Essentially, the ability, through the appropriate choice of ISO, to detect either only electrical, or only magnetic, or both influences electromagnetic field on charges and currents was known in classical, pre-relativistic electrodynamics (i.e. before the creation of SRT).
Really, classic formula for the Lorentz force, it breaks down into two terms: the first determines the electrical part of this force, the second - the magnetic part. Since only a moving charge experiences magnetic action, then when moving to ISO, in which this charge will be stationary, the instruments will not detect magnetic * action. But no disappearance (or emergence) of matter occurs in this case: in no ISO it is possible to simultaneously eliminate both electrical and magnetic influence The fact is that there is a single electromagnetic field, but historically it has developed so that its various manifestations (depending on the observation conditions, on the choice of ISO) received independent names: electrical influence (in this case the electromagnetic field is called electric), magnetic influence (in this In this case, the electromagnetic field is called magnetic). We are actually talking about stationary or static fields. It is in this case that Maxwell's equations break down into two groups of equations, some of which describe the electrical manifestations of the electromagnetic field, others - the magnetic ones. In the non-stationary case, such a separation is no longer possible, and with any change in time of the electric (magnetic) field, vortices of the magnetic (electric) field are excited. Such an interconnected process can propagate in space in the form of electromagnetic waves. And in any ISO it will be possible to detect a single electromagnetic field as a single material environment.
All this, in principle, was known before the creation of SRT (except that the electromagnetic field was considered not one of the types of matter, but special condition electromagnetic ether). The main difference between the SRT results and the formulas before relativistic physics consists of various analytical expressions to transform the characteristics of the electromagnetic field
To illustrate the relativity of dividing a single electromagnetic field into electric and magnetic, consider the following problem: a direct current flows through a conductor, consider the field of this current based on two ISOs “Conductor” and “Electron”, connecting each of them with the corresponding object
In ISO "Explorer" crystal cell The conductor is stationary, but conduction electrons move at a certain speed. Since a direct current flows through a conductor, the number of electrons that “enter” the conductor is the same number that “comes out,” this follows from the definition of direct current. Therefore, both before and after the circuit is closed, the conductor as a whole turns out to be neutral. Mathematically, this can be written as follows: or, where are the volume densities positive charges crystal lattice and electrons, creating in a given ISO an electric current with a density and the sign (-) takes into account the sign of the electron charge, n - bulk density electrons, u– the speed of their directional movement.
In ISO “Electron” the conduction electrons are stationary, but the crystal lattice moves at a speed of (- u) . In this ISO, the volume density of both positive and negative charges according to formulas *:
where, since positive ions in ISO “Explorer” they are motionless.
Respectively,
Let's make an expression
which is greater than zero, the conductor in the Electron ISO acquires a positive charge. And if in the ISO “Conductor” a magnetic field can be detected around the conductor with the help of instruments (i.e. objectively), then in the ISO “Electron” the instruments will record both the electric field (from a charged conductor) and the magnetic field (from the current associated with movement of lattice ions in this ISO).
Let us note once again that no creation of matter occurs; in both ISOs there is a single electromagnetic field. But by choosing the ISO, i.e., the conditions for observing this material object, we discover from him different manifestations, different properties.
Since when moving from one ISO to another, not only the magnitude, but also the current density changes, and these characteristics of charges and currents are directly related to the characteristics of the electromagnetic field, its vectors and, which indicates the relative nature of these quantities.
Where v – speed relative motion two ISOs.
From the above formulas it follows that if in one ISO there is only an electric field, then in the other ISO not only an electric, but also a magnetic field is detected.
We are once again convinced that the division of a single electromagnetic field into electric and magnetic is relative.
* Until recently, it was believed that only the magnetic field is a relativistic object. This, of course, stemmed from ignorance of the history of physics and A. Einstein’s principle of relativity. A relativistic object is a single electromagnetic field, and how will it manifest itself in the chosen reference frame (electric or magnetic effect) does not allow us to consider only the magnetic field as relativistic, and the electric field as non-relativistic.
* The reader will find the derivation of the formulas used in the author’s book “Special Theory of Relativity”, published by POIPKRO, 1995, p. 85.
