﻿ Electromagnetic induction

# Electromagnetic induction

Under electromagnetic induction ( Faraday's induction even after Michael Faraday induction short ) refers to the occurrence of an electric field by changing the magnetic flux density.

In many cases, it can be shown directly by measuring an electrical voltage the electric field. A typical example is shown in the adjacent image: Due to the movement of the magnet a voltage is induced, which can be measured at the terminals of the coil and is ready for further applications.

The electromagnetic induction was discovered in 1831 by Michael Faraday in the effort, the operation of an electromagnet ( " current produces magnetic field" ) to invert ( " magnetic field produces electricity "). The relationship is one of the four Maxwell's equations. The induction effect is technically used mainly in electrical equipment such as generators, electric motors and transformers. In these applications always occur alternating voltages.

• 5.1 induction law in differential form
• 5.2 Transition from the differential form of the integral form
• 5.3 's law of induction in integral form
• 5.4 Derivation of the law of induction for a conductor loop
• 5.5 Ohm's law for moving conductors
• 6.1 induction in quiescent systems 6.1.1 Broken metallic conductor loop
• 6.1.2 Closed ideal -type conductor loop
• 6.1.3 conductor loop with finite resistance
• 6.2.1 Moving conductor rod in a magnetic field

## Historic development and context

The electromagnetic induction as part of Maxwell's equations and classical electrodynamics ( KED ) reflects the state of knowledge at the end of the 19th century. At the time, some other terms and nomenclatures used for the presentations, the basic ideas about the induction process, however, were present.

As the discoverer of the law of induction are Michael Faraday, Joseph Henry and Hans Christian Ørsted, who discovered the Law of Induction in 1831 independently of each other, Faraday published his results first.

In Faraday's first demonstration building for the induction of 29 August 1831, he wrapped two conductor wires on the opposite sides of an iron core; an arrangement that resembles modern toroidal transformers. He expected because of his knowledge of permanent magnets whose properties were being investigated at this time only that a kind of wave along the ring spreads once in one of the two lines, a current begins to flow, and a current flow in the line on the other side of the ring leads. In the experiment, he joined one of the two wires to a galvanometer and looked at the pointer deflection, as he joined the other wire to a battery. In fact, he watched every time a transient current flow if the conductor was connected or disconnected with the battery. The cause of this induction phenomenon was the change of the magnetic flux in the plane of the conductor loop. In the following time Faraday identified further examples of electromagnetic induction. He observed, for example, transient currents in a coil arrangement when a permanent magnet rapidly into the coil and then moved out. From the historical research was also called Faradayscheibe, a direct current generator, produced, described from today's perspective as so-called movement induction and has its roots in the movement of the conductor and the entrained charges in a magnetic field. Faraday published the law, beginning with "The relation Which holds in between the magnetic pole, the moving wire or metal, and the direction of the current evolved, i.e. the law Which Governs the evolution of electricity by magneto- electric induction, is very simple, Although rather difficult to express. "

Beginning of the 20th century was the inclusion of relativistic law of induction in the context of special relativity. Unlike in mechanics, in which the special theory of relativity noticeable difference only at speeds nearing the speed of light, relativistic effects in the electrodynamics can be observed even at very low speeds. It could be described within the framework of the theory of relativity, such as change, for example, the magnitudes of the electric and magnetic field components as a function of the motion between an observer and an observed electric charge. These dependencies in the relative movement between each other different reference systems are described by the Lorentz transformation. It is shown that the law of induction in combination with the rest of Maxwell 's equations " Lorentz invariant " is. That is, the structure of the equations is not changed by the Lorentz transformation between different frames. This is also very clear that the electric and magnetic fields are only two manifestations of the same phenomenon.

In the mid-20th century in the context of electrodynamics mastered the combination of quantum mechanics with special relativity, and it was also the induction law formulated in the framework of a quantum field theory of electromagnetism. This quantum field theory is called quantum electrodynamics ( QED). It represents today, also because of the large -scale application domain, one of the experiments most accurately verified theories of physics dar.

