# What is an induced current

## Electromagnetic induction

Under the **electromagnetic induction** (short: **induction**) one understands the creation of an electrical voltage along a conductor loop due to the change in the magnetic flux. Electromagnetic induction was discovered by Michael Faraday in 1831 while attempting to reverse the function of an electromagnet ("current generates magnetic field") ("magnetic field generates electricity"). The context in its integrated form is also called that *Faraday's law* and is part of Maxwell's equations.

The induction effect is mainly used technically in electrical machines such as generators, electric motors and transformers. Most of these applications involve AC voltages. However, there are also applications in which direct voltages arise directly and without rectification through electromagnetic induction, as is the case with unipolar induction.

Two different ways of looking at induction are common: The first explains induction with the help of the Lorentz force and the effect of the force on moving electrical charge carriers such as electrons. However, in certain situations, such as with magnetic shields or unipolar induction, this idea can be associated with problems in understanding. The second common model uses methods from field theory and explains the induction process with the help of changes in magnetic fluxes and the associated magnetic flux densities.

### General

The law of electromagnetic induction, or induction law for short, describes the conditions under which an electrical voltage is produced (induced) along an electrical conductor loop. A conductor loop can be implemented, for example, in the form of a coil used in electrical engineering.

The vector description, which is useful for understanding, is divided into two possible forms of representation:

- The integral form or also
*global shape*of the law of induction. The global properties of a spatially extended field area are described via the integration path. - The differential form or also
*local shape*of the law of induction. The properties of individual local field points are described in the form of densities. The volumes of the global form tend towards zero and the field strengths that occur are differentiated.

Both forms of representation describe one and the same facts. Depending on the specific application and problem, it can make sense to use one or the other form. Both forms of representation are described below.

### Law of induction in integral form

A magnetic flux that changes over time induces a voltage in a conductor loop surrounding it *U*_{ind}. This voltage can be measured by means of a voltage measuring device when the closed conductor loop is opened. The actual induction effect comes about because a time-changing flow dΦ / dt is surrounded by an electrical vortex field with self-contained field lines. This vortex electric field *E.*_{tan}, only the tangential field component plays a role, leads along an infinitesimal piece d*r* the conductor loop to an induced voltage d*U*_{ind}:

- d
*U*_{ind}=*E.*_{tan}d*r*

The summation of all infinitesimal voltages along the conductor loop results in the one between the points *r*_{A.} and *r*_{B.} induced voltage:

If you now mentally draw the two points *r*_{A.} and *r*_{B.} together, it strives in the closed conductor loop with the contour *C.* (*r*_{A.} = *r*_{B.}) induced voltage to the induced circulating voltage *U*_{uind}:

The third term is the equivalent vector representation of the tangential field strength component with the help of the scalar product. The integral represents a so-called ring integral and should indicate that over a complete revolution along the loop *C.* is integrated.

The circulating voltage *U*_{uind} is identical to that used in transformer calculations *Winding voltage* which indicates the induced voltage for one turn. The practical importance of the circulating voltage is that it can be used to calculate the voltages induced in the windings of electrical machines. The circulating voltage is directly proportional to the rate of change of the magnetic flux over time and is therefore also a measure of the eddy strength of the electrical eddy field.

The negative sign in the above equation results from Lenz's rule: The induced field strength is always directed in such a way that a current change caused by it or an associated change in the magnetic field counteracts the causing current or magnetic field change.

Because rivers are not individual points in space but only areas *A.* assign, the magnetic flux is usually expressed by the corresponding magnetic flux density*B.* out:

so that the law of induction can be written in the usual integral form as:

The order of integration or differentiation can be reversed for static conductors.

With time-varying magnetic fluxes, closed electrical vortex fields exist not only along the conductor loop but also, for example, inside the iron core of a transformer. There they are the cause of the mostly undesired eddy currents and the so-called eddy current losses.

### Law of induction in differential form

The transition from the integral form to the differential form is nothing more than the transfer of the global vortex and source strengths into local, discrete vortex or source densities which are assigned to individual spatial points (points of a vector field). In the integral form there are numerical values with dimensions such as the magnetic flux, in the differential form of the law of induction the transition to functions is carried out in relation to function theory.

