What does the sign of electromagnetic induction show? S.A.

In the first experimental demonstration of electromagnetic induction (August 1831), Faraday wrapped two wires around opposite sides of an iron torus (a design similar to a modern transformer). Based on his assessment of the newly discovered property of the electromagnet, he expected that when a current was turned on in one wire, a special kind of 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 he connected the other wire 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 current surges when he quickly inserted a magnet into a coil and pulled it back out; it generated a direct current in a copper disk rotating near the magnet with a sliding electric wire (“Faraday disk”).

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

Faraday's law as two different phenomena

Some physicists note that Faraday's law describes two different phenomena in one equation: motor EMF, generated by the action of a magnetic force on a moving wire, and transformer EMF, generated by the action of electric force due to changes in the magnetic field. James Clerk Maxwell drew attention to this fact in his work About physical force lines 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 available 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 magnetic flux through the circuit applies regardless of the reason for the flux change: whether 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 "moving chain" and ∇ x E = − ∂ t B (\displaystyle (\stackrel (\mathbf (\nabla \ x\ E\ =\ -\partial _(\ t)B) )())) for a "changing field".

We know of no analogous situation in physics where such simple and precise general principles would require, for their real understanding, analysis from the point of view of two different phenomena.

Reflecting this apparent dichotomy was one of the main paths that led Einstein to develop the special theory of 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 that does not seem to be 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 the usual opinion draws a sharp distinction 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 a conductor, however, we find an electromotive force for which there is no corresponding energy in itself, but which produces—assuming equality of relative motion in the two cases under discussion—electric currents in the same direction and the same intensity as in the first case.

Examples of this kind, together with the unsuccessful attempt to detect any movement of the Earth relative to the “luminiferous medium,” suggest that the phenomena of electrodynamics, as well as mechanics, do not possess 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:

Where d S - area of ​​the surface element Σ( t), B- 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, work is done when moving a unit positive test charge along the closed curve ∂Σ E (\displaystyle (\mathcal (E))), the value of which is determined by the formula:

Where | E | (\displaystyle |(\mathcal (E))|) is 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 Lenz's law.

In Fig. 4 shows a spindle formed by two disks with conductive rims, and conductors located vertically between these rims. The current is supplied by sliding contacts to the conductive rims. This structure 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 magnetic induction form a right-handed 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 conducting rims is directed normal to the magnetic induction vector, then the force acting on the charge in the conductor will be equal

where v = speed of moving charge

Therefore, the force acting on the conductors

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 Biot-Savart-Laplace Law. This effect is also used in another device called the Railgun.

Faraday's law

An intuitive but flawed approach to using the flow rule expresses the flow through the circuit using the formula Φ B = Bwℓ, where w- 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 a rim. As can be seen in the figure, the current flows through both conductors in the same direction, i.e. there is no concept here "closed loop"

The simplest and most understandable explanation for this effect is given by the concept of ampere force. Those. there may be only one vertical conductor, 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 the Ampere force is not zero.

Faraday - Maxwell equation

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

This equation is present in the modern system of Maxwell's equations, often called Faraday's law. However, since it contains only partial derivatives with respect to time, its use 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 Kelvin-Stokes theorem:

A time-independent surface is required to perform integration Σ (considered in this context as part of the interpretation of partial derivatives). As shown in Fig. 6:

Elements d ℓ and d A have indefinite signs. To establish 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 the curve ∂Σ is determined by the right hand rule, according to which the four fingers of the right hand point in 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 through Σ . Note that the non-zero 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 the scalar field, which is a solution to the Poisson equation and has zero path integral.

The integral equation is valid for any ways ∂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)(Gauss series), B × v = − v × B (\displaystyle \mathbf (B) \times \mathbf (v) =-\mathbf (v) \times \mathbf (B) )(cross 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 )))(Kelvin-Stokes theorem), we find that the total derivative of the magnetic flux can be expressed

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:

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

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

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

and if we add up 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, this term is associated with motor EMF.

Example 3: Moving Observer's Point of View

Returning to the example in Fig. 3, in a moving reference frame a close connection is revealed between E- And B-fields, as well as between motor And induced EMF. Imagine an observer moving 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, that is, a zero value v×B. However, since the field B changes at a point x, a moving observer sees a time-varying magnetic field, namely:

Where k - unit vector in the direction z.