## General

In the electric voltage generated by induction due to a magnetic flux density change is called a circulation voltage or induced voltage. This is characterized in that it is represented by closed lines of electric flux (eddy field). This results in the induction voltage of different voltages, as they occur ( potential field ), for example in a battery. The field lines of the so-called Urspannungsquellen emf of a battery (see electromotive forces ) always proceed from positive to negative charges and therefore are never closed.

In mathematical form, can the law of induction by each of the following three equally important equations describe:

In the equations represents the electric field strength and the magnetic flux density. The size of the oriented sheet and the edge ( the contour ) of the observed surface of integration; is the local speed of the contour line with respect to the underlying reference. The emerging line integral leads along a closed line and therefore ends at the starting point. A point multiplication between two vectors marked their scalar product.

## Initial experiments

In the following, several popular experiments to demonstrate electromagnetic induction are described whose importance lies primarily in school and university teaching. A basic induction experiment is taken up already in the introductory text. If you move the permanent magnet shown in the introductory text in the coil up and down, it can be tapped on the terminals of the coil with an oscilloscope, an electrical voltage.

Exactly analogously can be connected to the terminals of a conductor loop or coil an alternating electric voltage tap off when you turn the conductor loop in a temporally constant magnetic field, as shown in the picture. According to the principle shown there (but a fundamentally improved arrangement ) function the voltage generators are typically used in power plants for supplying electrical energy in the power system. In the experiment shown, the direction of action can in principle be reversed: Creates one of the terminals of the conductor loop is rotatably mounted an alternating electric voltage, so the conductor loop rotates about its axis in a magnetic field (electric motor).

The movement of a conductor in the magnetic field can also be used to generate an electrical DC voltage. This is exemplified in the figure " Moving conductors in the field." Moving the conductor bar along the rails, which are connected by a sliding contact or the wheels is electrically connected to the conductor rod, as can be measured by a DC voltage, which depends on the speed of the conductor bar, of the magnetic flux density and the distance between the rails on the voltmeter.

Instead of a linear motion can also demonstrate the experiment with a twisting motion, as shown by the example of Faradayscheibe. In the experiment shown, the aluminum plate takes over the function of the moving conductor rod from the experiment with the moving conductor rod in a magnetic field. ( Relative to the Earth's magnetic field moving or rotating conductive layers of air also exhibit the phenomenon of electric induction and be perceived as a thunderstorm or lightning generally at certain temperature and humidity conditions. )

By turning the aluminum plate in the magnetic field, it can be on the oscilloscope, which picks up the voltage by means of a sliding contact on the outer edge of the aluminum plate and to the axis of rotation, to demonstrate an electric voltage, with which for example, can operate an electric incandescent lamp. The voltage at the terminals depends on the strength of the magnetic flux density, the rotational speed and the diameter of the disc.

To the amazement of Faraday, however, such a Unipolargenerator unexpected properties that were discussed according to Faraday's discovery in the literature for a long time and led to a long -lasting controversy over the question of whether to speak as a material object could assign a velocity to the magnetic field and concrete whether the magnetic field rotates with the magnet. The main discovery was that the measurable voltage on the oscilloscope against an obvious intuitive assumption demonstrably not dependent solely on the relative motion between the permanent magnet and the aluminum disc. After you turn in the experiment shown, for example, only the permanent magnet and leaves the aluminum disc resting (), as can be observed despite the presence of relative motion between magnet and conductor on the oscilloscope no voltage. Turning the other hand, both discs at the same speed ( ), the oscilloscope will display a voltage even though the two wheels do not move relative to each other. Similarly, a voltage indicator can be observed when the voltage taps instead of the aluminum disc on the right is assumed to be an electrically conductive permanent magnet.

Although the controversy surrounding this issue in the context of special relativity Einstein's theory can be solved and it has been proven not arrive alone on the relative velocity between the magnet and conductor, is even today sometimes still used in schools teaching the so-called hedgehog model of the magnetic field, according to which the magnetic field lines as Hedgehog spines are attached to the magnet. Induction kick the model accordingly always on when the head of the field lines " cut " ( relative motion between the conductor and magnetic field). As part of the seminar teacher conference " Physics" in Dillingen 2002 Hubel pointed explicitly to the difficulties associated with the hedgehog model and emphasized the hedgehog model should not be misunderstood as a causal explanation of induction; it is rather untenable and could lead to misconceptions.

