In the first experimental demonstration of electromagnetic induction (August 1831), Faraday wrapped two wires around opposite sides of an iron torus (the design is similar to a modern transformer). Based on his assessment of a recently discovered property of an electromagnet, he expected that when a current was turned on in one wire of a special kind, a wave would pass through the torus and cause some electrical influence on its opposite side. He connected one wire to the galvanometer and looked at it while the other wire was connected to the battery. Indeed, he saw a brief surge of current (which he called a "wave of electricity") when he connected the wire to the battery, and another similar surge when he disconnected it. Within two months, Faraday found several other manifestations of electromagnetic induction. For example, he saw bursts of current when he quickly inserted a magnet into the coil and pulled it back out, he generated a direct current in a copper disk rotating near the magnet with a sliding electrical wire ("Faraday's disk").

Faraday explained electromagnetic induction using the concept of so-called lines of force. However, most of the scientists of the time rejected his theoretical ideas, mainly because they were not formulated mathematically. The exception was Maxwell, who used Faraday's ideas as the basis for his quantitative electromagnetic theory. In Maxwell's works, the aspect of the change in time of electromagnetic induction is expressed in the form of differential equations. Oliver Heaviside called this Faraday's law, although it differs somewhat in form from the original version of Faraday's law and does not take into account the induction of EMF during movement. The Heaviside version is a form of the group of equations recognized today, known as the Maxwell equations.

Faraday's law as two different phenomena

Some physicists note that Faraday's law in one equation describes two different phenomena: motor emf generated by the action of a magnetic force on a moving wire, and transformer EMF generated by the action of an electric force due to a change in the magnetic field. James Clerk Maxwell drew attention to this fact in his work On physical lines of force in 1861. In the second half of part II of this work, Maxwell gives a separate physical explanation for each of these two phenomena. Reference to these two aspects of electromagnetic induction is found in some modern textbooks. As Richard Feynman writes:

Thus, the "flux rule" that the EMF in a circuit is equal to the rate of change of the magnetic flux through the circuit applies regardless of the reason for the change in flux: either because the field is changing, or because the circuit is moving (or both) .... In our explanation of the rule, we used two completely different laws for two cases  – v × B (\displaystyle (\stackrel (\mathbf (v\times B) )())) for the "moving chain" and ∇ x E = − ∂ t B (\displaystyle (\stackrel (\mathbf (\nabla \ x\ E\ =\ -\partial _(\ t)B) )())) for the "changing field".

We do not know of any analogous situation in physics where such simple and precise general principles would require analysis in terms of two different phenomena for their real understanding.

Reflecting this apparent dichotomy was one of the main ways that led Einstein to develop special relativity:

It is known that Maxwell's electrodynamics - as it is usually understood at the present time - when applied to moving bodies leads to an asymmetry, which, as it seems, is not inherent in this phenomenon. Take, for example, the electrodynamic interaction of a magnet and a conductor. The observed phenomenon depends only on the relative motion of the conductor and the magnet, while conventional wisdom draws a sharp difference between the two cases, in which either one or the other body is in motion. For if the magnet is in motion and the conductor is at rest, an electric field with a certain energy density arises in the vicinity of the magnet, creating a current where the conductor is located. But if the magnet is at rest and the conductor is moving, then no electric field arises in the vicinity of the magnet. In the conductor, however, we find an electromotive force for which there is no corresponding energy in itself, but which causes - assuming equality of relative motion in the two cases under discussion - electric currents in the same direction and of the same intensity as in the first case.

Examples of this kind, together with an unsuccessful attempt to detect any movement of the Earth relative to the "light-bearing medium", suggest that the phenomena of electrodynamics, as well as mechanics, do not have properties corresponding to the idea of ​​absolute rest.

- Albert Einstein, On the electrodynamics of moving bodies

Flux through the surface and EMF in the circuit

Faraday's law of electromagnetic induction uses the concept of magnetic flux Φ B through the closed surface Σ, which is defined through the surface integral :

Φ = ∬ S B n ⋅ d S , (\displaystyle \Phi =\iint \limits _(S)\mathbf (B_(n)) \cdot d\mathbf (S) ,)

where d S - surface element area Σ( t), B is the magnetic field, and B· dS- scalar product B and dS. It is assumed that the surface has a "mouth" outlined by a closed curve, denoted ∂Σ( t). Faraday's law of induction states that when the flow changes, then when a unit positive test charge moves along a closed curve ∂Σ, work is done E (\displaystyle (\mathcal (E))), the value of which is determined by the formula:

| e | = | d d t | , (\displaystyle |(\mathcal (E))|=\left|((d\Phi ) \over dt)\right|\ ,)

where | e | (\displaystyle |(\mathcal (E))|)- the magnitude of the electromotive force (EMF) in volts, and Φ B- magnetic flux in webers. The direction of the electromotive force is determined by the Lenz law.