The starting point is the global law of induction for bodies at rest, whereby the restriction to bodies at rest is essential in the following:

Now becomes the area of the contour *C.* continuously reduced in that equation, if you let it strive towards zero in order to obtain a point, the associated vortex strength also strives towards zero. One way out is to establish a relationship between the vortex strength and the associated bordered area before the transition and to work with this relationship further on. The value of this point then remains finite and is a characteristic property of the considered field point, with which the local shape is formed.

Mathematically, this path is followed in such a way that the above equation in relation (ratio) with a vector surface element *A.*′, Whose normal vector **n** points in the direction of flow is set, and then a limit value determination for *A.*′ → 0 takes place. This eliminates the need for integration through the area integral:

The sequence that occurs and initially appears complicated, starting from the left with integration, followed by division and limit value formation, is summarized in vector analysis as a special vector operator called red (for *rotation*, engl. *curl*), with which the differential representation of the law of induction has the form:

accepts. The local vortex density is red**E.** of the electric field **E.** is therefore equal to the negative change in the local magnetic flux density over time **B.**.

For practical application it is important that the vector operations, like red, relate to a specific coordinate system. This was implicit in the above equation by the normal vector of *A.*'Anticipated. Depending on the freely selectable coordinate system, there are different representations for the vector operation.

### Application to different geometries

The law of induction describes the occurrence of induction with different geometries:

- A conductor loop rotates in a constant B-field.
- A conductor loop is penetrated by a changing B-field.
- A conductor loop changes the area penetrated by the B-field.

### Broken metallic conductor loop

In the simplest case, there is a metallic conductor loop with an interruption. Since the metal has a very good electrical conductivity compared to the interruption, no electrical voltage drops along the conductor loop (from 1 'to 1). The total circulating voltage drops across the terminals as voltage

from.

If the B-field increases during the time step dt, there is an increase in the magnetic flux before, since the B-field and the area d*A.* point in the same direction. The voltage corresponds to the minus sign in the law of induction *u*_{11'} negative.

If the B-field decreases during the time step dt, there is a decrease in the flow in front. The voltage corresponds to the minus sign in the law of induction *u*_{11'} positive.

In the description, an integration in positive time direction (i.e.*t* > 0) required.

### Closed, ideally conductive conductor loop

A closed conductor loop with ideal conductivity prevents the magnetic flux through the conductor loop from changing, because because of the ideal conductivity of the metal, the rotation integral of the voltages is zero, and the following applies:

The change in flux is prevented by the currents induced in the conductor loop.

In practice, the electrical resistance of the short-circuit loop is slightly greater than zero. If the electrical resistance of the conductor is R, the following applies:

Because of the low resistance of the electrical conductor, high electrical currents flow, which reduce the momentary power to the magnetic field remove and heat the conductor loop. Working according to this principle, inter alia. Induction brakes and induction cookers. With induction brakes, the energy required to maintain the B-field comes from the movement of the associated vehicle. This steadily decreases until the vehicle has slowed down. With induction stoves, the energy to maintain the magnetic field comes from the household network.

The statement that the electricity *i*(*t*) counteracts its cause is problematic in terms of the selected description model. In fact, with increasing magnetic flux, due to the minus sign in the law of induction, a current flows in the opposite direction to the positive current direction shown *i*(*t*). This current generates a magnetic field strength according to the flow rate theorem *H*that points the other way around than the B-field. It should be noted, however, that the law of induction does not differentiate between self-excitation and external excitation. In this respect, the compensation effect of the induced current is already in the magnetic flux , which goes into the law of induction.

If the B-field is impressed from the outside, the flow does not change due to the induced current that is created. Rather, the source that generates the magnetic field instantaneously compensates for the change in flux generated by the induced current by providing additional energy to maintain the B field.

This situation is a very good approximation, for example, with a transformer with an impressed primary voltage. As soon as a current flows in the secondary circuit, the source current increases in the primary-side voltage source, so that the magnetic flux in the common core remains constant.