Lorentz's law

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

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

Using Lorentz's law, in which there is only an electric field component, an observer can calculate the emf along the loop in time t according to the formula:

and we see that exactly the same result is found for a stationary observer who sees that the center of mass x C has moved by the amount x C+ v t. However, the moving observer received the result under the impression that in Lorentz's law 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 with a point x C. He sees a change in the magnetic flux, but the loop seems motionless to him: the center of the loop x C is fixed because the observer moves with the loop. Then the flow:

where the minus sign arises due to the fact that the normal to the surface has the direction opposite to the applied field B. From Faraday's law of induction, the emf is equal to:

and we see the same result. The time derivative is used in integration because the limits of integration do not depend on time. Again, to convert the time derivative to the time derivative x methods for differentiating a complex function are used.

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

Electric generator

The phenomenon of the occurrence of EMF, generated according to Faraday's law of induction due to the relative movement of the circuit and the magnetic field, underlies the operation of electric generators. If a permanent magnet moves relative to a conductor, or vice versa, a conductor moves relative to a magnet, then an electromotive force occurs. If a conductor is connected to an electrical load, then 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 a Faraday disk, shown in a simplified form in Fig. 8. Please note that the analysis of Fig. 5, and 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 is that mechanical work is necessary to control this current. When the generated current flows through the conducting rim, according to Ampere's law, this current creates a magnetic field (in Fig. 8 it is labeled “Induced B”). The rim thus becomes an electromagnet that resists the rotation of the disk (an example of Lenz's rule). In the far part of the picture, 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 picture, 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 reduction flow caused by rotation. Thus, both sides of the circuit generate an emf that prevents rotation. The energy required to maintain the motion of the disk against this reactive force is exactly equal to the electrical energy generated (plus the energy to compensate for losses due to friction, due to Joule heat, etc.). This behavior is common to all generators that convert mechanical energy into electrical energy.

Although Faraday's law describes the operation of all electrical generators, the detailed mechanism may differ from case to case. When a magnet rotates around a stationary 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 is rotating, the moving charges are subject to 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 electrical generator can be run in reverse and become a motor. Consider, for example, a Faraday disk. Suppose a direct current flows through a conducting radial arm from some voltage. Then, according to the Lorentz force law, this moving charge is affected by a force in the magnetic field B, which will rotate the disk in the direction determined by the left-hand rule. In the absence of effects causing dissipative losses, such as friction or Joule heat, the disk will rotate at such a speed that dΦB/dt was equal to the voltage causing the current.

Electric transformer

The emf predicted by Faraday's law is also the reason for the operation of electrical transformers. When the electric current in a wire loop changes, the changing current creates an alternating magnetic field. A 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 / d t. The electromotive force arising in the second loop is called induced emf or Transformer EMF. If the two ends of this loop are connected through an electrical load, then current will flow through it.

The magnetic induction vector \(~\vec B\) characterizes the force properties of the magnetic field at a given point in space. Let us introduce another quantity that depends on the value of the magnetic induction vector not at one point, but at all points of an arbitrarily chosen surface. This quantity is called magnetic flux and is denoted by the Greek letter Φ (phi).

• Magnetic fluxΦ of a uniform field through a flat surface is a scalar physical quantity numerically equal to the product of the induction modulus B magnetic field, surface area S and the cosine of the angle α between the normal \(~\vec n\) to the surface and the induction vector \(~\vec B\) (Fig. 1):

The SI unit of magnetic flux is weber(Wb):

1 Wb = 1 T ⋅ 1 m 2.

• Magnetic flux 1 Wb is a magnetic flux of a uniform magnetic field with an induction of 1 T through a flat surface with an area of ​​1 m 2 perpendicular to it.

From formula (1) it follows that the magnetic flux can change:

• or only due to a change in the modulus of the induction vector B magnetic field, then \(~\Delta \Phi = (B_2 - B_1) \cdot S \cdot \cos \alpha\) ;
• or only by changing the contour area S, then \(~\Delta \Phi = B \cdot (S_2 - S_1) \cdot \cos \alpha\) ;
• or only due to rotation of the circuit in a magnetic field, then \(~\Delta \Phi = B \cdot S \cdot (\cos \alpha_2 - \cos \alpha_1)\) ;
• or simultaneously by changing several parameters, then \(~\Delta \Phi = B_2 \cdot S_2 \cdot \cos \alpha_2 - B_1 \cdot S_1 \cdot \cos \alpha_1\) .

