Chapter 1

ELECTROMAGNETISM

§1. Electric forces

§2. Electric and magnetic fields

§3. Characteristics of vector fields

§4. Laws of electromagnetism

§5. What is it - "fields"?

§6. Electromagnetism in science and technology

Repeat: ch. 12 (Issue 1) "Power Characteristics"

§ 1. Electric forces

Consider a force that, like gravity, varies inversely with the square of the distance, but only in million billion billion billion times stronger. And which differs in one more. Let there be two kinds of "substance" which can be called positive and negative. Let the same varieties repel, and different ones attract, in contrast to gravitation, in which only attraction occurs. What will happen then?

Everything positive will be repelled with terrible force and scattered in different directions. Everything negative, too. But something completely different will happen if positive and negative are mixed equally. Then they will be attracted to each other with great force, and as a result, these incredible forces will almost completely balance, forming dense "fine-grained" mixtures of positive and negative; between two piles of such mixtures there will be practically no attraction or repulsion.

There is such a force: it is electrical force. And all matter is a mixture of positive protons and negative electrons, attracting and repelling with incredible force. However, the balance between them is so perfect that when you stand near someone, you do not feel any effect of this force. And if the balance was disturbed even a little bit, you would immediately feel it. If there were only 1% more electrons in your body or in the body of your neighbor (standing at arm's length from you) than protons, then your repulsive force would be unimaginably large. How big? Enough to raise a skyscraper? More! Enough to lift Mount Everest? More! The repulsive force would be enough to lift a "weight" equal to the weight of our Earth!

Since such enormous forces in these subtle mixtures are so perfectly balanced, it is not difficult to understand that a substance, striving to keep its positive and negative charges in the finest balance, must have great rigidity and strength. The top of a skyscraper, say, only moves a couple of meters in gusts of wind, because electrical forces keep every electron and every proton more or less in place. On the other hand, if a sufficiently small amount of matter is considered so that there are only a few atoms in it, then there will not necessarily be an equal number of positive and negative charges, and large residual electric forces may appear. Even if the numbers of those and other charges are the same, a significant electric force can still act between neighboring areas. Because the forces acting between the individual charges vary inversely with the squares of the distances between them, and it may turn out that the negative charges of one part of the substance are closer to the positive charges (of the other part) than to the negative ones. The forces of attraction will then exceed the forces of repulsion, and as a result, there will be an attraction between the two parts of the substance in which there is no excess charge. The force that holds atoms together, and the chemical forces that hold molecules together, are all electric forces, acting where the number of charges is not the same or where the gaps between them are small.

You know, of course, that an atom has positive protons in the nucleus and electrons outside the nucleus. You may ask: “If these electrical forces are so great, then why don’t protons and electrons overlap each other? If they want to form a close company, why not get even closer? The answer has to do with quantum effects. If we try to enclose our electrons in a small volume surrounding the proton, then, according to the uncertainty principle, they should have an RMS momentum, the greater, the more we restrict them. It is this motion (required by the laws of quantum mechanics) that prevents electrical attraction from bringing the charges even closer together.

Here another question arises: “What holds the core together?” There are several protons in the nucleus, and they are all positively charged. Why don't they fly away? It turns out that in the nucleus, in addition to electric forces, there are also non-electric forces, called nuclear. These forces are more powerful than electrical forces, and they are capable, despite electrical repulsion,

hold the protons together. The action of nuclear forces, however, does not extend far; it falls much faster than 1/r 2 . And this leads to an important result. If there are too many protons in the nucleus, then the nucleus becomes too large and it can no longer hold on. An example is uranium with its 92 protons. Nuclear forces act primarily between a proton (or neutron) and its nearest neighbor, while electrical forces act over long distances and cause each proton in the nucleus to be repulsed from all the others. The more protons in the nucleus, the stronger the electrical repulsion, until (like uranium) the equilibrium becomes so precarious that it costs almost nothing for the nucleus to fly apart from the effect of the electrical repulsion. It is worth it to “push” it a little (for example, by sending a slow neutron inside) - and it falls apart in two, into two positively charged parts, flying apart as a result of electrical repulsion. The energy that is released in this case is the energy of the atomic bomb. It is commonly referred to as "nuclear" energy, although it is actually "electrical" energy, released as soon as the electrical forces overcome the nuclear forces of attraction.