On fig. 4 shows a spindle formed by two discs with conductive rims and conductors arranged vertically between these rims. current is supplied by sliding contacts to the conductive rims. This design rotates in a magnetic field that is directed radially outward and has the same value in any direction. those. the instantaneous speed of the conductors, the current in them and the magnetic induction, form the right triple, which causes the conductors to rotate.

Lorentz force

In this case, the Ampere Force acts on the conductors, and the Lorentz Force acts on a unit charge in the conductor - the flux of the magnetic induction vector B, the current in the conductors connecting the conductive rims is directed normally to the magnetic induction vector, then the force acting on the charge in the conductor will be equal to

F = q B v . (\displaystyle F=qBv\,.)

where v = the speed of the moving charge

Therefore, the force acting on the conductors

F = I B ℓ , (\displaystyle (\mathcal (F))=IB\ell ,)

where l is the length of the conductors

Here we used B as a given, in fact it depends on the geometric dimensions of the rims of the structure and this value can be calculated using the Law Bio - Savart - Laplace. This effect is also used in another device called the Railgun.

Faraday's Law

Intuitively attractive but flawed approach to using the flow rule expresses the flow through the circuit by the formula Φ B = Bwℓ, where w- the width of the moving loop.

The fallacy of this approach is that this is not a frame in the usual sense of the word. the rectangle in the figure is formed by individual conductors closed to the rim. As you can see in the figure, the current flows in both conductors in the same direction, i.e. there is no concept "closed loop"

The simplest and most understandable explanation of this effect is given by the concept of force ampere. Those. the vertical conductor can be generally one, so as not to be misleading. Or a conductor final thickness can be located on the axis connecting the rims. The diameter of the conductor must be finite and different from zero so that the moment of force ampere is not zero.

Faraday - Maxwell equation

An alternating magnetic field creates an electric field described by the Faraday-Maxwell equation:

∇ × E = − ∂ B ∂ t (\displaystyle \nabla \times \mathbf (E) =-(\frac (\partial \mathbf (B) )(\partial t)))

∇ × (\displaystyle \nabla \times ) stands for rotor E- electric field B- density magnetic flux.

This equation is present in the modern system of Maxwell's equations, it is often called Faraday's law. However, since it contains only partial derivatives with respect to time, its application is limited to situations where the charge is at rest in a time-varying magnetic field. It does not take into account [ ] electromagnetic induction in cases where a charged particle moves in a magnetic field.

In another form, Faraday's law can be written in terms of integral form theorem Kelvin-Stokes :

∮ ∂ Σ ⁡ E ⋅ d ℓ = − ∫ Σ ∂ ∂ t B ⋅ d A (\displaystyle \oint _(\partial \Sigma )\mathbf (E) \cdot d(\boldsymbol (\ell ))=-\ int _(\Sigma )(\partial \over (\partial t))\mathbf (B) \cdot d\mathbf (A) )

Integration requires a time-independent surface Σ (considered in this context as part of the interpretation of partial derivatives). As shown in fig. 6:

Σ - a surface bounded by a closed contour ∂Σ , and how Σ , and ∂Σ are fixed, independent of time, E- electric field, d ℓ - infinitesimal contour element ∂Σ , B- magnetic field , d A is an infinitesimal element of the surface vector Σ .

d elements ℓ and d A have undefined signs. To set the correct signs, the right-hand rule is used, as described in the article on the Kelvin-Stokes theorem. For a flat surface Σ, the positive direction of the path element dℓ curve ∂Σ is determined by the right hand rule, according to which four fingers of the right hand point to this direction when the thumb points in the direction of the normal n to the surface Σ.

Integral over ∂Σ called path integral or curvilinear integral. The surface integral on the right side of the Faraday-Maxwell equation is an explicit expression for the magnetic flux Φ B in terms of Σ . Note that the nonzero path integral for E differs from the behavior of the electric field created by charges. Charge generated E-field can be expressed as the gradient of a scalar field , which is a solution to the Poisson equation and has a zero path integral.

The integral equation is valid for any way ∂Σ in space and any surface Σ , for which this path is the boundary.