Electromagnetic induction (EMI)

Discovery of EMR

You already know that there is always a magnetic field around a current-carrying conductor. Is it not possible, on the contrary, to create a current in a conductor using a magnetic field? It was this question that interested the English physicist Michael Faraday, who in 1822 wrote in his diary: “Convert magnetism into electricity.” And only after 9 years this problem was solved by him.

Opening electromagnetic induction, as Faraday called this phenomenon, was made on August 29, 1831. Initially, induction was discovered in conductors stationary relative to each other when closing and opening a circuit. Then, clearly understanding that bringing current-carrying conductors closer or further away should lead to the same result as closing and opening a circuit, Faraday proved through experiments that current arises when the coils move relative to each other (Fig. 2).

On October 17, as recorded in his lab notebook, an induced current was detected in the coil while the magnet was being pushed in (or pulled out) (Figure 3).

Within one month, Faraday experimentally discovered that an electric current arises in a closed loop with any change in the magnetic flux through it. The current obtained in this way is called induction current I i.

It is known that an electric current arises in a circuit when external forces act on free charges. The work done by these forces when moving a single positive charge along a closed loop is called electromotive force. Consequently, when the magnetic flux changes through a surface limited by a contour, extraneous forces appear in it, the action of which is characterized by an emf, which is called induced emf and denoted by E i.

Induction current I i in the circuit and induced emf E i are related by the following relationship (Ohm's law):

\(~I_i = -\dfrac (E_i)(R),\)

Where R- circuit resistance.

• The phenomenon of the occurrence of induced emf when a magnetic flux changes through an area limited by a contour is called phenomenon of electromagnetic induction. If the circuit is closed, then along with the induced emf, an induced current also arises. James Clerk Maxwell proposed the following hypothesis: a changing magnetic field creates an electric field in the surrounding space, which leads free charges into directed motion, i.e. creates an induced current. The field lines of such a field are closed, i.e. electric field vortex. Induction currents arising in massive conductors under the influence of an alternating magnetic field are called Foucault currents or eddy currents.

Story

Here is a brief description of the first experiment, given by Faraday himself.

“A copper wire 203 feet long was wound on a wide wooden reel (a foot is equal to 304.8 mm), and between its turns was wound a wire of the same length, but insulated from the first cotton thread. One of these spirals was connected to a galvanometer, and the other to a strong battery consisting of 100 pairs of plates... When the circuit was closed, a sudden but extremely weak effect on the galvanometer was noticed, and the same was noticed when the current stopped. With the continuous passage of current through one of the spirals, it was not possible to notice either an effect on the galvanometer, or at all any inductive effect on the other spiral, despite the fact that the heating of the entire spiral connected to the battery and the brightness of the spark jumping between the coals indicated about battery power."

See also

1. Vasiliev A. Volta, Oersted, Faraday // Quantum. - 2000. - No. 5. - P. 16-17

Lenz's rule

Russian physicist Emilius Lenz formulated the rule in 1833 ( Lenz's rule), which allows you to set the direction of the induction current in the circuit:

• The induced current arising in a closed circuit has a direction in which the own magnetic flux created by it through the area limited by the circuit tends to prevent the change in the external magnetic flux that caused this current.

• the induced current has such a direction that it interferes with the cause that causes it.

For example, when the magnetic flux through the turns of the coil increases, the induced current has such a direction that the magnetic field it creates prevents the increase in the magnetic flux through the turns of the coil, i.e. the induction vector \((\vec(B))"\) of this field is directed against the induction vector \(\vec(B)\) of the external magnetic field. If the magnetic flux through the coil weakens, then the induced current creates a magnetic field with induction \ ((\vec(B))"\), increasing the magnetic flux through the turns of the coil.

See also

EMR Law

Faraday's experiments showed that the induced emf (and the strength of the induced current) in a conducting circuit is proportional to the rate of change of the magnetic flux. If in a short time Δ t magnetic flux changes by ΔΦ, then the rate of change of magnetic flux is equal to \(\dfrac(\Delta \Phi )(\Delta t)\). Taking into account Lenz's rule, D. Maxwell in 1873 gave the following formulation of the law of electromagnetic induction:

• The induced emf in a closed circuit is equal to the rate of change of the magnetic flux penetrating this circuit, taken with the opposite sign

• This formula can only be applied when the magnetic flux changes uniformly.