IN previous chapter we found out that electrical and magnetic field must always be considered together as one complete electromagnetic field. The division of the electromagnetic field into electric and magnetic is relative in nature: such a division in decisive degree depends on the frame of reference in which the phenomena are considered. In this case, a field that is constant in one frame of reference, in the general case, turns out to be variable in another frame. Let's look at some examples.
The charge moves in the inertial K-frame of reference with constant speed v. In this reference frame, we will observe both the electric and magnetic fields of a given charge, and both fields are variable in time. If we go to an inertial K¢-system moving with the charge, then the charge is at rest in it and we will observe only the electric field.
Two identical charges move in the K-frame of reference towards each other at the same speed v. In this frame of reference we will observe both electric and magnetic fields, both variable. Find a K¢-system where only one of the fields would be observed, in in this case it is forbidden.
In the K-system there is a constant non-uniform magnetic field (for example, the field of a stationary permanent magnet). Then in the K¢-system moving relative to the K-system, we will observe alternating magnetic and electric fields.
Thus, it becomes clear that the relationship between the electric field and the magnetic field is different in various systems countdown. When moving from one reference system to another, the fields and are transformed in a certain way. The laws of this transformation are established in special theory relativity, and quite in a complex way. For this reason, we will not reproduce the relevant findings here.
Since the vectors and characterizing the electromagnetic field depend on the reference system, a natural question arises about invariants, i.e. independent of the reference system quantitative characteristics electromagnetic field (the invariant is denoted by inv; see, for example, (43.1)).
It can be shown that there are two such invariants, which are combinations of vectors and , this is
Inv; E 2 - c 2 B 2 = inv, (43.1)
Where With– speed of light in vacuum.
The invariance of these quantities (with respect to Lorentz transformations) is a consequence of the field transformation formulas when passing from one inertial reference system to another.
The use of these invariants allows in some cases to quickly and easily find a solution and make appropriate conclusions and predictions. Here are the most important of them:
From invariance dot product it immediately follows that in the case when in any reference frame ^, i.e. = 0, then in all other inertial frames of reference ^ ;
From the invariance of E 2 - c 2 B 2 it follows that in the case when E = c B (i.e. when E 2 - c 2 B 2 = 0), then in any other inertial reference frame E¢ = c B¢;
If in any reference system the angle between vectors and is acute (or obtuse) - this means that it is greater (or less) than zero - then the angle between vectors and will also be acute (or obtuse) in any other reference system;
If in any reference frame E > c B (or E< c B) – this means that E 2 - c 2 B 2 > 0 (or E 2 - c 2 B 2< 0), то и в любой другой системе отсчета будет также E¢ > c B¢ (or E¢< c B¢);
If both invariants are equal to zero, then in all inertial frames of reference ^ and E = c B, this is exactly what is observed in an electromagnetic wave;
If equal to zero only invariant, then one can find a reference system in which either E¢ = 0 or B¢ = 0; which one is determined by the sign of the other invariant. The converse statement is also true: if in any reference system E = 0 or B = 0, then in any other reference system ^.
And one last thing. It must be remembered that the fields and , generally speaking, depend on both coordinates and time. Therefore, each of the invariants (43.1) refers to the same space-time point of the field, the coordinates and time of which in different systems references are connected by Lorentz transformations.
Only Einstein’s principle of relativity is applicable to the electromagnetic field, since the fact of the propagation of electromagnetic waves in a vacuum in all reference systems with the same speed With is not compatible with Galileo's principle of relativity.
According to Einstein's principle of relativity, mechanical, optical and electromagnetic phenomena in all inertial reference systems proceed in the same way, that is, they are described by the same equations. Maxwell's equations are invariant under Lorentz transformations: their form does not change when moving from one inertial reference system to another, although the quantities in them are transformed according to certain rules.
From the principle of relativity it follows that separate consideration electric and magnetic fields has relative meaning. So, if the electric field is created by the system stationary charges, then these charges, being stationary relative to one inertial frame of reference, move relative to another and, therefore, will generate not only an electric, but also a magnetic field. Similarly, a conductor with a constant current, stationary relative to one inertial reference frame, excites a constant magnetic field at each point in space, moves relative to other inertial frames, and the alternating magnetic field it creates excites a vortex electric field.
Thus, Maxwell's theory, its experimental confirmation, as well as Einstein's principle of relativity lead to unified theory electrical, magnetic and optical phenomena, based on the concept of an electromagnetic field.