D d t ∫ A B d A = ∫ A (∂ B ∂ t + v div B + rot (B × v)) d A (\displaystyle (\frac (\text(d))((\text(d))t ))\int \limits _(A)(\mathbf (B) )(\text( d))\mathbf (A) =\int \limits _(A)(\left((\frac (\partial \mathbf (B) )(\partial t))+\mathbf (v) \ (\text(div))\ \mathbf (B) +(\text(rot))\;(\mathbf (B) \times \mathbf (v))\right)\;(\text(d)))\mathbf (A) )

and taking into account div B = 0 (\displaystyle (\text(div))\mathbf (B) =0)(Series Gauss), B × v = − v × B (\displaystyle \mathbf (B) \times \mathbf (v) =-\mathbf (v) \times \mathbf (B) )(Vector product) and ∫ A rot X d A = ∮ ∂ A ⁡ X d ℓ (\displaystyle \int _(A)(\text(rot))\;\mathbf (X) \;\mathrm (d) \mathbf (A) = \oint _(\partial A)\mathbf (X) \;(\text(d))(\boldsymbol (\ell )))(theorem Kelvin - Stokes), we find that the total derivative of the magnetic flux can be expressed

∫ Σ ∂ B ∂ t d A = d d t ∫ Σ B d A + ∮ ∂ Σ ⁡ v × B d ℓ (\displaystyle \int \limits _(\Sigma )(\frac (\partial \mathbf (B) )(\ partial t))(\textrm (d))\mathbf (A) =(\frac (\text(d))((\text(d))t))\int \limits _(\Sigma )(\mathbf (B) )(\text( d))\mathbf (A) +\oint _(\partial \Sigma )\mathbf (v) \times \mathbf (B) \,(\text(d))(\boldsymbol (\ell)))

By adding a member ∮ ⁡ v × B d ℓ (\displaystyle \oint \mathbf (v) \times \mathbf (B) \mathrm (d) \mathbf (\ell ) ) to both sides of the Faraday-Maxwell equation and introducing the above equation, we get:

∮ ∂ Σ ⁡ (E + v × B) d ℓ = - ∫ Σ ∂ ∂ t B d A displaystyle \oint \limits _(\partial \Sigma )((\mathbf (E) +\mathbf (v) \times \mathbf (B)))(\text(d))\ell =\underbrace (-\int \limits _(\Sigma )(\frac (\partial )(\partial t))\mathbf (B) (\text(d))\mathbf (A) ) _((\text(induced))\ (\ text(emf)))+\underbrace (\oint \limits _(\partial \Sigma )(\mathbf (v) )\times \mathbf (B) (\text(d))\ell ) _((\text (motional))\ (\text(emf)))=-(\frac (\text(d))((\text(d))t))\int \limits _(\Sigma )(\mathbf (B ) )(\text( d))\mathbf (A) ,)

which is Faraday's law. Thus, the Faraday law and the Faraday-Maxwell equations are physically equivalent.

Rice. 7 shows the interpretation of the contribution of the magnetic force to the EMF on the left side of the equation. Area swept by segment dℓ crooked ∂Σ during dt while moving at speed v, is equal to:

d A = − d ℓ × v d t , (\displaystyle d\mathbf (A) =-d(\boldsymbol (\ell \times v))dt\ ,)

so that the change in magnetic flux ΔΦ B through the part of the surface bounded by ∂Σ during dt, equals:

d Δ Φ B d t = − B ⋅ d ℓ × v = − v × B ⋅ d ℓ , (\displaystyle (\frac (d\Delta \Phi _(B))(dt))=-\mathbf (B) \cdot \ d(\boldsymbol (\ell \times v))\ =-\mathbf (v) \times \mathbf (B) \cdot \ d(\boldsymbol (\ell ))\ ,)

and if we add these ΔΦ B -contributions around the loop for all segments dℓ , we get the total contribution of the magnetic force to Faraday's law. That is, the term is associated with motor EMF.

Example 3: the point of view of a moving observer

Returning to the example in Fig. 3, in a moving frame of reference, a close connection is revealed between E- and B fields, as well as between motor and induced EMF. Imagine an observer moving along with the loop. The observer calculates the EMF in the loop using both Lorentz's law and Faraday's law of electromagnetic induction. Since this observer is moving with the loop, he does not see any movement of the loop, i.e. the zero magnitude v×B. However, since the field B changes at a point x, a moving observer sees a time-varying magnetic field, namely:

B = k B (x + v t) , (\displaystyle \mathbf (B) =\mathbf (k) (B)(x+vt)\ ,)

where k is the unit vector in the direction z.