• The minus sign in the law follows from Lenz's law. With an increase in magnetic flux (ΔΦ > 0), the emf is negative (E i < 0), т.е. индукционный ток имеет такое направление, что вектор магнитной индукции индукционного магнитного поля направлен против вектора магнитной индукции внешнего (изменяющегося) магнитного поля (рис. 4, а). При уменьшении магнитного потока (ΔΦ < 0), ЭДС положительная (Ei> 0) (Fig. 4, b).

In the International System of Units, the law of electromagnetic induction is used to establish the unit of magnetic flux. Since the induced emf E i expressed in volts, and time in seconds, then from Weber’s EMR law can be determined as follows:

• magnetic flux through a surface bounded by a closed loop is equal to 1 Wb if, with a uniform decrease in this flux to zero in 1 s, an induced emf equal to 1 V arises in the loop:

Induction emf in a moving conductor

When moving a conductor with a length l at a speed \(\vec(\upsilon)\) in a constant magnetic field with an induction vector \(\vec(B)\) an induced emf occurs in it

\(~E_i = B \cdot \upsilon \cdot l \cdot \sin \alpha,\)

where α is the angle between the direction of velocity \(\vec(\upsilon)\) of the conductor and the magnetic induction vector \(\vec(B)\).

The reason for the appearance of this EMF is the Lorentz force acting on free charges in a moving conductor. Therefore, the direction of the induction current in the conductor will coincide with the direction of the Lorentz force component on this conductor.

Taking this into account, we can formulate the following to determine the direction of the induction current in a moving conductor ( left hand rule):

• you need to position your left hand so that the magnetic induction vector \(\vec(B)\) enters the palm, four fingers coincide with the direction of the speed \(\vec(\upsilon)\) of the conductor, then the thumb set at 90° will indicate the direction induction current (Fig. 5).

If the conductor moves along the magnetic induction vector, then there will be no induced current (the Lorentz force is zero).

Literature

1. Aksenovich L. A. Physics in secondary school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P.344-351.
2. Zhilko V.V. Physics: textbook. allowance for 11th grade. general education institutions with Russian language Training with a 12-year period of study (basic and advanced levels) / V.V. Zhilko, L.G. Markovich. - Mn.: Nar. Asveta, 2008. - pp. 170-182.
3. Myakishev, G.Ya. Physics: Electrodynamics. 10-11 grades: textbook. for in-depth study of physics / G.Ya. Myakishev, A.3. Sinyakov, V.A. Slobodskov. - M.: Bustard, 2005. - P. 399-408, 412-414.

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

Faraday's law. Induction

Electromagnetic induction is the phenomenon of the occurrence of 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 it says that:

How did scientists manage to derive such a formula and formulate this law? You and I already know that there is always a magnetic field around a conductor carrying current, and electricity has magnetic force. Therefore, at the beginning of the 19th century, the problem arose about 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. He spent almost 10 years, starting in 1822, on various experiments, but without success. And only on August 29, 1831, triumph came.

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

Faraday's experiments

Faraday's experiments consisted of the following:

Firstly, if you take a permanent magnet and move it inside a coil to which a galvanometer is attached, an electric current will arise 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 is in the opposite direction.

Now let's try to change this experience a little. To do this, we will try to put a coil on and off a stationary magnet. And what do we ultimately see? What we observe is that as the coil moves relative to the magnet, current appears again in the circuit. And if the coil stops flowing, then the current immediately disappears.

Now let's do another experiment. To do this, we will take and place a flat circuit without a conductor in a magnetic field, and we will try to connect its ends to a galvanometer. And what are we seeing? As soon as the galvanometer circuit is rotated, 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 current appears in the coil when the magnetic flux that penetrates this coil changes.

And here the question arises: with any movements of the magnet and coil, can an electric current arise? It turns out not always. No current will 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 induced 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 reason for the change in the magnetic flux. In this case, only the speed, which affects changes in the magnetic flux, is of paramount importance. From Faraday's experiments it follows that the faster the magnet moves in the coil, the more the galvanometer needle deflects.

Now we can summarize this lesson and conclude that the law of electromagnetic induction is one of the basic 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. Such famous scientists as Lenz, Jacobi, and others made a huge contribution to the development of electrical engineering.

Today we will reveal such a physics phenomenon as the “law of electromagnetic induction”. We'll tell you why Faraday conducted the experiments, give the formula, and explain the importance of the phenomenon for everyday life.