Finally, one may ask, how is a negatively charged electron held together (after all, there are no nuclear forces in it)? If the electron is all of the same kind of matter, then each part of it must repel the rest. Then why don't they scatter in different directions? Does an electron really have "parts"? Maybe we should consider the electron as just a point and say that electric forces act only between different point charges, so that the electron does not act on itself? Maybe. The only thing that can be said now is that the question of how the electron is held together has caused many difficulties in trying to create a complete theory of electromagnetism. And we have not received an answer to this question. We will discuss it a little later.

As we have seen, it is hoped that the combination of electrical forces and quantum mechanical effects will determine the structure of large amounts of matter and hence their properties. Some materials are hard, others are soft. Some are electrical "conductors" because their electrons are free to move; others are "insulators", their electrons are each tied to their own atom. Later we will find out where such properties come from, but this question is very complicated, so we will first consider electrical forces in the simplest situations. Let us begin by studying the laws of electricity alone, including here also magnetism, since both are really phenomena of the same nature.

We said that electric forces, like gravitational forces, decrease in inverse proportion to the square of the distance between charges. This relation is called Coulomb's law. However, this law ceases to hold exactly if the charges are moving. Electric forces also depend in a complex way on the movement of charges. One of the parts of the force acting between moving charges, we call magnetic by force. In fact, this is only one of the manifestations of electrical action. That's why we talk about "electromagnetism".

There is an important general principle that makes it relatively easy to study electromagnetic forces. We find experimentally that the force acting on an individual charge (regardless of how many more charges there are or how they move) depends only on the position of this individual charge, on its speed and magnitude. The force F acting on the charge q ,

moving at a speed v, we can write it as:

here E- electric field at the location of the charge, and B - a magnetic field. It is essential that the electric forces acting from all other charges of the Universe add up and give just these two vectors. Their meanings depend on where there is a charge, and can change with time. If we replace this charge with another, then the force acting on the new charge changes exactly in proportion to the magnitude of the charge, unless all other charges in the world change their motion or position. (In real conditions, of course, each charge acts on all other charges in its neighborhood and can cause them to move, so sometimes when one given charge is replaced by another, the fields may change.)

From the material presented in the first volume, we know how to determine the motion of a particle if the force acting on it is known. Equation (1.1) combined with the equation of motion gives

So, if E and B are known, then the movement of charges can be determined. It remains only to find out how E and B are obtained.

One of the most important principles that simplifies the derivation of field values ​​is as follows. Let a certain number of charges moving in some way create a field E 1 , and another set of charges - a field E 2 . If both sets of charges act simultaneously (keeping their positions and motions the same as they had when considered separately), then the resulting field is exactly the sum

E \u003d E 1 + E 2. (1.3)

This fact is called overlay principle fields (or superposition principle). It also holds for magnetic fields.

This principle means that if we know the law for the electric and magnetic fields formed solitary a charge moving in an arbitrary way, then, therefore, we know all the laws of electrodynamics. If we want to know the force acting on the charge BUT, we only need to calculate the magnitude of the fields E and B created by each of the charges B, C, D etc., and add up all these E and B; thus we will find the fields, and from them - the forces acting on BUT. If it turned out that the field created by a single charge is simple, then this would be the most elegant way to describe the laws of electrodynamics. But we have already described this law (see Issue 3, Chapter 28), and, unfortunately, it is rather complicated.

It turns out that the form in which the laws of electrodynamics become simple is not at all what one might expect. She is not is simple if we want to have a formula for the force with which one charge acts on another. True, when the charges are at rest, the law of force - Coulomb's law - is simple, but when the charges move, the relations become more complicated due to time delay, the influence of acceleration, etc. As a result, it is better not to try to build electrodynamics using only the laws of forces acting between charges; much more acceptable is another point of view, in which the laws of electrodynamics are easier to manage.

§ 2. Electric and magnetic fields

First of all, we need to slightly expand our understanding of the electric and magnetic vectors E and B. We have defined them in terms of the forces acting on the charge. Now we intend to talk about electric and magnetic fields in point, even if there is no charge.

Fig. 1.1. A vector field represented by a set of arrows, the length and direction of which indicate the magnitude of the vector field at the points where the arrows come from.