Chapter 13
MAGNETOSTATICS
§1.Magnetic field
§2. Electric current; charge conservation
§Z. Magnetic force acting on current
§4. Magnetic field of direct currents; Ampere's law
§5. Magnetic field of straight wire and solenoid; atomic currents
§6. Relativity of magnetic and electric fields
§7.Transformation of currents and charges
§8.Superposition; rule right hand
Repeat: Ch. 15 (issue 2) “Special theory of relativity”
§ 1. Magnetic field
The force acting on electric charge, depends not only on where it is, but also on how fast it is moving. Each point in space is characterized by two vector quantities, which determine the force acting on any charge. Firstly, there is electric force, giving that part of the force that does not depend on the movement of the charge. We describe it using the electric field E. Secondly, there is an additional force component called magnetic force, which depends on the charging speed. This magnetic force has amazing property: at any given point in space, as direction, so and magnitude forces depend on the direction of particle motion; at every moment the force is always perpendicular to the velocity vector; in addition, at any location the force is always perpendicular a certain direction in space(Fig. 13.1), and finally, the magnitude of the force is proportional component speed perpendicular to this selected direction. All these properties can be described by introducing the magnetic field vector B, which determines the selected direction in space and at the same time serves as a constant of proportionality between force and speed, and writing the magnetic force in the form qvXB. The total electromagnetic force acting on the charge can then be written as follows:
F=q(E+vXB), (13.1)
It is called by Lorentz force.
Fig. 13.1. The velocity-dependent component of the force on a moving charge is directed perpendicular to V and vector B. It is also proportional to the component V perpendicular to B, i.e. vsinq.
Magnetic force can be easily demonstrated by bringing a magnet close to the cathode tube. The deflection of the electron beam indicates that the magnet excites forces acting on the electrons perpendicular to the direction of their movement (we already talked about this in Issue 1, Chapter 12).
The unit of magnetic field B is obviously 1 newton second divided by coulomb meter. Ta same the unit can be written as volt-second per square meter. It is also called Weber per square meter.
§ 2. Electric current; charge conservation
Let us now think about why magnetic forces act on wires through which electric current flows. To do this, we define what is meant by current density. Electric current consists of moving electrons or other charges that form a net current, or flow. We can represent the flow of charges as a vector that determines the number of charges that passes per unit time through a unit area perpendicular to the flow (exactly as we did when determining the flow of heat). Let's call this quantity current density and denote it by vector j. It is directed along the movement of charges. If we take a small area Da in this place material, then the number of charges flowing through the area per unit time is equal to
j n Da, (13.2)
Where n - unit vector normal to Da.
Current density is related to the average speed of charge flow. Let us assume that there is a distribution of charges drifting on average with speed v. When this distribution passes through a surface element Da, then the charge Dq passing through Yes during Dt, equal to the charge contained in a parallelepiped with base D A and height vDt(Fig. 13.2).
Fig. 13.2. If the charge distribution with density r moves at speed v, then the amount of charge passing per unit time through the area Da, there is rv·nDа.
The volume of a parallelepiped is the product of the projection Yes, perpendicular to v, on vDt, and multiplying it by the charge density r, we get Dq. Thus,
Dq = r v·nDaDt.
The charge passing per unit time is then equal to рv·nDа, where do we get it from
j = pv. (13.3)
If the charge distribution consists of individual charges, say electrons with charge q , moving with an average speed v, then the current density is equal to
j = Nqv,(13.4)
Where N- number of charges per unit volume.
The total amount of charge passing through a surface per unit time S, called electric current I. He equal to the integral from the normal flow component over all surface elements (Fig. 13.3):
Fig. 13.3. The current I through the surface S is equal to
Fig. 13.4. The integral of j·n no of a closed surface is equal to the rate of change of the total charge Q inside.
Current I from a closed surface S represents the speed at which charges leave the volume V, surrounded by a surface 5. One of the basic laws of physics says that electric charge is indestructible; it is never lost or created. Electric charges can move from place to place, but they never appear out of nothing. We say that the charge is maintained. If a resultant current arises from a closed surface, then the amount of charge inside should decrease accordingly (Fig. 13.4). Therefore, we can write the law of conservation of charge in this form:
The charge inside can be written as a volume integral of the charge density
Applying (13.6) to a small volume DV, we can take into account that the integral on the left is C·jDV. The charge inside is equal to rDV, so the conservation of charge can also be written like this:
(again Gauss's theorem from mathematics!).