Lorenz law

The Faraday-Maxwell equation says that a moving observer sees an electric field E y in axis direction y, determined by the formula:

∇ × E = k d E y d x (\displaystyle \nabla \times \mathbf (E) =\mathbf (k) \ (\frac (dE_(y))(dx))) = − ∂ B ∂ t = − k d B (x + v t) d t = − k d B d x v , (\displaystyle =-(\frac (\partial \mathbf (B) )(\partial t))=-\mathbf ( k) (\frac (dB(x+vt))(dt))=-\mathbf (k) (\frac (dB)(dx))v\ \ ,) d B d t = d B d (x + v t) d (x + v t) d t = d B d x v . (\displaystyle (\frac (dB)(dt))=(\frac (dB)(d(x+vt)))(\frac (d(x+vt))(dt))=(\frac (dB )(dx))v\ .)

Solution for E y up to a constant that adds nothing to the loop integral:

E y (x , t) = − B (x + v t) v . (\displaystyle E_(y)(x,\ t)=-B(x+vt)\ v\ .)

Using the Lorentz law, in which there is only an electric field component, the observer can calculate the EMF along the loop in time t according to the formula:

E = − ℓ [ E y (x C + w / 2 , t) − E y (x C − w / 2 , t) ] (\displaystyle (\mathcal (E))=-\ell ) = v ℓ [ B (x C + w / 2 + v t) − B (x C − w / 2 + v t) ] , (\displaystyle =v\ell \ ,)

and we see that exactly the same result is found for a stationary observer who sees that the center of mass x C has shifted by x C+ v t. However, the moving observer got the result under the impression that only electric component, while the stationary observer thought that it acted only magnetic component.

Faraday's law of induction

To apply Faraday's law of induction, consider an observer moving along with a point x C. He sees a change in the magnetic flux, but the loop seems to him to be motionless: the center of the loop x C is fixed because the observer is moving along with the loop. Then the flow:

Φ B = − ∫ 0 ℓ d y ∫ x C − w / 2 x C + w / 2 B (x + v t) d x , (\displaystyle \Phi _(B)=-\int _(0)^(\ell )dy\int _(x_(C)-w/2)^(x_(C)+w/2)B(x+vt)dx\ ,)

where the minus sign occurs because the normal to the surface has a direction opposite to the applied field B. From Faraday's law of induction, the EMF is:

E = − d Φ B d t = ∫ 0 ℓ d y ∫ x C − w / 2 x C + w / 2 d d t B (x + v t) d x (\displaystyle (\mathcal (E))=-(\frac (d \Phi _(B))(dt))=\int _(0)^(\ell )dy\int _(x_(C)-w/2)^(x_(C)+w/2)(\ frac (d)(dt))B(x+vt)dx) = ∫ 0 ℓ d y ∫ x C − w / 2 x C + w / 2 d d x B (x + v t) v d x (\displaystyle =\int _(0)^(\ell )dy\int _(x_(C) -w/2)^(x_(C)+w/2)(\frac (d)(dx))B(x+vt)\ v\ dx) = v ℓ [ B (x C + w / 2 + v t) − B (x C − w / 2 + v t) ] , (\displaystyle =v\ell \ \ ,)

and we see the same result. The time derivative is used in the integration because the integration limits are independent of time. Again, to convert the time derivative to the derivative with respect to x methods of differentiation of a complex function are used.

A stationary observer sees the EMF as motor , while the moving observer thinks it is induced EMF.

Electric generator

The phenomenon of the emergence of an EMF generated according to the Faraday law of induction due to the relative motion of the circuit and the magnetic field underlies the operation of electric generators. If the permanent magnet moves relative to the conductor, or vice versa, the conductor moves relative to the magnet, then an electromotive force arises. If the conductor is connected to an electrical load, then a current will flow through it, and therefore, the mechanical energy of movement will be converted into electrical energy. For example, disk generator built on the same principle as shown in Fig. 4. Another implementation of this idea is the Faraday disk, shown in a simplified form in fig. 8. Please note that the analysis of fig. 5 and a direct application of the Lorentz force law show that solid the conductive disk works in the same way.