## Induction in a conductor loop

Although the general formulation of the law of induction does not require a conductor loop, should be considered on an existing thin, highly conductive wire conductor loop initially, as in many introductory textbooks usual induction. This enables a large number of technical applications such as motors and generators for rotation and alternating describe and understand, without the need for a treatment of the relativistic aspects of field theory or the application of the Lorentz transformation would be required.

For between the two wire ends ( for example, with the oscilloscope ) measurable electrical voltage is valid under these conditions in general:

Wherein the magnetic flux

Is passing through one (any ) of the conductor loop, which leads to the measuring device and the lines in the measurement device defined area. It can be shown that it is essential for the calculation of the flow not to the exact shape of the surface but only on the boundary.

The sign given is for the case that the orientation of the surface to each other (indicated by the arrow labeled ) and the direction of the arrow voltage as in the picture corresponding to a "left -hand rule ".

In the calculation, it is not necessary to distinguish whether the voltage of the arrangement is produced by a change in the flux density, or by a movement of the conductor.

### Example: Moving conductor rod in a magnetic field

The in the picture sketched measurement setup consists of a stationary electrically conductive rail assembly with the speed slides a conductor bar on the. The arrangement is in a spatially and temporally constant magnetic field with the flux density, which is caused by a stationary permanent magnet or a static DC powered coil assembly. The voltage between the two rails is measured with a voltmeter.

The voltage depends on the strength of the magnetic flux density, the speed and distance from the rail:

This will be explained in the following with the law of induction for the conductor loop:

• First it is checked if the directions of the B field is given as positive, and the voltage are arranged in the sense of the arrow left -hand rule. The arrow shown in the figure (x ) indicates that in the plane of the screen shows inside. Both sizes are therefore actually linked together in terms of a left -hand rule, so that the sign can be taken out of the equation.
• The area enclosed by the conductor and the meter surface is flat and has the surface area. Since the magnetic flux lines that area pierced vertically, applies.
• The voltage is calculated by means of the law of induction for the conductor loop. The first term is caused by the change in the flux density, and is also called the rest inductance. Because the magnetic flux density does not change with time, the first term in this example is zero. The second term is caused by the movement of the conductor bar and the concomitant increase in area. This term is also known as motion induction. It is, and in this case the voltage at the terminal significantly. Using the product rule for derivatives yields:

### Example: Moving conductor rod in the magnetic field ( with current flow )

The circuit has a finite resistance, there is used in the movement of the conductor bar to a current flow. For the current applies:

It should be taken into account on the one hand, that for the calculation of the flux change the entire active in the conductor loop flux density must always be considered. This is composed of the flux density of the permanent magnet and the flux density that comes into existence by the current flow.

With a large resistance, the effect of current flow to the magnetic field can be ignored, and the following applies:

### Example: conducting loop in a magnetic field

Business is a conductor loop with the angular velocity in a viewed from the laboratory frame temporally constant magnetic field, so the view from the conductor loop, the magnetic flux density is constantly changing, and there is a change in magnetic flux through the conductor loop.

The measured at the terminals in the rotating system voltage can be calculated as follows:

• The bounded through the conductor loop flat surface has the surface area.
• The magnetic flux density is constantly changing in the coordinate system of the comoving observer its magnitude and its direction. Assuming that the image shows the area at the time, so the fraction occurring perpendicularly to the surface of the flux density.
• The round thrusting through the surface magnetic flux is accordingly.
• Thus follows for the voltage using the chain rule:

### Induction in an electrical coil having a plurality of turns

#### Description by using the derivative of the magnetic flux

The induction law is also applicable for electrical coils having a plurality of turns. The area required for the calculation of the magnetic flux in the adjacent figure illustrates ( see also ). The law of induction, in its general form therefore does not require a factor of the number of turns of the coil, even if the coil wire is rotated several times in a specific case a cylinder.

In most publications on electromagnetic induction in electrical coils of the factor for the number of turns is the simplicity introduced, and the induction law is in the form

Specified. Herein, the flow through a bounded by the coil wire and the connections area, enclosed by a single turn magnetic flux, and is the measured voltage.