Therefore, we assert that since forces “act” on the charge, then in the place where it stood, “something” remains even when the charge is removed from there. If a charge located at a point (x, y, z), at the moment t feels the action of the force F, according to equation (1.1), then we connect the vectors E and B with a dot (x, y, z) in space. We can assume that E (x, y, z, t) and B (x, y, z, t) give forces, the effect of which will be felt at the moment t charge located in (x, y, z), provided that that placing a charge at that point won't disturb neither the location nor the movement of all other charges responsible for the fields.

Following this notion, we associate with each dot (x, y, z) space, two vectors E and B, capable of changing over time. The electric and magnetic fields are then considered as vector functions from x, y, z and t. Since the vector is determined by its components, then each of the fields E (x, y, 2, t) and B (x, y, z, t) is three mathematical functions of x, y, z and t.

It is precisely because E (or B) can be defined for every point in space that it is called a "field". A field is any physical quantity that takes on different values ​​at different points in space. Let's say temperature is a field (scalar in this case) that can be written as T(x, y, z). In addition, the temperature can also change with time, then we say that the temperature field depends on time, and write T (x, y, z, t). Another example of a field is the "velocity field" of a flowing fluid. We record the velocity of the fluid at any point in space at the moment t v (x, y, z, t). The field is vector.

Let's get back to electromagnetic fields. Although the formulas by which they are created by charges are complex, they have the following important property: the relationship between the values ​​of the fields in some point and their values ​​in neighboring point very simple. A few such relations (in the form of differential equations) are sufficient to completely describe the fields. It is in this form that the laws of electrodynamics look particularly simple.

Fig. 1.2. A vector field represented by lines tangent to the direction of the vector field at each point.

Line density indicates the magnitude of the field vector.

A lot of ingenuity has been expended in helping people visualize the behavior of fields. And the most correct point of view is the most abstract: you just need to consider the fields as mathematical functions of coordinates and time. You can also try to get a mental picture of the field by drawing a vector at many points in space so that each of them shows the strength and direction of the field at that point. Such a representation is shown in Fig. 1.1. You can go even further: draw lines that at any point will be tangent to these vectors. They seem to follow the arrows and keep the direction of the field. If this is done, then information about lengths vectors will be lost, but they can be saved if, in those places where the field strength is low, lines are drawn less often, and where it is large, thicker. Let's agree that number of lines per unit area, located across the lines will be proportional to field strength. This is, of course, only an approximation; sometimes we have to add new lines to match the field strength. The field shown in Fig. 1.1 is represented by the field lines in Fig. 1.2.

§ 3. Characteristics of vector fields

Vector fields have two mathematically important properties that we will use to describe the laws of electricity from a field point of view. Let us imagine a closed surface and ask the question, does “something” follow from it, i.e., does the field have the property of “outflow”? For example, for a velocity field, we can ask whether the velocity is always directed away from the surface, or, more generally, whether more fluid flows out of the surface (per unit time) than flows in.

Fig. 1.3. The flux of a vector field through a surface, defined as the product of the mean value of the perpendicular component of the vector and the area of ​​that surface.

The total amount of liquid flowing through the surface we will call the "flow of velocity" through the surface per unit time. The flow through a surface element is equal to the velocity component perpendicular to the element times its area. For an arbitrary closed surface total flow is equal to the average value of the normal component of the velocity (counted outward) multiplied by the surface area:

Flux = (Mean Normal Component)·(Surface Area).

In the case of an electric field, a concept similar to the source of a liquid can be mathematically defined; we, too

Fig. 1.4. Velocity field in liquid (a).

Imagine a tube of constant cross section laid along an arbitrary closed curve(b). If the liquid suddenly freeze everywhere, except for the tube then liquid in the tube will begin to circulate (c).