### Induction voltage from moving an electrical conductor in a magnetic field (1st induction phenomenon)

When an electrical conductor is moved in a magnetic field, an induction voltage only arises if it does not move in the direction of the field (parallel to the field lines). It should be noted that in this context under the term “electrical conductor” the contour (conductor loop) that is fixed with the conductor is essential and not a specific, mostly metallic conductor in the form of a wire or something similar. This conceptual separation is necessary because otherwise certain induction phenomena and general application of the law of induction which are triggered by mechanical movements in a magnetic field, such as unipolar induction, will lead to difficulties in understanding.

A maximum voltage is reached when the electrical conductor moves perpendicular to the magnetic field (see also the three-finger rule). Since both speed and magnetic flux density are vector quantities, the vector product must be formed. (Hence the factor .)

According to Lenz's rule, the induction voltage arises in order to generate a magnetic field through the current flow caused by it, which compensates for the external magnetic field change, so that an induction always counteracts its cause. This is reflected in the negative sign of the following equations.

The voltage generated is:

*U*= induced voltage in V*n*= Number of turns in a moving coil*v*= Speed of the electrical conductor in m / s*B.*= Magnetic flux density in T*l*= Length of the electrical conductor in m- = Angle between velocity vector and magnetic flux density

As soon as the electrical conductor moves in the magnetic field, a force, the Lorentz force, acts on the charges in the electrical conductor. Thus they are separated. The tension then only depends on the magnitude of this force. The force (on a charge) in turn depends only on the speed and the magnetic field. The above equation results.

### Induction voltage by changing the magnetic flux (2nd induction phenomenon)

Inductions occur not only when electrical conductors move in a magnetic field, but also when the magnetic field changes. To understand this, one has to introduce a model quantity: the magnetic flux. This is defined for a homogeneous field as:

in which *A.* the area of the current loop, the spatial orientation of which is given by its normal vector.

The following applies to the induced voltage:

Here n is the number of turns of the coil. If either the magnetic field changes (weaker or stronger) or the area becomes smaller or larger, the magnetic flux also changes. This change is represented by the derivative with respect to time and this results in the induced voltage. The factor -n indicates, on the one hand, the increase in voltage the more turns a coil has in the field and, on the other hand, the sign of Lenz's rule, which states that the induction current (or the voltage) always counteracts the cause.

*Sentence:* If the magnetic flux changes:

- ,

of a coil of *n* Windings interspersed in the span of time *d**t* around *d*Φ, the mean voltage in this coil is:

induced (according to Faraday), which is polarized in such a way that the current it drives can counteract the cause of the induction process (according to Lenz). Instantaneous value by deriving according to time:

### Technical applications

- Induction loop for vehicles to control traffic lights and barriers
- dynamic microphone
- dynamic (magnetic) pickup system for turntables
- Pickups for electric string instruments (e.g. electric guitar and electric bass)
- Sound head for scanning magnetic tapes
- Generator = dynamo = alternator
- RFID tag (e.g. ski pass)
- Induction lamp
- Induction transmitter

A voltage is induced as long as the magnetic field enclosed by the coil changes. An induction voltage is only present when the magnetic flux changes. Since the flux is the product of the flux density and the area, either the flux density *B.* or the area *A.* to change. A change in the area is achieved by z. B. rotates the coil in a constant magnetic field or a magnet in a coil. The area penetrated by the magnetic field is zero when the coil is perpendicular to the magnetic field; it is maximum when the field penetrates the coil axially. According to this principle, electricity is generated in a generator (dynamo machine).

A change in the flux density can be achieved through a variable magnetic field. According to this principle, when the primary winding is fed with an alternating voltage, an alternating voltage is induced in the secondary winding of a transformer, the magnitude of which is proportional to the ratio of the number of turns.

This also includes all types of induction heating using eddy currents: the induction furnace, induction hardening and the induction cooker, etc. Inductive heating of materials: Induction furnaces are mainly used in industry for hardening, soldering, melting, etc. This technology is increasingly used in private applications, for example in the kitchen as an induction cooker.