#### Time- integrated form, voltage-time area

By integrating over time the specified equation can be transformed as follows:

This relationship describes the course of the river as an integral function of the voltage profile.

Looking at the process in a time interval from 0 to T at constant surface through which the magnetic flux occurs - the time interval may extend, for example a half cycle of an alternating voltage -, it follows for the then resulting flow

For the case that means that the magnetic flux be always caused by a conductor loop or a flux change in this, as they are set by applying a voltage according to the given time t there from the voltage -time integral within the specified limits 0 to T, and this also must comply. The voltage is relevant for each of the induced voltage. This corresponds to the applied voltage less resistive voltage drops (I * R), where these are not negligible.

To illustrate, the voltage-time integral and the surface tension between the graph and the time axis over the interval [0; T], which is why you sometimes referred to it as a voltage-time or voltage-time sum, mostly in older literature based on the concept of the force pulse as a voltage surge. ( The reason for this is the fact that metrological earlier integration of induced voltage pulses was performed using a ballistic galvanometer, see also illustration of the magnetic force flow! )

Example of 50 Hz at Vrms = 230 V: determined on graphical manner by counting the small squares, one obtains the result of about 1.05 volt-second picture on the right, for a half-sine thus 2.1 volts seconds. That is, the voltage-time area, which transports the induction in the iron core of a transformer from one end to the other end of the hysteresis curve. If a transformer is designed to fit 230 V at 50 Hz, induction runs in continuous operation mainly in the vertical range of the hysteresis curve. Higher voltages or lower frequency leads to oversteer the hysteresis curve in the horizontally extending portions of the core saturation, which in practice is then clearly observable by the increase of the magnetizing current.

As another example, serve a widely practiced measuring principle for the magnetic flux: Here the flow to be measured is detected by a sensing coil, and if the voltage at the coil on an integrator as a result directly indicates the flow at its output.

### Detecting the change in flux

When a voltage is tapped off at the terminals of a rigid conductor loop, it can always be returned in accordance with the law of induction for the conductor loops a flux change in the conductor loop.

Hubel has the keyword " horseshoe paradox" suggests that this flux change remains in some cases hidden from the untrained eye, and discusses the problems using various arrangements with horseshoe magnets, such as those typically used in the classroom ( see adjacent photo ). While the change in flux in the conductor loop in the first arrangement for beginners is usually easily recognizable, this fails many students in the second image. Learners focus on the satisfied air of the assembly and do not consider that the flux density at the pole of the permanent magnet to the outside area decreases continuously (see third picture ).

Interestingly, two different observers who are asked for the cause of the stress, give different answers depending on their frame of reference:

• An observer for the horseshoe magnet at rest ( " observer sitting on the magnet " ), provides for negligible current in the conductor loop, a time constant field in the magnetic flux density (). It is also said that from the perspective of the observer exists an electric potential field. The measurable at the terminals of the loop voltage it is logically attributed to the effect of the Lorentz force.
• An observer for the conductor loop rests ( " observer sits on the circuit loop " ), but sees that changes the magnetic flux density in the conductor loop, because from his perspective, the horseshoe magnet moves in the conductor loop in out and out of her. This observer can be tapped on the terminals voltage is explained thus by an electric vortex field, which results from the change of the magnetic flux density. Since the conductor loop in his view, their position does not change (), he can finally realize no Lorentz force.

Although the questioning of the reference system ( inertial frame ) from which an observation is made, does not play a significant role in connection with the special case of " induction law for a conductor loop ," it must be strictly observed in the application of the general law. The reason is that electromagnetic quantities ( in particular, the electric field strength and hence the voltage ) of the reference system depend in which they are measured. The conversion is performed in each case using the Lorentz transformation.

### Flow control

The flow control is a possible generalization of the law of induction for a conductor loop dar. We considered doing a circuit represented by the curve and forms the boundary curve of a surface. If you disconnect the power circuit at a point on, we measures the voltage

Here, the size is the field strength that prevails in the rest frame of the co-moving with the speed waylet (that is, in the comoving system ). The equation is valid in the specified form solely for non-relativistic speeds. Since in almost all industrial applications, the objects are moving relatively slowly, this is usually not a relevant constraint.

