Law of magnetic induction. Electromagnetic induction

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 a recently discovered property of an 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 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 aspect of time variation of electromagnetic induction is expressed as 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 group of equations recognized today 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 electric force due to changes in the magnetic field. James Clerk Maxwell drew attention to this fact in his work About 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 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 various laws for two cases  –      for a “moving chain” and     for a “changing field”.
We do not know of any analogous situation in physics, when such simple and precise general principles would require for their real understanding analysis from the point of view of two different phenomena.

-Richard Feynman,   Feynman lectures in physics

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 relative motion conductor and magnet, whereas common opinion draws a sharp difference between these 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 unsuccessful attempt to detect any movement of the Earth relative to the “luminiferous medium” suggests 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

The definition of surface integral assumes that the surface Σ is divided into small elements. Each element is associated with a vector dA, the magnitude of which is equal to the area of ​​the element, and the direction is normal to the element to the outside.

Vector field F(r, t) is defined throughout space, and the surface Σ is limited by the curve ∂Σ moving with speed v. The field is integrated over this surface.

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 dA- area of ​​the surface element Σ( t), B- magnetic field, and B· dA- dot product B And dA. It is assumed that the surface has a “mouth” delineated by a closed curve designated ∂Σ( t). Faraday's law of induction states that when the flow changes, then when moving a unit positive test charge along a closed curve ∂Σ, work is done, the value of which is determined by the formula:

where 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.

Therefore, EMF

where v = speed of conductor or magnet, A l= vertical loop length. In this case, the speed is related to the angular velocity of rotation v = rω, where r= radius of the cylinder. note that same job carried out according to anyone a path that rotates with the loop and connects the upper and lower rims.

Faraday's law

An intuitively attractive but flawed approach to using the flow rule expresses the flow through a circuit as Φ B = Bwℓ, where w- width of the moving loop. This expression is independent of time, so it incorrectly follows that no emf is generated. The error in this statement is that it does not take into account the entire path of the current through the closed loop.

For correct use flow rules we must consider the entire current path, which includes the path through the rims on the upper and lower rims. We can choose an arbitrary closed path through the rims and the rotating loop, and using the flow law, find the emf along this path. Any path that includes a segment adjacent to a rotating loop takes into account the relative motion of the parts of the chain.

As an example, consider a path passing at the top of the chain in the direction of rotation of the upper disk, and at the bottom of the chain - in opposite direction in relation to the lower disk (shown by arrows in Fig. 4). In this case, if the rotating loop has deviated by an angle θ from the collector loop, then it can be considered as part of a cylinder with an area A = rℓθ. This area is perpendicular to the field B, and its contribution to the flow is equal to:

where the sign is negative because according to the rule right hand field B , generated by a loop with current, opposite in direction to the applied field B". Since this is only the time-dependent part of the flux, according to the flux law the emf is:

in accordance with the formula of Lorentz's law.

Now consider another path, in which we choose to pass along the rims of the disks through opposite segments. In this case the associated thread will be decrease with increasing θ, but according to the right-hand rule, the current loop adds attached field B, therefore the EMF for this path will be exactly the same value as for the first path. Any mixed return path produces the same result for the emf value, so it doesn't really matter which path you choose.

Direct estimation of flux change

Rice. 5. Simplified version of Fig. 4. The loop slides at speed v in a stationary uniform field B.

Using a closed path to calculate EMF, as done above, depends on the detailed geometry of the path. In contrast, the use of Lorentz's law is not subject to such restrictions. The following discussion is intended to provide a better understanding of path equivalence and to avoid having to figure out the details of the chosen path when using the flow law.

Rice. Figure 5 is an idealization of Figure 4; it shows the projection of a cylinder onto a plane. The same analysis along the linked path is valid, but some simplifications are made. Time-independent circuit components cannot influence the rate of change of flow. For example, when constant speed When the loop slips, the current flow through the loop does not depend on time. Instead of considering the details of a selected closed loop when calculating the emf, you can focus on the field region B, swept by a moving loop. The proposal boils down to finding the speed at which the flow crosses the chain. This concept provides a direct estimate of the rate of change of flow, eliminating the need to think about the more time-dependent details of different path paths along the circuit. Just as when applying Lorentz's law, it becomes clear that any two paths associated with a sliding loop, but differing in the way they cross the loop, create a flow with the same rate of change.