In the Faraday disk example, the disk rotates in a uniform magnetic field perpendicular to the disk, resulting in a current in the radial arm due to the Lorentz force. It is interesting to understand how it turns out that in order to control this current, mechanical work is necessary. When the generated current flows through the conductive rim, according to the Ampère law, this current creates a magnetic field (in Fig. 8 it is signed "induced B" - Induced B). The rim thus becomes an electromagnet that resists the rotation of the disc (an example of Lenz's rule). In the far part of the figure, the reverse current flows from the rotating arm through the far side of the rim to the bottom brush. The B field created by this reverse current is opposite to the applied field, causing reduction flow through the far side of the chain, as opposed to increase flow caused by rotation. On the near side of the figure, the reverse current flows from the rotating arm through the near side of the rim to the bottom brush. Induced field B increases flow on this side of the chain, as opposed to decrease flow caused by rotation. Thus, both sides of the circuit generate an emf that opposes rotation. The energy needed to keep the disk moving against this reactive force is exactly equal to the generated electrical energy (plus the energy to compensate for losses due to friction, due to heat generation Joules, etc.). This behavior is common to all generators for converting mechanical energy into electrical energy.

Although Faraday's law describes the operation of any electrical generator, the detailed mechanism may vary from case to case. When a magnet rotates around a fixed conductor, the changing magnetic field creates an electric field, as described in the Maxwell-Faraday equation, and this electric field pushes charges through the conductor. This case is called induced EMF. On the other hand, when the magnet is stationary and the conductor rotates, the moving charges are affected by a magnetic force (as described by Lorentz's law), and this magnetic force pushes the charges through the conductor. This case is called motor EMF.

electric motor

An electric generator can work in "reverse" and become an engine. Consider, for example, the Faraday disk. Suppose a direct current flows through the conductive radial arm from some voltage. Then, according to the Lorentz force law, this moving charge is affected by a force in a magnetic field B, which will rotate the disk in the direction determined by the left hand rule. In the absence of effects that cause dissipative losses, such as friction or heat Joules, the disk will rotate at such a speed that d Φ B / dt was equal to the voltage causing the current.

electrical transformer

The EMF predicted by Faraday's law is also the reason electrical transformers work. When the electric current in the wire loop changes, the changing current creates an alternating magnetic field. The second wire in the magnetic field available to it will experience these changes in the magnetic field as changes in the magnetic flux associated with it. dΦ B / dt. The electromotive force generated in the second loop is called induced emf or EMF transformer. If the two ends of this loop are connected through an electrical load, then current will flow through it.

Empirically, M. Faraday showed that the strength of the induction current in a conducting circuit is directly proportional to the rate of change in the number of magnetic induction lines that pass through the surface limited by the circuit in question. The modern formulation of the law of electromagnetic induction, using the concept of magnetic flux, was given by Maxwell. The magnetic flux (Ф) through the surface S is a value equal to:

where is the modulus of the magnetic induction vector; - the angle between the magnetic induction vector and the normal to the contour plane. The magnetic flux is interpreted as a quantity that is proportional to the number of magnetic induction lines passing through the considered surface area S.

The appearance of an induction current indicates that a certain electromotive force (EMF) arises in the conductor. The reason for the appearance of EMF induction is a change in the magnetic flux. In the system of international units (SI), the law of electromagnetic induction is written as follows:

where is the rate of change of the magnetic flux through the area that the contour limits.

The sign of the magnetic flux depends on the choice of the positive normal to the contour plane. In this case, the direction of the normal is determined using the rule of the right screw, connecting it with the positive direction of the current in the circuit. So, the positive direction of the normal is arbitrarily assigned, the positive direction of the current and the EMF of induction in the circuit are determined. The minus sign in the basic law of electromagnetic induction corresponds to Lenz's rule.

Figure 1 shows a closed loop. Assume that the positive direction of the contour traversal is counterclockwise, then the normal to the contour () is the right screw in the direction of traversal of the contour. If the magnetic induction vector of the external field is co-directed with the normal and its modulus increases with time, then we get:

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In this case, the induction current will create a magnetic flux (F '), which will be less than zero. The lines of magnetic induction of the magnetic field of the induction current () are shown in fig. 1 dotted line. The induction current will be directed clockwise. The induction emf will be less than zero.

Formula (2) is a record of the law of electromagnetic induction in the most general form. It can be applied to fixed circuits and conductors moving in a magnetic field. The derivative, which is included in expression (2), generally consists of two parts: one depends on the change in the magnetic flux over time, the other is associated with the movement (deformations) of the conductor in a magnetic field.

In the event that the magnetic flux changes in equal time intervals by the same amount, then the law of electromagnetic induction is written as:

If a circuit consisting of N turns is considered in an alternating magnetic field, then the law of electromagnetic induction will take the form:

where the quantity is called flux linkage.