## General law of induction in differential form and in integral form

The law of electromagnetic induction, short- law of induction describes the relationship between electric and magnetic fields. It states that if a change of the magnetic flux ring strain caused by an area on the edge of this area. In a particularly commonly used formulations, the law of induction will be described by the edge line of the surface is shown as a broken conductor loop at the open ends thereof, the voltage can be measured.

The understanding for meaningful description is divided into two possible sets of tables:

Both representations describe the same facts. Depending on the specific application and the problem it may be useful to use one or the other form.

In the application of the law of induction is important to note that all quantities appearing in the equations, that is, the electric field strength, the magnetic flux density, the oriented surface, the contour line of this area and the local velocity of a point on the surface or the edge line of any but for all sizes same reference system ( inertial ) from are described.

Does the contour line through matter, is also to be observed:

• The contour line is an imaginary line. Since it has no physical counterpart, a possible movement of the time contour line has basically no influence on the physical processes taking place. In particular, a movement of the contour line does not change the field sizes and. In the integral form I is the movement of the contour line is therefore not taken into account. In the integral form II, the movement of the imaginary contour line affects both sides of the equation to the same extent, so you when calculating, for example, an electrical voltage with integral form I to the same result occurs as when calculating the same voltage using integral form II
• Basically, the velocity of the contour line (e.g., conductor loop, magnets) may be the speed of the body used in the experiment differ. The speed of the contour line with respect to the observer is characterized by the frame of the article, while the speed of objects is described by the letter.
• In contrast to the movement of the contour line of the speed of the body in general has an influence on the physical processes taking place. This is especially true for the field variables and the measures of each observer.

### Induction law in differential form

The induction law in differential form is:

The presence of electric vertebrae or a time-varying magnetic flux density is the primary characteristic of the induction. In electric fields without induction (e.g., motionless in the field charges ) are no closed field lines of the electric field strength, and the contour integral of the electric field always gives zero.

Its main application is the induction law in differential form on the one hand with theoretical derivations and in the numerical field calculation, on the other hand (but rarely) in the analytical calculation of specific technical issues.

As was shown in Einstein's first work on the special theory of relativity, the Maxwell equations are in differential form in accordance with the special theory of relativity. An adapted to today's parlance this derivation can be found in the now out of print textbook by Simonyi.

### Transition from the differential form of the integral form

The relationship between the integral form, and the differential form can be described mathematically by means of the set of Stokes. Here, the global vortex and source strengths in local, discrete vortex and source densities, the individual spatial points ( points of a vector field ) are assigned, transferred.

The starting point is the induction law in differential form:

For conversion into the integral form of Stokes' theorem is used, which is formulated for obvious reasons with the variables:

Substituting in the right term of Stokes' law, the vector field corresponding to the induction law in differential form by the term, we obtain

This is a possible general form of the law of induction in integral form, which can be applied contrary to many otherwise claims for both contour lines in resting bodies as well as moving objects.

To obtain a formulation containing the magnetic flux is added to both sides of the equation the term. The findings indicate that

The right part of the equation corresponds to because of the negative rate of change of magnetic flux, so that the law of induction can also be listed in integral form in full generality as follows:

In many textbooks, these relationships are not recorded properly, which can be recognized that the prices quoted on the left side of the equation term is missing. The induction law is quoted correctly, however, for example in flow stream.

The error probably is that the missing term is slammed by mistake the electric field strength. ( Some authors refer to this as well as an effective electric field strength. ) In its consequence, the omission of the term means that the size of E is used inconsistently and depending on the context has a different meaning.

### Law of induction in integral form

In the following section, the first integral form of Faraday's Law is considered:

According to the mathematical formulation of the integral of the area shall be considered to be a constant time and does not consider its temporal changes.

With regard to the concept of induced voltage - the integral of the electric field strength - the in the picture drawn connecting line between points A and B will be considered first in an electric field.

