How does electromotive force arise in a conductor that is in an alternating magnetic field? What is a vortex electric field, its nature and causes of its occurrence? What are the main properties of this field? Today's lesson will answer all these and many other questions.

Topic: Electromagnetic induction

Lesson:Vortex electric field

Let us remember that Lenz's rule allows us to determine the direction of the induced current in a circuit located in an external magnetic field with an alternating flux. Based on this rule, it was possible to formulate the law of electromagnetic induction.

Law of Electromagnetic Induction

When the magnetic flux piercing the area of ​​the circuit changes, an electromotive force appears in this circuit, numerically equal to the rate of change of the magnetic flux, taken with a minus sign.

How does this electromotive force arise? It turns out that the EMF in a conductor that is in an alternating magnetic field is associated with the emergence of a new object - vortex electric field.

Let's consider experience. There is a coil of copper wire in which an iron core is inserted in order to enhance the magnetic field of the coil. The coil is connected through conductors to an alternating current source. There is also a coil of wire placed on a wooden base. An electric light bulb is connected to this coil. The wire material is covered with insulation. The base of the coil is made of wood, i.e., a material that does not conduct electric current. The coil frame is also made of wood. Thus, any possibility of contact of the light bulb with the circuit connected to the current source is eliminated. When the source is closed, the light bulb lights up, therefore, an electric current flows in the coil, which means that external forces do work in this coil. It is necessary to find out where outside forces come from.

A magnetic field penetrating the plane of a coil cannot cause the appearance of an electric field, since the magnetic field acts only on moving charges. According to the electronic theory of conductivity of metals, there are electrons inside them that can move freely within the crystal lattice. However, this movement in the absence of an external electric field is random. Such disorder leads to the fact that the total effect of the magnetic field on a current-carrying conductor is zero. This distinguishes the electromagnetic field from the electrostatic field, which also acts on stationary charges. Thus, the electric field acts on moving and stationary charges. However, the type of electric field that was studied earlier is created only by electric charges. The induced current, in turn, is created by an alternating magnetic field.

Suppose that the electrons in a conductor are set into ordered motion under the influence of some new kind of electric field. And this electric field is generated not by electric charges, but by an alternating magnetic field. Faraday and Maxwell came to a similar idea. The main thing in this idea is that a time-varying magnetic field generates an electric one. A conductor with free electrons in it makes it possible to detect this field. This electric field sets the electrons in the conductor in motion. The phenomenon of electromagnetic induction consists not so much in the appearance of an induction current, but in the appearance of a new type of electric field that sets in motion electric charges in a conductor (Fig. 1).

The vortex field differs from the static one. It is not generated by stationary charges, therefore, the intensity lines of this field cannot begin and end on the charge. According to research, the vortex field strength lines are closed lines similar to the magnetic field induction lines. Consequently, this electric field is a vortex - the same as a magnetic field.

The second property concerns the work of the forces of this new field. By studying the electrostatic field, we found out that the work done by the forces of the electrostatic field along a closed loop is zero. Since when a charge moves in one direction, the displacement and the effective force are co-directed and the work is positive, then when the charge moves in the opposite direction, the displacement and the effective force are oppositely directed and the work is negative, the total work will be zero. In the case of a vortex field, the work along a closed loop will be different from zero. So, when a charge moves along a closed line of an electric field that has a vortex character, the work in different sections will maintain a constant sign, since the force and displacement in different sections of the trajectory will maintain the same direction relative to each other. The work of the vortex electric field forces to move a charge along a closed loop is non-zero, therefore, the vortex electric field can generate an electric current in a closed loop, which coincides with the experimental results. Then we can say that the force acting on the charges from the vortex field is equal to the product of the transferred charge and the strength of this field.

This force is the external force that does the work. The work done by this force, related to the amount of charge transferred, is the induced emf. The direction of the vortex electric field intensity vector at each point of the intensity lines is determined by Lenz's rule and coincides with the direction of the induction current.