Examples of problem solving

EXAMPLE 1

Exercise	What is the rate of change of the magnetic flux in the solenoid, which has N = 1000 turns, if an induction EMF equal to 200 V is excited in it?
Solution	The basis for solving this problem is the law of electromagnetic induction in the form: where is the rate of change of the magnetic flux in the solenoid. Therefore, we find the desired value as: Let's do the calculations:
Answer

EXAMPLE 2

Exercise	A square conducting frame is in a magnetic field that changes according to the law: (where and are constants). The normal to the frame makes an angle with the direction of the field magnetic induction vector. frame wall b. Get an expression for the instantaneous value of the induction emf ().
Solution	Let's make a drawing. As a basis for solving the problem, we take the basic law of electromagnetic induction in the form:

>>Physics and Astronomy >>Physics grade 11 >> Law of electromagnetic induction

Faraday's law. Induction

Electromagnetic induction is called such a phenomenon as the occurrence of an electric current in a closed circuit, subject to a change in the magnetic flux that passes through this circuit.

Faraday's law of electromagnetic induction is written as follows:

And says that:

How did scientists manage to derive such a formula and formulate this law? We already know that there is always a magnetic field around a conductor with current, and electricity has a magnetic force. Therefore, at the beginning of the 19th century, the problem arose of the need to confirm the influence of magnetic phenomena on electrical ones, which many scientists tried to solve, and the English scientist Michael Faraday was among them. Almost 10 years, starting in 1822, he spent on various experiments, but to no avail. And only on August 29, 1831 did the triumph come.

After intense searches, research and experiments, Faraday came to the conclusion that only a magnetic field that changes over time can create an electric current.

Faraday's experiments

Faraday's experiments were as follows:

First, if you take a permanent magnet and move it inside the coil to which the galvanometer is attached, then an electric current arises in the circuit.
Secondly, if this magnet is pulled out of the coil, then we observe that the galvanometer also shows a current, but this current has the opposite direction.

Now let's try to change this experience a little. To do this, we will try to put on and remove the coil on a fixed magnet. And what do we end up seeing? And we observe that during the movement of the coil relative to the magnet, current appears in the circuit again. And if the coil stops, then the current immediately disappears.

Now let's do another experiment. To do this, we will take and place in a magnetic field a flat circuit without a conductor, and we will try to connect its ends with a galvanometer. And what are we seeing? As soon as the galvanometer circuit turns, we observe the appearance of an induction current in it. And if you try to rotate the magnet inside it and next to the circuit, then in this case a current will also appear.

I think you have already noticed that the current appears in the coil when the magnetic flux that permeates this coil changes.

And here the question arises, with any movements of the magnet and the coil, can an electric current arise? It turns out not always. Current will not occur when the magnet rotates around a vertical axis.

And from this it follows that with any change in the magnetic flux, we observe that an electric current arises in this conductor, which existed throughout the entire process, while changes in the magnetic flux occurred. This is precisely the phenomenon of electromagnetic induction. And the induction current is the current that was obtained by this method.

If we analyze this experience, we will see that the value of the induction current is completely independent of the cause of the change in the magnetic flux. In this case, only the speed is of paramount importance, which affects the changes in the magnetic flux. From Faraday's experiments it follows that the faster the magnet moves in the coil, the more the galvanometer needle deviates.

Now we can summarize this lesson and conclude that the law of electromagnetic induction is one of the fundamental laws of electrodynamics. Thanks to the study of the phenomena of electromagnetic induction, scientists from different countries have created various electric motors and powerful generators. A huge contribution to the development of electrical engineering was made by such famous scientists as Lenz, Jacobi, and others.

Fedun V.I. Abstract of lectures on the physics of Electromagnetism

Lecture 26

Electromagnetic induction. Faraday's discovery .

In 1831, M. Faraday made one of the most important fundamental discoveries in electrodynamics - he discovered the phenomenon electromagnetic induction .

In a closed conducting circuit, with a change in the magnetic flux (vector flux) covered by this circuit, an electric current arises.

This current is called induction .

The appearance of an induction current means that when the magnetic

	flow in the circuit arises emf induction (work on the transfer of a unit charge along a closed circuit). Note that the value completely independent of how the change in the magnetic flux is carried out , and is determined only by the rate of its change, i.e. magnitude . Changing the sign of the derivative leads to a sign change emf induction .
Figure 26.1.