The voltage between points A and B ( "outer poles " a "socket" ) can be approximately calculated by dividing the path into many small sections of the path. Since one can approximately assume a constant electric field strength along such a path Tückes due to the only small length, the result for the partial voltage along a waylet inside the value

The total voltage between two points thus obtained

The exact representation is defined using an integral. This can be thought of as the limit of an infinite number of path sections with infinitesimal length. For the calculation is defined A., a function of a parameter related to the points along the path described in the field ( ie in the arrow direction in the interior ). The voltage between two points can then be formally recorded via a line integral:

You can now migrate further the point along the contour of a total circulation until it has surrounded the enclosed area exactly once and again the same starting point (B = A), calculated as the total value of the voltage induced in the closed conductor loop circulation Voltage:

With respect to the sign is taken into account that while the contour in accordance with the right-hand rule, the circled area.

The third term of the above equations is the second term equivalent vectorial representation of the tangential field strength component by using the inner product, and the two integrals are so-called ring integrals are always used, if (as here) is integrated along a closed path, in this case along the contour of the loop C.

The induced voltage can be measured in a non- moving conductor loop approximated as a voltage drop with a voltmeter when you walk along the closed line attaching a conductor loop and this separates at one point. As to the conductor wire drops almost no electric voltage is all the induced voltage between the terminals.

### Derivation of the law of induction for a conductor loop

To the law of induction for the conductor loop

Derive the law of induction to be used in integral form II. For as in the law of induction for the conductor loop ( exceptionally) left-handed combination of surface normal and the direction of rotation eliminates the negative sign, and the law of induction is:

The contour line should state the path of the wire trace and be completed by the short path through the meter to a complete circulation. For this reason, the speed of the contour line and the speed of the conductor loop are always identical.

First it will be shown that the integral is within the wire to zero. Without loss of generality it may be assumed that the semiconductor element is moved in the positive direction. To transform this differential voltage in the reference system of the voltmeter is assumed to be dormant, the Lorentz transformation is applied. For movement of the primed reference system with the speed in the positive x direction, this is:

Since the metallic wire, by convention, highly conductive, according to Ohm's law applies to the field strength in the moving system. Using the Lorentz transformation follows

The term is therefore given as a point within the conductor wire:

The integral ring is fully visible from this base, in the form of the voltage at the terminals of the assembly.

### Ohm's law for moving conductors

In contrast to a stationary conductor, in which only the electric field strength affects current sourcing, acts on the charges in a moving conductor the complete Lorentz force

For nonrelativistic velocities measured in the stationary reference frame Lorentz force is equal to the force experienced by the charge in the comoving system.

For moving materials, Ohm's law is valid for the specific conductivity can be represented by the equation

Are defined by the electric field intensity, the speed of the respective conductive element and the magnetic flux density. Ohm's law is then as in the case of non-moving materials

## Applications of the general law of induction

In applications of the law of induction must be taken strictly on it, in which reference its associated operations are described. Because the electromagnetic field sizes change with a change of the reference system. This will be explained with an example:

• An observer looking at a charge that does not move relative to it. He notes that an electric field exists, the magnetic flux density is everywhere equal to zero:
• The observer moves away from the charge, as he recognizes in the relatively moving him to charge an electric current, which leads to a magnetic field with it. Thus, it is observed, an electric field and a magnetic field.

The interconversion of the fields into one another is described by the so-called Lorentz transformation, which arises from the special theory of relativity.

The charges moving relative to the observer only with low speeds, so the observer is in each such reference system is the same size of the Lorentz force

Measure. This can be interpreted to mean that when a change of the reference system electric and magnetic parts of the force merge.

At relativistic velocities is taken into account that the term ( three-component ) force from the classical ( non-relativistic ) mechanics and two comes very quickly against each other observers moving at the same physical process always observe different forces. For a correct description of relativistic forces, the so-called four forces that are elements of a Minkowski space are.

### Induction in quiescent systems

By way of introduction, two examples are considered with dormant components. Since the position of the electrical lines and components with time does not change when resting array ( ), it is convenient to choose the law of induction for a resting surface contour (). In this specific case the surface integral of the time derivative of the flux density and the total flux change are identical, and:

#### Broken metallic conductor loop

In the simplest case, a metallic conductor loop occurs with interruption. Since the interior of a conductor with negligible current flow or high electrical conductivity is field-free (), the entire circumferential stress occurs at the terminals voltage

On.