§ 3. Magnetic force acting on current
Now we are sufficiently prepared to determine the force acting on a wire in a magnetic field through which current flows. The current consists of charged particles moving along a wire with speed v. Each charge feels a transverse force F = qvXB (Fig. 13.5, a).
Fig. 13.5. The magnetic force on a wire carrying current is equal to the sum of the forces on individual moving charges
If in a unit volume of such charges there is N, then their number in a small volume inside the wire DV is equal to N D V. The total magnetic force DV acting on the volume DV is . sum of forces on individual charges
Ho Nqv after all, it’s exactly equal to j, so
(Fig. 13.5, b). The force acting per unit volume is equal to JXB.
If along a wire with a cross section A If current flows evenly across the cross-section, then we can take a cylinder with base A and length DL as a volume element. Then
DF = jXBDL. (13.10)
Now we can call jA the vector of current I in the wire. (Its magnitude is the electric current in the wire, and its direction coincides with the direction of the wire.) Then
DF=IXBDL. (13.11)
The force acting per unit length of the wire is IXB.
This equation contains important result- magnetic
The force acting on the wire and arising from the movement of charges in it depends only on the total current, and not on the amount of charge carried by each particle (and does not even depend on its sign!). The magnetic force acting on a wire near a magnet is easily detected by the deflection of the wire when the current is turned on, as we described in Chapter. 1 (see Fig. 1.6, page 20).
§ 4. Magnetic field of direct current; Ampere's law
We saw that a force acts on a wire in a magnetic field created by, say, a magnet. From the law that action equals reaction, we can expect that when a current flows through a wire, a force is generated acting on the source of the magnetic field, i.e., the magnet. Such forces do exist; This can be verified by the deflection of the compass needle near the current-carrying wire. Further, we know that magnets experience forces from other magnets, and from this it follows that when a current flows through a wire, it creates its own magnetic field. This means that moving charges create a magnetic field. Let's try to understand the laws that such magnetic fields obey. The question is posed like this: given a current, what magnetic field will it create? The answer to this question was obtained experimentally by three experiments and confirmed by Ampere’s brilliant theoretical proof. We won't stop there interesting story, but let's just say that big number experiments clearly demonstrated the validity of Maxwell's equations. We will take them as a starting point. Omitting terms with time derivatives in the equations, we obtain the equations magnetostatics
These equations are valid only under the condition that all electric charge densities and all currents are constant, so that electric and magnetic fields do not change with time - all fields are "static".
It can be noted here that believing in the existence of a static magnetic field is quite dangerous, because in general, currents are needed to produce a magnetic field, and currents arise only from moving charges. Consequently, “magnetostatics” is only an approximation.
It is associated with a special case of dynamics when it moves big number charges, which can be approximately described as constant flow of charges. Only in this case can we talk about the current density j, which does not change with time. More accurately, this area should be called the study of direct currents. Assuming that all fields are constant, we discard terms with d E/dt And dB/dt V complete equations Maxwell [equations (2.41)] and we obtain the two equations (13.12) and (13.13) written above. Note also that since the rotor divergence of any vector is always zero, equation (13.13) requires that C·j=0. By virtue of equation (13.8), this is true only if dr/dt=0. But this can happen if E does not change over time, therefore, our assumptions are internally consistent.
The condition that C·J= 0 means that we can only have charges flowing along closed paths. They can, for example, flow through wires that form closed loops called chains. The circuits may, of course, contain generators or batteries to maintain the current of the charges. But they should not have capacitors that charge or discharge. (We will, of course, extend the theory to include alternating fields, but first we want to take the simpler case of direct currents.)
Let us now turn to equations (13.12) and (13.13) and see what they mean. The first says that the divergence of B is zero. Comparing it with a similar electrostatics equation, according to which С·Э=r/e 0, we can conclude that there is no magnetic analogue of the electric charge. Doesn't happen magnetic charges, from which lines B could emanate. If we talk about “lines” vector field B, then they do not begin anywhere and do not end anywhere. But then where do they come from? Magnetic fields "appear" in the presence currents; rotor, taken from them is proportional to the current density. When there are currents, there are magnetic field lines that form loops around the currents. Since B lines have neither an end nor a beginning, they often return to their starting point, forming closed loops. But there may be more complex cases, when the lines are not simple loops. However, no matter how they go, they never come from points. No one has ever found any magnetic charges, so C·B=0. The same statement is true not only for magnetostatics, but also Always - even for dynamic fields.