Faraday discovered that an induction current can be induced in two different ways, which can be conveniently explained with a diagram.

1st method: moving the frame in the magnetic field of a fixed coil (see fig.26.1).

2nd method: changing the magnetic field generated by the coil , due to its movement or due to a change in the strength of the current in it (or both). Frame while being immobile.

In both these cases, the galvanometer will show the presence of induction current in the frame .

The direction of the induction current and, accordingly, the sign of the emf. induction determined by the Lenz rule.

Lenz's rule.

The induction current is always directed in such a way as to counteract the cause that causes it. .

Lenz's rule expresses an important physical property - the desire of a system to counteract a change in its state. This property is called electromagnetic inertia .

The law of electromagnetic induction (Faraday's law).

Whatever the reason for the change in the magnetic flux covered by a closed conducting circuit, which occurs in the emf circuit. induction is defined by the formula

Nature of electromagnetic induction.

In order to clarify the physical causes that lead to the emergence of emf. Induction, we consider two cases successively.

1. The circuit moves in a constant magnetic field.

act force

The electromotive force generated by this field is called electromotive force induction . In our case

.

Here the minus sign is put because the third-party field directed against the positive loop bypass defined by the right screw rule. Work is the rate of increment of the area of ​​the contour (increment of the area per unit time), therefore

,

where
- increment of the magnetic flux through the circuit.

.

The result obtained can be generalized to the case of an arbitrary orientation of the magnetic field induction vector relative to the contour plane and on any contour moving (and/or deforming) in an arbitrary way in a constant inhomogeneous external magnetic field.

So, the excitation of emf. induction during the movement of the circuit in a constant magnetic field is explained by the action of the magnetic component of the Lorentz force, proportional to
, which occurs when the conductor is moved.

2. The circuit is at rest in an alternating magnetic field.

The experimentally observed occurrence of an inductive current indicates that in this case, external forces appear in the circuit, which are now associated with a time-varying magnetic field. What is their nature? The answer to this fundamental question was given by Maxwell.

Since the conductor is at rest, the speed of the ordered movement of electric charges
and hence the magnetic force proportional to
, is also equal to zero and can no longer set the charges in motion. However, in addition to the magnetic force, only a force from the electric field equal to . Therefore, it remains to conclude that induced current due to electric field arising when the external magnetic field changes in time. It is this electric field that is responsible for the appearance of the emf. induction in a fixed circuit. According to Maxwell, a time-varying magnetic field generates an electric field in the surrounding space. The appearance of an electric field is not associated with the presence of a conductive circuit, which only makes it possible to detect the existence of this field by the appearance of an induction current in it.

Wording law of electromagnetic induction , given by Maxwell, is one of the most important generalizations of electrodynamics.

Any change in the magnetic field in time excites an electric field in the surrounding space .

The mathematical formulation of the law of electromagnetic induction in the understanding of Maxwell has the form:

Tension vector circulation this field along any fixed closed contour is defined by the expression

,

where - magnetic flux penetrating the circuit .

Used to indicate the rate of change of the magnetic flux, the sign of the partial derivative indicates that the circuit is stationary.

Vector flow through a surface bounded by a contour , is equal to
, so the expression for the law of electromagnetic induction can be rewritten as follows:

This is one of the equations of Maxwell's system of equations.

The fact that the circulation of the electric field excited by a time-varying magnetic field is non-zero means that the considered electric field not potential.It, like a magnetic field, is eddy.

In general, the electric field can be represented by the vector sum of the potential (the field of static electric charges, the circulation of which is zero) and the vortex (due to the time-varying magnetic field) electric fields.

At the basis of the phenomena we have considered, which explain the law of electromagnetic induction, there is no general principle that makes it possible to establish the commonality of their physical nature. Therefore, these phenomena should be considered as independent, and the law of electromagnetic induction - as a result of their joint action. All the more surprising is the fact that the emf. induction in the circuit is always equal to the rate of change of the magnetic flux through the circuit. In cases where the field also changes and the location or configuration of the circuit in a magnetic field, emf. induction should be calculated by the formula

The expression on the right side of this equality is the total derivative of the magnetic flux with respect to time: the first term is associated with the change in the magnetic field over time, the second with the movement of the circuit.

It can be said that in all cases the induction current is caused by the total Lorentz force

.

What part of the induction current is caused by the electric, and what part of the magnetic component of the Lorentz force depends on choice of reference system.

On the work of the Lorentz and Ampère forces.