With an increase of the B- field at time step is an increase in the magnetic flux

Before, since the B-field and the surface normal of pointing in the same direction. According to the minus sign in the induction law, the voltage is negative. Decrease in the B- field at time step there is a reduction in flux. According to the minus sign in the induction law, the voltage is positive.

#### Closed ideal -type conductor loop

A closed loop circuit with ideal conductivity prevents changes the magnetic flux through the plane spanned by the conductor loop, because due to the ideal conductivity of the metal is the contour integral of the electric field equal to zero, and we have:

The formation of the flow change is prevented by the current induced in the conductor loop flows, A., a local change of the flux density generated because the magnetic field of the induced currents in conductors near the largest, and thus in the vicinity of the circuit occurs, the largest compensation effect. The total flow, in other words, the integrated over the entire loop surface flux density, but this does not change.

#### Conductor loop with finite resistance

In practice the electrical resistance of a conductor loop is always larger than zero. R is the electric resistance of the conductor, the following applies

Due to the resistance of the electric conductor an electric current flows, the instantaneous power cut off the magnetic field, and heats the conductor loop. According to this principle work, inter alia, induction cookers, where the energy for constant change of the magnetic field from the household power comes from.

The statement that the power of its cause counteracts is problematic in the sense of the selected model description. Indeed, flows with increasing magnetic flux due to the minus sign in the law of induction, a current opposite to the marked positive current direction. This current generates according to the flux rate, a magnetic field strength of H, the other way round as is the B field. It should however be noted that the induction law does not distinguish between self-excitation and external excitation. Insofar as the effect of compensation of the induced current is already contained in the magnetic flux which is used in the calculation.

### Induction in moving systems

In measuring systems with moving components occur even at low speeds to relativistic effects. This fundamental fact is illustrated by a simple thought experiment:

• An observer observing a ( relative to it is not moving) charge, to measure an electric field, however, due to the lack of current flow no magnetic field.
• The observer moves, however, on the charge to or away from it, he is on the one hand notice that the electric field changes due to the movement. This means that the observer measures a different E-field at the same distance from the load, but other relative speed to the load. On the other hand, the observer interprets the charge as well as a current moving towards or away from him to him. The observer is thus additionally to the electric field detect a magnetic field.

In order to prevent misunderstandings arising in measurements with moving components, is the indication of the reference system, relative to which the observations are described, absolutely necessary.

#### Moving conductor rod in a magnetic field

It will find the same conditions as in paragraph 2.1.

Consider the adjacent conductor rod in a magnetic field. Since the conductor loop is open, and therefore does:

In the moving with the speed of conductor bar, this results from the perspective of an observer in the rest frame of the field strength

Whereas in the region of the stationary conductor with a field strength of

Prevails.

If one calculates the ring integral over the electric field strength along the entire circuit, the result is

The ring integral can be but also via the induction law in the integral form I

Formulate what to due

Simplified.

The result for the measurable voltage at the terminals is accordingly

As the example given in many representations is displayed as an example of electromagnetic induction, to be explicitly confirmed:

• The terminal voltage can not be attributed to vortex of the electric field viewed from the rest frame out, since there are no such account.
• From the rest frame of view, so there is an electric potential field. The current driving force caused by the magnetic part of the Lorentz force.

The experiment shown standing next to Hering's paradox, named after Carl Hering, shows that the voltage meter does not detect rash, although a change in flux is present.

Arrangement: An electrically conductive ideal permanent magnet is moved with the velocity in a conductor loop. The upper and lower contact surface of the magnet are connected via fixed wheels electrically conductively connected to the lead wires drawn.

Intuitive approach: Outside of a formal approach, the terminal voltage is immediately obvious. After all, the arrangement comprises one of a voltmeter interrupted resting conductor loop in field-free space. The fact that the two ends of the loop come in contact with a moving permanent magnet does not change anything, since the charge of the permanent magnet this experience, do not leave without the action of an external force.