Fig. 1.5. Circulation vector wow fields equal to the product

the average tangent component of the vector (taking into account its sign

with respect to the bypass direction) by the length of the contour.

we call it a flow, but, of course, it is no longer a flow of some kind of liquid, because the electric field cannot be considered the speed of something. It turns out, however, that the mathematical quantity defined as the average normal component of the field still has a useful value. Then we're talking about the flow of electricity also defined by equation (1.4). Finally, it is useful to talk about the flow not only through a closed, but also through any limited surface. As before, the flux through such a surface is defined as the average normal component of the vector multiplied by the area of ​​the surface. These representations are illustrated in Fig. 1.3. Another property of vector fields concerns not so much surfaces as lines. Imagine again the velocity field describing the fluid flow. An interesting question can be asked: does the liquid circulate? This means: is there a rotational motion along some closed contour (loop)? Imagine that we have instantly frozen the liquid everywhere, except for the inside of a tube of constant cross section closed in the form of a loop (Fig. 1.4). Outside the tube, the liquid will stop, but inside it can continue to move if momentum is preserved in it (in the liquid), that is, if the momentum that drives it in one direction is greater than the momentum in the opposite direction. We define a quantity called circulation, as the velocity of the fluid in the tube multiplied by the length of the tube. Again, we can expand our notions and define "circulation" for any vector field (even if there is nothing moving there). For any vector field circulation in any imaginary closed circuit is defined as the average tangent component of the vector (taking into account the direction of the bypass), multiplied by the length of the contour (Fig. 1.5):

Circulation = (Mean tangent component)·(Length of traversal path). (1.5)

You see that this definition does indeed give a number proportional to the circulation velocity in a tube drilled through a quick-frozen liquid.

Using only these two concepts - the concept of flow and the concept of circulation - we are able to describe all the laws of electricity and magnetism. It may be difficult for you to clearly understand the meaning of the laws, but they will give you some idea of ​​how the physics of electromagnetic phenomena can ultimately be described.

§ 4. Laws of electromagnetism

The first law of electromagnetism describes the flow of an electric field:

where e 0 is some constant (read epsilon zero). If there are no charges inside the surface, but there are charges outside it (even very close to it), then all the same average the normal component of E is zero, so there is no flow through the surface. To show the usefulness of this type of statement, we will prove that equation (1.6) coincides with Coulomb's law, if only we take into account that the field of an individual charge must be spherically symmetric. Draw a sphere around a point charge. Then the average normal component is exactly equal to the value of E at any point, because the field must be directed along the radius and have the same magnitude at all points on the sphere. Our rule then states that the field on the surface of the sphere times the area of ​​the sphere (i.e., the flux flowing out of the sphere) is proportional to the charge inside it. If you increase the radius of a sphere, then its area increases as the square of the radius. The product of the average normal component of the electric field and this area must still be equal to the internal charge, so the field must decrease as the square of the distance; thus the field of "inverse squares" is obtained.

If we take an arbitrary curve in space and measure the circulation of the electric field along this curve, then it turns out that in the general case it is not equal to zero (although this is the case in the Coulomb field). Instead, the second law holds for electricity, stating that

And, finally, the formulation of the laws of the electromagnetic field will be completed if we write two corresponding equations for the magnetic field B:

And for the surface S, bounded curve WITH:

The constant c 2 that appeared in equation (1.9) is the square of the speed of light. Its appearance is justified by the fact that magnetism is essentially a relativistic manifestation of electricity. And the constant e o was set in order for the usual units of electric current strength to arise.

Equations (1.6) - (1.9), as well as equation (1.1) - these are all the laws of electrodynamics.

As you remember, Newton's laws were very easy to write, but many complex consequences followed from them, so it took a long time to study them all. The laws of electromagnetism are incomparably more difficult to write, and we must expect the consequences of them to be much more complicated, and now we will have to understand them for a very long time.

We can illustrate some of the laws of electrodynamics with a series of simple experiments that can show us at least qualitatively the relationship between electric and magnetic fields. You get to know the first term in equation (1.1) by combing your hair, so we won't talk about it. The second term in equation (1.1) can be demonstrated by passing a current through a wire suspended over a magnetic bar, as shown in Fig. 1.6. When the current is turned on, the wire moves due to the fact that a force F = qvXB acts on it. When a current flows through the wire, the charges inside it move, that is, they have a speed v, and the magnetic field of the magnet acts on them, as a result of which the wire moves away.

When the wire is pushed to the left, the magnet itself can be expected to experience a push to the right. (Otherwise, this whole device could be mounted on a platform and get a reactive system in which momentum would not be conserved!) Although the force is too small to notice the movement of a magnetic wand, the movement of a more sensitive device, say a compass needle, is quite noticeable.