In a stationary circuit located in an alternating magnetic field, an induced electric current arises. The magnetic field itself cannot be a source of external forces, since it can only act on orderly moving electric charges. There cannot be an electrostatic field, since it is generated by stationary charges. After the assumption that a time-varying magnetic field generates an electric field, we learned that this alternating field is of a vortex nature, i.e. its lines are closed. The work of the vortex electric field along a closed loop is different from zero. The force acting on the transferred charge from the vortex electric field is equal to the value of this transferred charge multiplied by the intensity of the vortex electric field. This force is the external force that leads to the occurrence of EMF in the circuit. The electromotive force of induction, i.e. the ratio of the work of external forces to the amount of transferred charge, is equal to the rate of change of magnetic flux taken with a minus sign. The direction of the vortex electric field intensity vector at each point of the intensity lines is determined by Lenz's rule.

1. Kasyanov V.A., Physics 11th grade: Textbook. for general education institutions. - 4th ed., stereotype. - M.: Bustard, 2004. - 416 pp.: ill., 8 l. color on
2. Gendenstein L.E., Dick Yu.I., Physics 11. - M.: Mnemosyne.
3. Tikhomirova S.A., Yarovsky B.M., Physics 11. - M.: Mnemosyne.
   

1. Electronic physics textbook ().
2. Cool physics ().
3. Xvatit.com ().

1. How to explain the fact that a lightning strike can melt fuses and damage sensitive electrical appliances and semiconductor devices?
2. * When the ring was opened, a self-induction emf of 300 V arose in the coil. What is the intensity of the vortex electric field in the coil turns, if their number is 800, and the radius of the turns is 4 cm?

From Faraday’s law (see (123.2)) it follows that any a change in the magnetic induction flux associated with the circuit leads to the emergence of an electromotive force of induction and, as a result, an induction current appears. Consequently, the occurrence of emf. electromagnetic induction is possible in a stationary circuit,

located in an alternating magnetic field. However, the e.m.f. in any circuit occurs only when external forces act on current carriers in it - forces of non-electrostatic origin (see § 97). Therefore, the question arises about the nature of external forces in this case.

Experience shows that these extraneous forces are not associated with either thermal or chemical processes in the circuit; their occurrence also cannot be explained by Lorentz forces, since they do not act on stationary charges. Maxwell hypothesized that any alternating magnetic field excites an electric field in the surrounding space, which is the cause of the appearance of induced current in the circuit. According to Maxwell's ideas, the circuit in which the emf appears plays a secondary role, being a kind of only a “device” that detects this field.

So, according to Maxwell, a time-varying magnetic field generates an electric field E B, the circulation of which, according to (123.3),

where E B l - projection of the vector E B onto the direction dl.

Substituting the expression (see (120.2)) into formula (137.1), we obtain

If the surface and contour are stationary, then the operations of differentiation and integration can be swapped. Hence,

(137.2)

where the partial derivative symbol emphasizes the fact that the integral is a function of time only.

According to (83.3), the circulation of the electrostatic field strength vector (let’s denote it E Q) along any closed contour is zero:

(137.3)

Comparing expressions (137.1) and (137.3), we see that there is a fundamental difference between the fields under consideration (E B and E Q): the circulation of the vector E B in contrast to

circulation of vector E Q is not equal to zero. Therefore, the electric field E B, excited by a magnetic field, like the magnetic field itself (see § 118), is vortex.

Bias current

According to Maxwell, if any alternating magnetic field excites a vortex electric field in the surrounding space, then the opposite phenomenon should also exist: any change in the electric field should cause the appearance of a vortex magnetic field in the surrounding space. To establish quantitative relationships between a changing electric field and the magnetic field it causes, Maxwell introduced into consideration the so-called displacement current .

Consider an alternating current circuit containing a capacitor (Fig. 196). There is an alternating electric field between the plates of a charging and discharging capacitor, therefore, according to Maxwell, displacement currents “flow” through the capacitor, hidden in those areas where there are no conductors.