From the very definition of work, it follows that the force acting in a magnetic field on an electric charge and perpendicular to its velocity cannot do work. However, when a current-carrying conductor moves, carrying charges along with it, the Ampere force still does work. Electric motors serve as a clear confirmation of this.

This contradiction disappears if we take into account that the movement of a conductor in a magnetic field is inevitably accompanied by the phenomenon of electromagnetic induction. Therefore, along with the Ampère force, work on electric charges is also performed by the electromotive force of induction arising in the conductor. Thus, the total work of the magnetic field forces consists of the mechanical work due to the Ampère force and the work of the emf induced when the conductor moves. Both works are equal in absolute value and opposite in sign, so their sum is equal to zero. Indeed, the work of the ampere force during the elementary displacement of a current-carrying conductor in a magnetic field is equal to
, during the same time emf induction does work

,

then full work
.

The Ampere forces do the work not due to the energy of the external magnetic field, which can remain constant, but due to the emf source that maintains the current in the circuit.

In 1821, Michael Faraday wrote in his diary: "Turn magnetism into electricity." After 10 years, this problem was solved by him. In 1831, Michael Faraday established that in any closed conducting circuit, when the flux of magnetic induction changes through the surface bounded by this circuit, an electric current arises. This phenomenon is called electromagnetic induction, and the resulting current induction(Fig. 3.27).

Rice. 3.27 Faraday's experiments

An inductive current always occurs when there is a change in the flux of magnetic induction coupled to the circuit. The strength of the induction current does not depend on the method of changing the flux of magnetic induction, but is determined only by the rate of its change.

Faraday's law: the strength of the induction current that occurs in a closed conducting circuit (emf of induction that occurs in the conductor) is proportional to the rate of change of the magnetic flux coupled to the circuit (penetrating through the surface bounded by the circuit), and does not depend on the method of changing the magnetic flux.

Lenz established a rule by which you can find the direction of the induction current. Lenz's rule: the induction current is directed in such a way that its own magnetic field prevents a change in the external magnetic flux crossing the circuit surface(Fig. 3.28).

Rice. 3.28 Illustration of Lenz's rule

According to Ohm's law, an electric current in a closed circuit can only occur if an EMF appears in this circuit. Therefore, the induction current discovered by Faraday indicates that an EMF of induction occurs in a closed circuit located in an alternating magnetic field. Further research showed that the EMF of electromagnetic induction in the circuit is proportional to the change in the magnetic flux through the surface bounded by this contour.

The instantaneous value of the induction emf is expressed Faraday-Lenz law)

where is the flux linkage of a closed conducting circuit.

Discovery of the phenomenon of electromagnetic induction:

1. showed the relationship between electric and magnetic fields;

2. proposed a method for generating electric current using a magnetic field.

Thus, the occurrence of an EMF of induction is possible in the case fixed circuit located in variable magnetic field. However, the Lorentz force does not act on immobile charges, so it cannot be used to explain the occurrence of induction EMF.

Experience shows that the induction EMF does not depend on the type of substance of the conductor, on the state of the conductor, in particular on its temperature, which may even be unequal along the conductor. Consequently, external forces are not associated with a change in the properties of the conductor in a magnetic field, but are due to the magnetic field itself.

To explain the EMF of induction in fixed conductors, the English physicist Maxwell suggested that an alternating magnetic field excites a vortex electric field in the surrounding space, which is the cause of the induction current in the conductor. The vortex electric field is not electrostatic (i.e., potential).

EMF of electromagnetic induction occurs not only in a closed current-carrying conductor, but also in a segment of the conductor that crosses the lines of magnetic induction during its movement (Fig. 3.29).

Rice. 3.29 Formation of induction emf in a moving conductor

Let a straight line segment of a conductor with a length l moves from left to right with speed v(Fig. 3.29). Magnetic field induction AT directed away from us. Then the electrons moving with speed v Lorentz force works

Under the action of this force, the electrons will be displaced towards one of the ends of the conductor. Consequently, there is a potential difference and an electric field inside the conductor with intensity E. From the side of the arisen electric field, a force will act on the electrons qE, the direction of which is opposite to the Lorentz force. When these forces balance each other, the movement of electrons will stop.

The circuit is open, which means, but there is no galvanic cell or other current sources in the conductor, which means it will be an induction EMF

.

When moving in a magnetic field of a closed conducting circuit, the EMF of induction is located in all its sections that intersect the lines of magnetic induction. The algebraic sum of these emfs is equal to the total induction emf of the closed loop.