How does the current in the wire push the magnet? The current flowing through the wire creates its own magnetic field around it, which acts on the magnet. In accordance with the last term in equation (1.9), the current should lead to circulation vector B; in our case, field lines B are closed around the wire, as shown in fig. 1.7. It is this field B that is responsible for the force acting on the magnet.

Fig.1.6. Magnetic stick that creates a field near the wire AT.

When current flows through the wire, the wire is displaced due to the force F = q vxb.

Equation (1.9) tells us that for a given amount of current flowing through the wire, the circulation of the field B is the same for any curve surrounding the wire. Those curves (circles, for example) that lie far from the wire have a longer length, so the tangent component B must decrease. You can see that B should be expected to decrease linearly with distance from a long straight wire.

We said that the current flowing through the wire forms a magnetic field around it, and that if there is a magnetic field, then it acts with some force on the wire through which the current flows.

Fig.1.7. The magnetic field of the current flowing through the wire acts on the magnet with some force.

Fig. 1.8. Two wires carrying current

also act on each other with a certain force.

So, one should think that if a magnetic field is created by a current flowing in one wire, then it will act with some force on the other wire, through which the current also flows. This can be shown by using two freely suspended wires (Fig. 1.8). When the direction of the currents is the same, the wires attract, and when the directions are opposite, they repel.

In short, electric currents, like magnets, create magnetic fields. But then what is a magnet? Since magnetic fields are created by moving charges, can it not turn out that the magnetic field created by a piece of iron is actually the result of the action of currents? Apparently, that's the way it is. In our experiments it is possible to replace the magnetic stick with a coil of wound wire, as shown in Fig. 1.9. When the current passes through the coil (as well as through a straight wire above it), exactly the same movement of the conductor is observed as before, when a magnet was in place of the coil. Everything looks as if a current circulates continuously inside a piece of iron. Indeed, the properties of magnets can be understood as a continuous current within the iron atoms. The force acting on the magnet in Fig. 1.7 is explained by the second term in equation (1.1).

Where do these currents come from? One source is the movement of electrons in atomic orbits. In iron this is not the case, but in some materials the origin of magnetism is precisely this. In addition to rotating around the nucleus of an atom, the electron also rotates around its own axis (something similar to the rotation of the Earth); it is from this rotation that a current arises, which creates a magnetic field of iron. (We said "something like the rotation of the Earth" because, in fact, the matter in quantum mechanics is so deep that it does not fit well into classical concepts.) In most substances, some electrons spin in one direction, some in the other, so that magnetism disappears, and in iron (for a mysterious reason, which we will discuss later) many electrons rotate so that their axes point in the same direction and this is the source of magnetism.

Since the fields of magnets are generated by currents, there is no need to insert additional terms into equations (1.8) and (1.9) that take into account the existence of magnets. These equations are about all currents, including circular currents from rotating electrons, and the law turns out to be correct. It should also be noted that, according to equation (1.8), there are no magnetic charges similar to electric charges on the right side of equation (1.6). They have never been discovered.

The first term on the right side of equation (1.9) was discovered theoretically by Maxwell; he is very important. He says change electrical fields causes magnetic phenomena. In fact, without this term, the equation would lose its meaning, because without it the currents in open circuits would disappear. But in fact, such currents exist; the following example speaks of this. Imagine a capacitor made up of two flat plates.

Fig. 1.9. The magnetic stick shown in Fig. 1.6

can be replaced by a coil that flows

The force will still be acting on the wire.

Fig. 1.10. The circulation of the field B along the curve C is determined either by the current flowing through the surface S 1 or by the rate of change of the flow, the field E through the surface S 2 .

It is charged by current flowing into one of the plates and outflowing from the other, as shown in Fig. 1.10. Draw a curve around one of the wires With and stretch a surface over it (surface S 1 , that crosses the wire. In accordance with equation (1.9), the circulation of the field B along the curve With is given by the amount of current in the wire (multiplied by with 2 ). But what happens if we pull on the curve another surface S 2 in the form of a cup, the bottom of which is located between the plates of the capacitor and does not touch the wire? No current passes through such a surface, of course. But a simple change in the position and shape of an imaginary surface should not change the real magnetic field! The circulation of field B must remain the same. Indeed, the first term on the right side of equation (1.9) is combined with the second term in such a way that the same effect occurs for both surfaces S 1 and S 2 . For S 2 the circulation of the vector B is expressed in terms of the degree of change in the flow of the vector E from one plate to another. And it turns out that the change in E is connected with the current just so that equation (1.9) is satisfied. Maxwell saw the need for this and was the first to write the complete equation.