Let us find a quantitative relationship between the changing electric and the magnetic fields it causes. According to Maxwell, an alternating electric field in a capacitor at each moment of time creates such a magnetic field as if there were a conduction current between the plates of the capacitor equal to the current in the supply wires. Then we can say that the conduction currents (I) and displacement (I cm) are equal: I cm =I.

Conduction current near the capacitor plates

,(138.1)

(surface charge density s on the plates is equal to the electrical displacement D in the capacitor (see (92.1)). The integrand in (138.1) can be considered as a special case of the scalar product when and dS are mutual

parallel. Therefore, for the general case we can write

Comparing this expression with (see (96.2)), we have

Expression (138.2) was called by Maxwell the displacement current density.

Let's consider the direction of the conductivity and displacement current density vectors j and j cm. When charging a capacitor (Fig. 197, c) through the conductor connecting the plates, the current flows from the right plate to the left; the field in the capacitor is enhanced, therefore, , i.e. the vector is directed in the same direction as D . It can be seen from the figure that the directions of the vectors and j coincide. When the capacitor is discharged (Fig. 197, b) through the conductor connecting the plates, current flows from the left

facings to the right; the field in the capacitor is weakened; hence,<0, т. е.

the vector is directed opposite to vector D. However, the vector is directed again

the same as vector j. From the examples discussed, it follows that the direction of vector j, therefore, of vector j cm coincides with the direction of vector , as follows from formula (138.2).

We emphasize that of all the physical properties inherent in conduction current. Maxwell attributed only one thing to the displacement current - the ability to create a magnetic field in the surrounding space. Thus, the displacement current (in a vacuum or substance) creates a magnetic field in the surrounding space (the induction lines of the magnetic fields of the displacement currents when charging and discharging a capacitor are shown in Fig. 197 by dashed lines).

In dielectrics, the bias current consists from two terms. Since, according to (89.2), D= , where E is the electrostatic field strength, and P is the polarization (see § 88), then the displacement current density

, ( 138.3)

where is the displacement current density in vacuum, is the polarization current density - the current caused by the ordered movement of electric charges in the dielectric (displacement of charges in non-polar molecules or rotation of dipoles in polar molecules). Excitation of a magnetic field by polarization currents is legitimate, since polarization currents by their nature do not differ from conduction currents. However, the fact that the other part of the displacement current density, not associated with the movement of charges, but due to only a change in the electric field over time, also excites a magnetic field, is a fundamentally new statement Maxwell. Even in a vacuum, any change in time of the electric field leads to the appearance of a magnetic field in the surrounding space.

It should be noted that the name “displacement current” is conditional, or rather, historically developed, since the displacement current is inherently an electric field that changes over time. Displacement current therefore exists not only in vacuum or dielectrics, but also inside conductors through which alternating current passes.

However, in this case it is negligible compared to the conduction current. The presence of displacement currents was confirmed experimentally by A. A. Eikhenvald, who studied the magnetic field of the polarization current, which, as follows from (138.3), is part of the displacement current.

Maxwell introduced the concept full current, equal to the sum of conduction currents (as well as convection currents) and displacement. Total current density

Introducing the concepts of displacement current and total current. Maxwell took a new approach to considering the closed circuits of alternating current circuits. The total current in them is always closed, that is, at the ends of the conductor only the conduction current is interrupted, and in the dielectric (vacuum) between the ends of the conductor there is a displacement current that closes the conduction current.

Maxwell generalized the theorem on the circulation of the vector H (see (133.10)), introducing the total current into its right side through surface S , stretched over a closed contour L . Then the generalized theorem on the circulation of the vector H will be written in the form

(138.4)

Expression (138.4) is always true, as evidenced by the complete correspondence between theory and experience.

An induced emf occurs either in a stationary conductor placed in a time-varying field, or in a conductor moving in a magnetic field that may not change with time. The value of the EMF in both cases is determined by the law (12.2), but the origin of the EMF is different. Let's consider the first case first.

Let there be a transformer in front of us - two coils placed on a core. By connecting the primary winding to the network, we get a current in the secondary winding (Fig. 246) if it is closed. The electrons in the wires of the secondary winding will begin to move. But what forces make them move? The magnetic field itself, penetrating the coil, cannot do this, since the magnetic field acts exclusively on moving charges (this is how it differs from the electric one), and the conductor with the electrons in it is motionless.