### Self induction

Based on the law of induration, externally acting, temporally variable magnetic fluxes in temporally constant conductor loops generate temporally variable electrical voltages. With a closed circuit of the conductor loop, temporally variable electrical currents arise which in turn, as with any current, cause temporally variable magnetic fluxes. Which are superimposed on the externally applied magnetic flux. This fact is commonly referred to as self-induction.

In principle, self-induction can also be fully described by the law of induction and does not require any formal additions or adjustments.However, the network theory common in electrical engineering, which is used, for example, to describe electrical machines such as transformers, may lead to difficulties in understanding, since network theory does not know any field variables such as magnetic flux.

Instead, voltages and currents that vary over time are used in equivalent circuit diagrams with passive components such as coils and electrical resistors. The induced voltages are modeled as voltage sources, which historically also as *electromotive force* (EMK). Since induced voltages are not a force in the physical sense, this term should be avoided.

In the network model, as shown, among other things, in circuit diagrams, counting arrows and certain orientations are also used, as shown in the illustration on the right.

For clarification, the external magnetic flux acting on the conductor loop is Φ_{ext} and the induced voltages caused thereby *u*_{ext} with the index *ext* Mistake. Current flowing when the externally closed loop is loaded *i* creates a magnetic flux Φ_{I.}, which one with the index *I.* is marked. The self-induced source voltage can be used as a voltage source with the magnitude *u*_{i} model, as shown in the first figure, and is that of the coil current *i* driving tension *u*_{ext} opposite. It is therefore also referred to as *Counter tension*. This representation is used, for example, when describing the so-called magnetizing current in a transformer.

The model of the *inductive voltage drop*As shown in the second figure, there is no need for an additional voltage source. The one on the drawn coil *L.* occurring voltage points in the same direction as the current *i* which by the externally driving voltage *u*_{ext} is caused. This representation has the advantage that the relationships in the network model for harmonic processes can be described more easily using Ohm's law with reactances. The special case of harmonic processes, which is important in electrical engineering, reduces the time derivatives in the law of induction to multiplications by jω (dΦ / d*t* ≡ jωΦ), which corresponds to a rotation of 90 ° in the complex plane.

### Applications

Self-induction is used, among other things, to generate the ignition spark in gasoline engines or the required high ignition voltage in fluorescent lamps with an ignition coil.

### Extensions and historical development

The electromagnetic induction shown as part of Maxwell's equations and classical electrodynamics (KED) reflects the state of knowledge from the end of the 19th century. At that time only different terms and nomenclatures were sometimes used for the representations, the basic ideas about the induction process were available.

At the beginning of the 20th century, the relativistic incorporation of the law of induction took place within the framework of the special theory of relativity and the conditions at speeds close to the speed of light were taken into account. For example, the magnitudes of the electric and magnetic field components change as a function of the movement between an observer and an observed electric charge. These dependencies in the relative movement between different reference systems are described by the Lorentz transformation. This shows that the law of induction, like the rest of Maxwell's equations, is “Lorentz invariant”. This means that the equations are not changed by the Lorentz transformation between different reference systems. It also becomes particularly clear that the electric and magnetic fields are only two manifestations of one and the same phenomenon.

In the middle of the 20th century, within the framework of electrodynamics, the connection between quantum mechanics and the special theory of relativity was achieved and the law of induction was also formulated within the framework of a quantum field theory of electromagnetism. This quantum field theory is known as quantum electrodynamics (QED). Today, also due to its large technical field of application, it represents one of the most precisely tested theory of physics through experiments.

### literature

- Karl Küpfmüller, Gerhard Kohn:
*Theoretical electrical engineering and electronics.*14 edition. Springer, 1993,__ISBN 3-540-56500-0__. - Adolf J. Schwab:
*Conceptual world of field theory, electromagnetic fields, Maxwell's equations.*6 edition. Springer, 2002,__ISBN 3-540-42018-5__.

- Induction at LeiFi Physics

Categories: Magnetism | Electrodynamics

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