With the device shown in Fig. 1.6, another law of electromagnetism can be demonstrated. Disconnect the ends of the hanging wire from the battery and attach them to a galvanometer - a device that records the passage of current through the wire. Stands only in the field of a magnet swing wire, as the current will immediately flow through it. This is a new consequence of equation (1.1): the electrons in the wire will feel the action of the force F=qvXB. Their speed is now directed to the side, because they deviate along with the wire. This v, together with the vertically directed field B of the magnet, results in a force acting on the electrons along wires, and the electrons are sent to the galvanometer.

Let us suppose, however, that we leave the wire alone and begin to move the magnet. We feel that there should be no difference, because the relative motion is the same, and indeed the current flows through the galvanometer. But how does a magnetic field act on charges at rest? In accordance with equation (1.1), an electric field should arise. A moving magnet must create an electric field. The question of how this happens is answered quantitatively by equation (1.7). This equation describes many practically very important phenomena occurring in electrical generators and transformers.

The most remarkable consequence of our equations is that, by combining equations (1.7) and (1.9), one can understand why electromagnetic phenomena propagate over long distances. The reason for this, roughly speaking, is something like this: suppose that somewhere there is a magnetic field that increases in magnitude, say, because a current is suddenly passed through the wire. Then it follows from equation (1.7) that the circulation of the electric field should occur. When the electric field begins to gradually increase for circulation to occur, then, according to equation (1.9), magnetic circulation must also occur. But the rise this the magnetic field will create a new circulation of the electric field, etc. In this way, the fields propagate through space, requiring neither charges nor currents anywhere but the source of the fields. It is in this way that we see each other! All this is hidden in the equations of the electromagnetic field.

§ 5. What is it - "fields"?

Let us now make a few remarks about the way we have adopted this question. You may say, “All these flows and circulations are too abstract. Let there be an electric field at every point in space, in addition, there are these same "laws". But what is there in fact happening? Why can't you explain all this by, say, something, whatever it is, flowing between the charges?" It all depends on your prejudices. Many physicists often say that direct action through the void, through nothing, is unthinkable. (How can they call an idea unthinkable when it's already made up?) They say, "Look, the only forces we know of are the direct action of one part of matter on another. It is impossible for there to be power without something to transmit it.” But what actually happens when we study the "direct action" of one piece of matter on another? We find that the first of them does not "rest" at all on the second; they are slightly spaced apart, and between them there are electrical forces acting on a small scale. In other words, we find that we are going to explain the so-called "action by direct contact" - with the help of a picture of electrical forces. Of course, it's unreasonable to try to argue that electrical force should look just like the old habitual muscle push-pull, if it turns out that all our attempts to pull or push result in electrical forces! The only reasonable question is to ask which way of considering electrical effects most convenient. Some prefer to represent them as the interaction of charges at a distance and use a complex law. Others like ley lines. They draw them all the time, and it seems to them that writing different E and B is too abstract. But field lines are just a crude way of describing a field, and it is very difficult to formulate strict, quantitative laws directly in terms of field lines. In addition, the concept of field lines does not contain the deepest of the principles of electrodynamics - the principle of superposition. Even if we know what the lines of force of one set of charges look like, then another set, we still will not get any idea about the picture of the lines of force when both sets of charges act together. And from a mathematical point of view, the imposition is easy to do, you just need to add two vectors. Force lines have their advantages, they give a clear picture, but they also have their drawbacks. The method of reasoning based on the concept of direct interaction (short-range interaction) also has great advantages when it comes to electric charges at rest, but it also has great disadvantages when dealing with fast motion of charges.

It is best to use the abstract representation of the field. It is a pity, of course, that it is abstract, but nothing can be done. Attempts to represent the electric field as the movement of some kind of gear wheels or with the help of lines of force or as stresses in some materials required more effort from physicists than would be needed to simply get the right answers to the problems of electrodynamics. Interestingly, the correct equations for the behavior of light in crystals were derived by McCulloch back in 1843. But everyone told him: “Excuse me, because there is not a single real material whose mechanical properties could satisfy these equations, and since light is vibrations which should take place in something so far we cannot believe these abstract equations. If his contemporaries did not have this bias, they would have believed in the correct equations for the behavior of light in crystals much earlier than it actually happened.