In addition to the magnetic field, the charges are also affected by the electric field. Moreover, it can also act on stationary charges. But the field that has been discussed so far (electrostatic and stationary field) is created by electric charges, and the induced current appears under the influence of an alternating magnetic field. This leads us to assume that electrons in a stationary conductor are driven by an electric field and this field is directly generated by an alternating magnetic field. This establishes a new fundamental property of the field: changing over time, the magnetic field generates an electric field. Maxwell first came to this conclusion.

Now the phenomenon of electromagnetic induction appears before us in a new light. The main thing in it is the process of generating an electric field by a magnetic field. In this case, the presence of a conducting circuit, for example a coil, does not change the essence of the matter. A conductor with a supply of free electrons (or other particles) only makes it possible to detect the resulting electric field. The field moves the electrons in the conductor and thereby reveals itself. The essence of the phenomenon of electromagnetic induction in a stationary conductor is not so much the appearance of an induction current, but rather the appearance of an electric field that sets electric charges in motion.

The electric field that arises when the magnetic field changes has a completely different structure than the electrostatic one. It is not directly connected with electric charges, and its lines of tension cannot begin and end on them. They do not begin or end anywhere at all, but are closed lines, similar to magnetic field induction lines. This is the so-called vortex electric field (Fig. 247).

The direction of its field lines coincides with the direction of the induction current. The force exerted by the vortex electric field on the charge is still equal to: But, unlike a stationary electric field, the work of the vortex field on a closed path is not zero. After all, when a charge moves along a closed line of tension

electric field (Fig. 247), the work on all sections of the path will have the same sign, since the force and displacement coincide in direction. The work of a vortex electric field to move a single positive charge along a closed path is an induced emf in a stationary conductor.

Betatron. When the magnetic field of a strong electromagnet changes rapidly, powerful electric field vortices are created that can be used to accelerate electrons to speeds close to the speed of light. The device of the electron accelerator - the betatron - is based on this principle. The electrons in the betatron are accelerated by the vortex electric field inside the annular vacuum chamber K, placed in the gap of the electromagnet M (Fig. 248).

So, let's capture what we have already learned. All our formulas can be derived from several statements.

Statement 1.

The mathematical formulation of this statement is the Ostrogradsky-Gauss theorem for the electric field strength

On the right side is the integral of the charge density over an arbitrary volume, which is equal to the total charge inside it. On the left side is the flow of the electric field strength vector through an arbitrary closed surface limiting this volume. As we have seen, Coulomb's law is also contained in this equation.

Statement 2.

Magnetic charges do not exist in nature.

The mathematical formulation of this statement is the Ostrogradsky-Gauss theorem for the magnetic induction vector, on the right side of which there is zero

Statement 3.

Mathematically, this is expressed as the circulation of the electrostatic field strength being equal to zero along an arbitrary contour

Statement 4.

The mathematical expression of this statement is the theorem on the circulation of the magnetic induction vector

On the left side is the circulation of the magnetic field along an arbitrary contour L, and on the right - the integral of the total current density over an arbitrary surface S, stretched over this contour. This integral is equal to the sum of the currents crossing the surface S. This equation contains the Biot-Savart-Laplace law.

These four equations must be supplemented with an expression for the Lorentz force acting on moving charges from electromagnetic fields

The attentive reader will notice that the headings for the last two statements are in a different font. This was not done by chance: these statements are subject to modification. The fact is that since we formulated these four statements, we have become acquainted with one more phenomenon - electromagnetic induction. It has not yet been reflected in the written equations. Let's do it.