Ancient Gods and Physics

Ancient people worshiped the unknown. And now man is afraid of the abyss of the sea and the depths of space. But science can explain why. Submarines are documenting the incredible life of the oceans at depths of more than a kilometer, and space telescopes are studying objects that existed only a few million years after the big bang.

But then people deified everything that fascinated and worried them:

• Sunrise;
• awakening of plants in spring;
• rain;
• birth and death.

In every object and phenomenon lived unknown forces that ruled the world. Until now, children tend to humanize furniture and toys. Left without adult supervision, they fantasize: a blanket will hug you, a stool will fit, a window will open on its own.

Perhaps the first evolutionary step of humanity was the ability to maintain a fire. Anthropologists suggest that the earliest fires were lit by a tree struck by lightning.

Thus, electricity has played a huge role in the life of mankind. The first lightning gave impetus to the development of culture, the basic law of electromagnetic induction led humanity to the modern state.

From vinegar to a nuclear reactor

Strange ceramic vessels were found in the Cheops pyramid: the neck was sealed with wax, and a metal cylinder was hidden in the depths. Residues of vinegar or sour wine were found on the inside of the walls. Scientists have come to a sensational conclusion: this artifact is a battery, a source of electricity.

But until 1600, no one undertook to study this phenomenon. Before moving electrons, the nature of static electricity was explored. The ancient Greeks knew that amber produces shocks when rubbed against fur. The color of this stone reminded them of the light of the star Electra from the Pleiades. And the name of the mineral became, in turn, a reason to christen a physical phenomenon.

The first primitive direct current source was built in 1800

Naturally, as soon as a sufficiently powerful capacitor appeared, scientists began to study the properties of the conductor connected to it. In 1820, the Danish scientist Hans Christian Oersted discovered that the magnetic needle deviates near a conductor connected to the network. This fact gave impetus to the discovery of the law of electromagnetic induction by Faraday (the formula will be given below), which allowed humanity to produce electricity from water, wind and nuclear fuel.

Primitive but modern

The physical basis for Max Faraday's experiments was laid by Oersted. If a switched-on conductor affects a magnet, then the reverse is also true: the magnetized conductor must cause a current.

The structure of the experiment, which helped to derive the law of electromagnetic induction (we will consider EMF as a concept a little later), was very simple. The wire wound into a spring was connected to a device that records the current. The scientist brought a large magnet to the coils. While the magnet moved next to the circuit, the device recorded the flow of electrons.

Since then, the technology has improved, but the basic principle of creating electricity at huge stations is still the same: a moving magnet excites a current in a conductor wound by a spring.

Development of the idea

The very first experiment convinced Faraday that electric and magnetic fields are interconnected. But it was necessary to find out exactly how. Does a magnetic field also arise around a current-carrying conductor, or are they simply capable of influencing each other? Therefore, the scientist went further. He wound one wire, supplied a current to it, and pushed this coil into another spring. And I also received electricity. This experiment proved that moving electrons create not only an electric, but also a magnetic field. Later, scientists figured out how they are located in space relative to each other. The electromagnetic field is also the reason why light exists.

By experimenting with different options for the interaction of live conductors, Faraday found out that current is transmitted best if both the first and second coils are wound on one common metal core. The formula expressing the law of electromagnetic induction was derived precisely on this device.

Formula and its components

Now that the history of the study of electricity has been brought to Faraday’s experiment, it’s time to write the formula:

Let's decipher:

ε is electromotive force (abbreviated emf). Depending on the value of ε, electrons move more intensely or weakly in the conductor. The EMF is affected by the power of the source, and it is influenced by the strength of the electromagnetic field.

Φ is the magnitude of the magnetic flux that is currently passing through a given area. Faraday rolled the wire into a spring because he needed a certain space through which the conductor would pass. Of course, it would be possible to make a very thick conductor, but this would be expensive. The scientist chose the shape of a circle because this flat figure has the greatest ratio of surface area to surface length. This is the most energy efficient form. Therefore, water droplets on a flat surface become round. In addition, a spring with a round cross-section is much easier to obtain: you just need to wind the wire around some round object.

t is the time during which the flow passed through the circuit.

The prefix d in the formula for the law of electromagnetic induction means that the quantity is differential. That is, a small magnetic flux must be differentiated over short periods of time in order to obtain the final result. This mathematical operation requires some preparation from people. To better understand the formula, we strongly encourage the reader to review differentiation and integration.