Paradox: The apparent contradiction of the experiment is clearly in a formal approach. For this purpose, it is convenient to put yourself in the frame of reference in which the voltmeter at rest ( laboratory system ). It should be investigated whether the experimental result contradicts the law of induction. To apply the law of induction, first a closed path of integration must be chosen. It should be agreed that this runs through the voltmeter and along the conductor loop and is completed by the red ( fixed ) line that passes through the magnet. The area through which the magnetic flux is defined, bounded by this circulation path, and should not change with time.

Obviously, the movement of the permanent magnets, results in a flow change in the plane surface, because prior to the introduction of the magnet, but does not penetrate the surface flow line, thereafter already. Due to the validity of the law of induction

Could now believe that the voltmeter should show a voltage of magnitude. After all, the wrong idea, can not exist in good conductive magnet, still in lead wire an E-field; the predicted from the law of induction electric field must, therefore, be between the terminals of the voltmeter and displayed by the voltmeter. In the experiment, however, the voltmeter shows, surprisingly, at no time a rash. This apparent contradiction describes the Hering's paradox.

Solution: The reasoning that leads to the apparent contradiction ( paradox ), does not consider the Lorentz transformation of the electric field strength and therefore comes to a wrong result. The following diagram is intended to show to the fact that the problem does not result in the correct application of the Lorentz transformation to a contradiction:

As before we consider the operations from the perspective of an observer at the voltmeter. The integration path runs along the conductor loop and is completed by the red line. For the electric field strength, the following applies:

We use the law of induction in the second integral form

Of. Because of integration time does not change (), it can also be

Be written.

The induced voltage is composed of the terminal voltage at the voltmeter and integrated over the length of the magnet in the electric field strength. According to the equation for the Lorentz transformation shows the electric field strength in the magnet, "upward". If the ring integral over the electric field strength, as usual, right-handed conducted to the surface normal (ie here: clockwise), we obtain

During the time of integration, the area enclosed by the surface of the magnet is increased by the magnetic flux and thus the right side of the equation thus

Substituting these values ​​into the law of induction a, it follows that is

The electric field, which one would expect to find actually in the voltmeter, is permanent magnet. This prepares learners many communication problems as they go out in a analogy to the induction law for a conductor loop implicitly assume that the temporal change of magnetic flux is always tapped on the terminals. The main difference to the induction law for a conductor loop is that in this case the speed of the contour line of the loop, the red line in the magnet and the speed of the conductor (in this case the magnet) differ from each other. The Hering's paradox shows no exception to the law of induction, but it is - as shown - easily compatible with the law of induction.

### Technical Applications

• Induction loop for the vehicle for the control of traffic signal systems and barriers
• Dynamic microphone
• Dynamic (magnetic ) pickup system for turntables
• Pickups for electric stringed instruments (eg electric guitar and electric bass)
• Head for scanning magnetic tape
• Generator = Dynamo = alternator
• RFID tag ( such as a ski pass)
• Transcranial magnetic stimulation
• Induction generator ( also inductive pulse ) as a speed sensor (eg in the automotive sector )
• Induction hardening
• Induction lamp
• Induction transmitter
• Transformer
• Induction loop system for the transmission of audio signals in hearing aids
• Up converter
• Betatron
• Induction linac

An electromagnetic force (EMF), that is, a measurable voltage at the terminals, is always present when the magnetic flux through the coil changes: As the river is the product of flux density and area, can intrude either the flux density B or change the area A; or both happen. A change in the surface is accomplished by the coil to rotate, for example, in a constant magnetic field and a magnet in a coil. Penetrated by the magnetic field surface is zero when the spool is transverse to the magnetic field, it is at a maximum when the field penetrates the coil axially. According to this principle ( dynamo ) is generated in a power generator, which can be picked up, for example, to non- co-rotating contact ribbons.

A change in the flux density is achieved, inter alia, by a varying magnetic field. According to this principle, an alternating voltage is induced in the secondary winding of a transformer in power supply to the primary winding with an alternating voltage whose level is proportional to the ratio of the number of turns.

This also includes all kinds of inductive heating by eddy current: the induction furnace, induction hardening, induction field, etc.

Inductive heating of materials: induction furnaces are mainly used in industry for hardening, brazing, melting, etc.. This technique is increasingly also in private before application, for example in the kitchen as an induction hob.

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