If the magnetic flux through the conducting coil L changes, then an induced emf occurs in the coil. What does this mean? The charges in the conductor will experience the force associated with this emf. But the appearance of a force acting on the charge means the appearance of some kind of electric field. The circulation of this field along the turn is exactly equal, by definition, to the induced emf

The difference between circulation and zero means that this electric field is not potential, but has vortex character, like a magnetic field. But if such a field has appeared, then what is the role of the coil? A coil is nothing more than a convenient detector for recording an eddy electric field from the resulting induction current. In order to part with the coil completely, let us express the induced emf in terms of the magnetic field flux. Let us rewrite Faraday's law in the form

Combining this equation with (9.6), we arrive at modified Statement 3 (Fig. 9.1).

Statement 5.

Rice. 9.1. The law of electromagnetic induction as interpreted by Maxwell:
a changing magnetic field generates a vortex electric field

Mathematically this is expressed as the equation

This equation contains Faraday's law of electromagnetic induction.

We need to be a little careful here: since we have an additional electric field, won't it change the first statement? Fortunately, the answer is negative: the flux of the vortex field through a closed surface is zero, so this field will not contribute to the left side of equation (9.1).

It would seem that we have already taken into account all the phenomena with which we are familiar. Why then did we mark the fourth equation as requiring modification? The fact is that the symmetry between electrical and magnetic phenomena is now broken. Let us assume that there are no charges or currents in the system. Could an electromagnetic field then exist? We know the answer from modern life: it can! There are electromagnetic waves that propagate in space and do not require any medium for this. In the absence of charges and currents, the first two equations (9.1) and (9.2) are completely symmetrical. The same cannot be said about the second pair of equations. Can an electric (vortex) field be generated without charges, simply by changing the magnetic field? Why can’t a magnetic field be generated not by currents, but by changing the electric field?

One of the questions that can often be found on the vastness of the global Internet is how a vortex electric field differs from an electrostatic one. In fact, the differences are dramatic. In electrostatics, the interaction of two (or more) charges is considered and, what is important, the strength lines of such fields are not closed. But the vortex electric field obeys completely different laws. Let's consider this issue in more detail.

One of the most common devices that almost every person encounters is a meter for metering consumed electrical energy. Not the modern electronic models, but the “old” ones, which use an aluminum rotating disk. It is “forced” to rotate by the induction of an electric field. As is known, in any conductor of large volume and mass (not a wire), which is penetrated by a changing magnetic flux, an electromotive force and an electric current called eddy current arise in accordance with it. Note that in this case it does not matter at all whether the magnetic field changes or the conductor itself moves in it. In accordance with the law of electromagnetic induction, closed vortex-shaped circuits are created in the mass of the conductor through which currents circulate. Their orientation can be determined using Lenz's rule. It states that the current is directed in such a way as to compensate for any change (both decrease and increase) of the initiating external magnetic flux. The counter disk rotates precisely due to the interaction of the external magnetic field and the currents generated within it.

How is the vortex electric field related to all of the above? In fact, there is a connection. It's all a matter of terms. Any change in the magnetic field creates a vortex electric field. Then everything is simple: a current is generated in the conductor and appears in the circuit. Its magnitude depends on the rate of change of the main flux: for example, the faster the conductor crosses the field strength lines, the greater the current. The peculiarity of this field is that its lines of tension have neither beginning nor end. Its configuration is sometimes compared to a solenoid (a cylinder with coils of wire on its surface). Another schematic representation uses a vector for explanation. Around each of them, lines are created that really resemble vortices. Important feature: the last example is true if the intensity of the magnetic flux changes. If you “look” along the induction vector, then as the flow increases, the vortex field lines rotate clockwise.

The property of induction is widely used in modern electrical engineering: measuring instruments, motors, and electron accelerators.

• this type of field is inextricably linked with charge carriers;
• the force acting on the charge carrier is created by the field;
• the field weakens as it moves away from the carrier;
• characterized by lines of force (or, which is also true, lines of tension). They are directed, so they represent a vector quantity.

To study the properties of the field at each arbitrary point, a test (trial) charge is used. At the same time, they strive to select a “probe” so that its introduction into the system does not affect the acting forces. This is usually the reference charge.

Note that Lenz's rule makes it possible to calculate only the electromotive force, but the value of the field vector and its direction are determined by another method. We are talking about Maxwell's system of equations.