Consequences from the law

Immediately after the discovery, they began to study the phenomenon of electromagnetic induction. Lenz's law, for example, was derived experimentally by a Russian scientist. It was this rule that added a minus to the final formula.

It looks like this: the direction of the induction current is not random; the flow of electrons in the second winding tends to reduce the effect of the current in the first winding. That is, the occurrence of electromagnetic induction is actually the resistance of the second spring to interference in “personal life”.

Lenz's rule has another consequence.

• if the current in the first coil increases, then the current of the second spring will also tend to increase;
• if the current in the inducing winding drops, then the current in the second winding will also decrease.

According to this rule, the conductor in which the induced current occurs actually tends to compensate for the effect of the changing magnetic flux.

Grain and donkey

People have long sought to use the simplest mechanisms for their own benefit. Grinding flour is a complicated matter. Some tribes grind the grain by hand: placing the wheat on one stone, covering it with another flat and round stone, and turning the millstone. But if you need to grind flour for an entire village, then you can’t do it with muscular labor alone. At first, people thought of tying a draft animal to a millstone. The donkey pulled the rope - the stone rotated. Then people probably thought: “The river flows all the time, it pushes all sorts of things downstream. Why don’t we use it for good?” This is how water mills appeared.

Wheel, water, wind

Of course, the first engineers who built these structures knew nothing about the force of gravity, due to which water always tends downward, or about the force of friction or surface tension. But they saw: if you put a wheel with blades in diameter in a stream or river, it will not only rotate, but will also be able to do useful work.

But this mechanism was also limited: not everywhere there is running water with sufficient current strength. So people moved on. They built mills that were powered by the wind.

Coal, fuel oil, gasoline

When scientists understood the principle of generating electricity, a technical task was set: to produce it on an industrial scale. At that time (mid-nineteenth century) the world was gripped by machine fever. They tried to entrust all the difficult work to the expanding steam.

But then they could only heat large volumes of water with fossil fuels - coal and fuel oil. Therefore, those that were rich in ancient carbons immediately attracted the attention of investors and workers. And the redistribution of people led to the industrial revolution.

Holland and Texas

However, this state of affairs has a bad impact on the environment. And scientists wondered: how to get energy without destroying nature? The well-forgotten old thing came to the rescue. The mill used torque to directly perform rough mechanical work. Hydroelectric turbines rotate magnets.

Currently, the cleanest electricity comes from wind energy. The engineers who built the first generators in Texas relied on experience from the windmills of Holland.

Law of electromagnetic induction (Faraday-Maxwell law). Lenz's rules

Summarizing the results of his experiments, Faraday formulated the law of electromagnetic induction. He showed that with any change in the magnetic flux in a closed conducting circuit, an induction current is excited. Consequently, an induced emf occurs in the circuit.

The induced emf is directly proportional to the rate of change of magnetic flux over time. The mathematical notation of this law was drawn up by Maxwell and therefore it is called the Faraday-Maxwell law (the law of electromagnetic induction).

4.2.2. Lenz's rule

The law of electromagnetic induction does not talk about the direction of the induced current. This question was solved by Lenz in 1833. He established a rule to determine the direction of the induction current.

The induced current has such a direction that the magnetic field it creates prevents a change in the magnetic flux penetrating a given circuit, i.e. induced current. It is directed in such a way as to counteract the cause that causes it. For example, let a permanent magnet NS be moved into a closed loop (Fig. 250).

Fig.250 Fig.251

The number of lines of force crossing a closed loop increases, therefore, the magnetic flux increases. An induced current occurs in the circuit I i, which creates a magnetic field, the lines of force of which (dotted lines perpendicular to the contour plane) are directed against the field lines of the magnet. When the magnet is extended, the magnetic flux passing through the circuit decreases (Fig. 251), and the induction current I i creates a field, the lines of force of which are directed towards the induction line of the magnet (dashed lines in Fig. 251).

Taking into account Lenz's rule, the Faraday-Maxwell law will be written in the form

To solve the physical problem, formula (568) is used.

The time-average value of the induced emf is determined by the formula

Let's find out ways to change the magnetic flux.

First way. В=const And α=const. Area changes S.

Example. Let in a uniform magnetic field В=const a conductor of length l moves perpendicular to the lines of force with a speed (Fig. 252) Then a potential difference arises at the ends of the conductor equal to the induced emf. Let's find her.

The change in magnetic flux is

In formula (570) α - this is the angle between the normal of the plane washed by the movement of the conductor and the induction vector.