dt}} = B \ {\ frac {R2}} { 2}} \ omega \,} Â Â$

with ? = d? / dt angular rotation level. This mark is selected based on Lenz's law: the field generated by the movement must oppose flux changes caused by rotation. For example, the circuit with radial segment in Figure 2 corresponding to the right-hand rule add to the applied B field, tends to increase the flux relationship. It shows that the flux through this path decreases due to rotation, so d? / dt is negative.

Hasil fluks-cutting untuk EMF ini dapat dibandingkan dengan menghitung pekerjaan yang dilakukan per satuan muatan membuat muatan uji yang sangat kecil melintasi garis hipotetis menggunakan muatan gaya/satuan Lorentz pada radius r , yaitu | v ÃÆ'— B | = Bv = Br? :

${\ displaystyle {\ mathcal {E}} = \ int_ {0} ^ {R} drBr \ omega = {\ frac {R ^ {2}} {2} } B \ omega \,}  ÂÂ$

which is the same result.

The above methodology for finding flux cuts by series formalized in flux law correctly treats the time derivative of the dividing surface? ( t ). Of course, the integral time derivative with the dependent timeout is not just a time derivative of the integrand only, a point that is often forgotten; see Leibniz's integral rules and Lorentz forces.

In choosing a surface? (I) t ), the restriction is that (i) it must be limited by the closed curve in which the EMF can be found, and (ii) must capture the relative motion of all moving parts of the circuit. Strictly not it is required that the boundary curve correspond to the physical line of the current flow. On the other hand, induction is all about relative motion, and its paths firmly must capture any relative motion. In the case of FIG. 1 in which a portion of the path is currently distributed to a region in space, the current-driven EMF can be found using multiple paths. Figure 2 shows two possibilities. All paths include clear loopbacks, but in two-way disks are shown: one is a simple geometric path, the other is tortuous. We are free to choose whatever path we like, but some of the acceptable paths are fixed in the disk itself and changed with disk. Flux is calculated through the entire path, returning the loop plus disk segment, and the rate of change is found.

In this example, all of these paths lead to the same flux change rate, and hence the same EMF. To give some intuition about the independence of this path, in Figure 3 the Faraday disk is opened to the strip, making it resemble the problem of a sliding rectangle. In the case of a sliding rectangle, it becomes clear that the current-flow pattern inside the rectangle is time-independent and therefore irrelevant to the rate of flux change connecting the circuit. No need to consider exactly how the current crosses a rectangle (or disc). Each selection of paths connecting the top and bottom of the rectangle (shaft-to-brush in the disk) and moving with a rectangle (rotating with the disk) sweeps the same flux change rate, and predicts the same EMF. For disks, the rate of change of flux estimation is the same as done above based on disk rotation over the line connecting the brush to the shaft.

Configure with back path

Whether the magnet "moves" is irrelevant in this analysis, since the flux is induced in the return path. The important relative movements are the movement of disks and return paths, not from discs and magnets. It becomes more obvious if the modified Faraday disk is used where the return path is not a wire but another disk. That is, mounting two disks do just next to each other on the same shaft and let them shift the electrical contacts at the center and around it. The current will be proportional to the relative rotation of the two disks and is independent of magnetic rotation.

Configuring without back path

Faraday disks can also be operated without galvanometer or backlash. When the disk is rotating, the electrons collect along the edges and leave the deficit near the axis (or vice versa). It is possible in principle to measure the charge distribution, for example, through the electromotive force generated between the rim and the shaft (though not always easy). This charge separation will be proportional to the relative rotational speed between disk and magnet.

The paradox where Faraday's law of induction seems to predict non-zero but actually predicts zero EMF EMF

This paradox is generally solved by determining that the real motion of the circuit is actually the deconstruction of the circuit followed by the reconstruction of the circuit on different paths.

Additional rules

Medan magnet dari kawat kedua diberikan oleh:

${\ displaystyle \ mathbf {B} _ {2} = {\ frac {\ mu_ {0}} {4 \ pi}} I_2} oint {C2}} {\ frac {(d \ mathbf {l_ {2}} \ \ mathbf {\ times} \ {\ hat {\ mathbf {r}}} 21 {21})} {r_ {21 } ^ 2}}} \} Â Â$

Jadi kita dapat menulis ulang gaya pada kawat 1 sebagai:

${\ displaystyle \ mathbf {F} 21 {1} \ oint C {1}} d \ mathbf {l_ {1} } \ \ mathbf {\ times} \ mathbf {B} _2}} Â Â$

Sekarang perhatikan segmen ${\ displaystyle d \ mathbf {l}}  ÂÂ$ dari konduktor mengungsi ${\ displaystyle d \ mathbf {r}}  ÂÂ$ dalam medan magnet konstan. Pekerjaan yang dilakukan ditemukan dari:

${\ displaystyle dW = d \ mathbf {F} \ cdot d \ mathbf {r}}  ÂÂ$

Jika kita memasukkan apa yang sebelumnya kita temukan untuk ${\ displaystyle d \ mathbf {F}}  ÂÂ$ yang kami dapatkan:

$            {\displaystyle         d        W        =        (        \mathrm{Saya}        d  }$

Source of the article : Wikipedia