In Fig. 5 sweep area per unit time is equal to dA/dt = vℓ, regardless of the details of the selected closed path, so that according to Faraday’s law of induction, the emf is equal to:

This path of independent emf shows that if the sliding loop is replaced by a solid conducting plate or even some complex curved surface, the analysis will be the same: find the flux in the sweeping area of ​​the moving parts of the circuit. Similarly, if the sliding loop in the generator drum in Fig. 4 is replaced by a solid conducting cylinder, the calculation of the swept area is done in exactly the same way as in the case of a simple loop. That is, the EMF calculated according to Faraday’s law will be exactly the same as in the case of a cylinder with solid conducting walls, or, if you prefer, a cylinder with walls made of grated cheese. Note, however, that the current flowing as a result of this emf will not be exactly the same, because the current also depends on the resistance of the circuit.

Faraday - Maxwell equation

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

This equation is present in modern system 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 (discussed in 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 install correct signs, the right-hand rule is used, as described in the article on the Kelvin–Stokes theorem. For flat surfaceΣ 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 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 a scalar field, which is a solution to Poisson's 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.

Rice. 7. Sweeping area of ​​a vector element dℓ crooked ∂Σ in time dt when moving at speed v.

and taking into account (Gauss Series), (Cross Product) and (Kelvin - Stokes Theorem), we find that the total derivative of the magnetic flux can be expressed

By adding a term 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 in time 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 in time 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 it is revealed close connection 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, i.e. 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 derivative with respect to x methods for differentiating complex functions are used.

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

Electric generator

Rice. 8. Electric generator based on a Faraday disk. The disk rotates with angular velocity ω, while a conductor located along the radius moves in a static magnetic field B. Magnetic Lorentz force v×B creates a current along the conductor towards the rim, then the circuit is completed through the lower brush and the disc support axis. Thus, due to mechanical movement current is generated.

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 the Faraday disk, shown in a simplified form in Fig. 8. Please note that the analysis of Fig. 5, and direct application Lorentz force law shows 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, it is necessary mechanical work. 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, which 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, exactly equal to the generated electrical energy(plus energy to compensate for losses due to friction, due to Joule heat release, etc.). This behavior is common to all generators that convert mechanical energy into electrical energy.

Although Faraday's law describes the work of any electric generators, detailed mechanism in different cases may vary. 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 a certain rule left hand. In the absence of effects that cause 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 dΦB/ 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.

Electromagnetic flow meters

Faraday's law is used to measure the flow of electrically conductive liquids and suspensions. Such devices are called magnetic flow meters. Induced voltage ℇ generated in a magnetic field B due to a conducting fluid moving at a speed v, is determined by the formula:

where ℓ is the distance between the electrodes in the magnetic flow meter.

In any metal object moving in relation to a static magnetic field, induced currents will arise, as in any stationary metal object in relation to a moving magnetic field. These energy flows most often undesirable, because of them an electric current flows in the metal layer, which heats the metal.

Eddy currents occur when solid mass metal rotates in a magnetic field, since the outer part of the metal intersects more lines of force than the inner one, therefore, the induced electromotive force is uneven and tends to create currents between points with the highest and lowest potentials. Eddy currents consume significant amount energy, and often lead to a harmful increase in temperature.

This example shows a total of five laminates or plates to demonstrate splitting. eddy currents. In practice, the number of plates or perforations ranges from 40 to 66 per inch, which leads to reduced losses by eddy currents to about one percent. Although the plates may be separated from each other by insulation, since the voltages encountered are extremely low, natural rust or oxide coating on the plates is sufficient to prevent current flow through the plates.

In this illustration, the solid copper bar of the inductor in the rotating armature simply passes under the tip of the N pole of the magnet. Please note uneven distribution lines of force through the rod. The magnetic field is more concentrated and therefore stronger at the left edge of the copper rod (a,b), while it is weaker at the right edge (c,d). Since the two edges of the rod will move at the same speed, this difference in field strength through the rod will create current vortices within the copper rod.