As for magnetic fields, the following remark can be made. Let's suppose that you finally managed to draw a picture of the magnetic field with some lines or some gears rolling through space. Then you will try to explain what happens to two charges moving in space parallel to each other and at the same speed. Since they are moving, they behave like two currents and have an associated magnetic field (like the currents in the wires in Fig. 1.8). But an observer who rushes along with these two charges will consider them stationary and say that no there is no magnetic field. Both "gears" and "lines" disappear when you race close to an object! All you have achieved is invented new problem. Where could these gears go?! If you drew lines of force, you will have the same concern. Not only is it impossible to determine whether these lines move with the charges or not, but in general they can completely disappear in some coordinate system.

We would also like to emphasize that the phenomenon of magnetism is in fact a purely relativistic effect. In the case just considered of two charges moving parallel to each other, one would expect that it would be necessary to make relativistic corrections to their motion of the order v 2 /c 2 . These corrections must correspond to the magnetic force. But what about the force of interaction between two conductors in our experience (Fig. 1.8)? After all, there is a magnetic force all acting force. It doesn't really look like a "relativistic correction". Also, if you estimate the speeds of the electrons in the wire (you can do it yourself), you will get that their average speed along the wire is about 0.01 cm/sec. So v 2 /c 2 is about 10 -2 5 . A completely negligible "correction". But no! Although in this case the magnetic force is 10 -2 5 of the "normal" electrical force acting between moving electrons, remember that the "normal" electrical forces have disappeared as a result of an almost perfect balance due to the fact that the numbers of protons and electrons in the wires are the same. This balance is much more accurate than 1/10 2 5 , and that small relativistic term we call the magnetic force is the only remaining term. It becomes dominant.

The almost complete mutual annihilation of electrical effects allowed physicists to study relativistic effects (i.e., magnetism) and discover the correct equations (with an accuracy of v 2 /c 2), without even knowing what was going on in them. And for this reason, after the discovery of the principle of relativity, the laws of electromagnetism did not have to be changed. Unlike mechanics, they were already correct up to v 2 /c 2 .

§ 6. Electromagnetism in science and technology

In conclusion, I would like to end this chapter with the following story. Among the many phenomena studied by the ancient Greeks, there were two very strange ones. First, a rubbed piece of amber could lift small scraps of papyrus, and second, near the city of Magnesia there were amazing stones that attracted iron. It is strange to think that these were the only phenomena known to the Greeks in which electricity and magnetism manifested themselves. And why only this was known to them is explained, first of all, by the fabulous accuracy with which the charges are balanced in the bodies (which we have already mentioned). Scientists who lived in later times discovered new phenomena one after another, in which some aspects of the same effects associated with amber and with a magnetic stone were expressed. Now it is clear to us that both the phenomena of chemical interaction and, ultimately, life itself must be explained using the concepts of electromagnetism.

And as the understanding of the subject of electromagnetism developed, such technical possibilities appeared that the ancients could not even dream of: it became possible to send signals by telegraph over long distances, to talk with a person who is many kilometers away from you, without the help of any communication lines, including huge power systems - large water turbines connected by many hundreds of kilometers of wire lines to another machine, which is set in motion by one worker with a simple turn of the wheel; many thousands of branching wires and tens of thousands of machines in thousands of places set in motion various mechanisms in factories and apartments. All this rotates, moves, works due to our knowledge of the laws of electromagnetism.

Today we use even more subtle effects. Giant electrical forces can be made very precise, controlled and used in any way. Our instruments are so sensitive that we are able to tell what a person is doing just by how he affects electrons trapped in a thin metal rod hundreds of kilometers away. To do this, you just need to adapt this twig as a television antenna!

In the history of mankind (if you look at it, say, in ten thousand years), the most significant event of the 19th century will undoubtedly be Maxwell's discovery of the laws of electrodynamics. Against the background of this important scientific discovery, the American Civil War in the same decade will look like a small provincial incident.

* It is only necessary to agree on the choice of the circulation sign.