The relationship between field B and currents is given by equation (13.13). The situation here is completely different, radically different from electrostatics, where we had CXE = 0. This equation meant that the line integral of E along any closed path is equal to zero:
Fig. 13.6. The contour integral of the tangential component B is equal to the surface integral of the normal component of the vector
(CX B).
We obtained this result using Stokes' theorem, according to which the integral over any closed path from any vector field is equal to surface integral from the normal component of the rotor of this vector (the integral is taken over any surface spanned by a given contour). Applying the same theorem to the magnetic field vector and using the notation shown in Fig. 13.6, we get
Having found rot B from equation (13.13), we have
Integral from j to S, according to (13.5), there is a total current I through the surface S. Since for direct currents the current through S does not depend on the form S, if it is limited by the curve Г, then they usually talk about “current through a closed loop Г”. We have, therefore, common law: circulation B along any closed curve
equal to the current I through the loop divided by e 0 s 2:
This law, called Ampere's law plays the same role in magnetostatics as Gauss's law does in electrostatics. Ampere's law alone does not determine B through currents; we must, generally speaking, also use C·B=0. But, as we will see in the next paragraph, it can be used to find a field in those special cases, which have some simple symmetry.
§ 5. Magnetic field of a straight wire and a solenoid; atomic currents
You can show how to use Ampere's law by determining the magnetic field near a wire. Let's ask the question: what is the field outside a long straight wire of cylindrical cross-section? We will make one assumption, perhaps not so obvious, but nevertheless correct: the field lines B go around the wire in a circle. If we make this assumption, then Ampere's law [equation (13.16)] tells us what the magnitude of the field is. Due to the symmetry of the problem, field B has the same size at all points of the circle concentric with the wire (Fig. 13.7). Then we can easily take the line integral of B·ds. It is simply equal to the value of B multiplied by the circumference. If the radius of the circle is r , That
The total current through the loop is simply the current I in the wire, so
The magnetic field strength decreases in inverse proportion to r, the distance from the wire axis. If desired, equation (13.17) can be written in vector form. Recalling that B is directed perpendicular to both I and r, we have
Fig. 13.7. Magnetic field outside a long wire carrying current I.
Fig. 13.8. Magnetic field of a long solenoid.
We highlighted the 1/4pe 0 multiplier with 2 because it appears frequently. It is worth remembering that it is exactly 10 -7 (in SI units), because an equation of the form (13.17) is used for definitions units of current, ampere. At a distance of 1 m a current of 1a creates a magnetic field equal to 2·10 -7 weber/m 2 .
Since the current creates a magnetic field, it will act with some force on the adjacent wire through which the current also passes. In ch. 1 we described a simple experiment showing the forces between two wires through which current flows. If the wires are parallel, then each of them is perpendicular to the B field of the other wire; then the wires will repel or attract each other. When currents flow in one direction, the wires attract; when currents flow in opposite directions, they repel.
Let's take another example, which can also be analyzed using Ampere's law, if we also add some information about the nature of the field. Let there be a long wire coiled into a tight spiral, the cross-section of which is shown in Fig. 13.8. This spiral is called solenoid. We observe experimentally that when the length of the solenoid is very large compared to the diameter, the field outside it is very small compared to the field inside. Using only this fact and Ampere's law, one can find the magnitude of the field inside.
Since the field remains inside (and has zero divergence), its lines should run parallel to the axis, as shown in Fig. 13.8. If this is the case, then we can use Ampere's law for the rectangular "curve" G in the figure. This curve travels a distance L inside the solenoid, where the field is, say, equal to IN 0 , then goes at right angles to the field and returns back along the outer region, where the field can be neglected.
Fig. 13.9. Magnetic field outside the solenoid.
The line integral of B along this curve is exactly B 0 L, and this must equal 1/e 0 c 2 times the total current inside G, i.e. NI(where N is the number of solenoid turns along the length L). We have
Or, by entering n - number of turns per unit length solenoid (so n=N/L), we get
What happens to the B lines when they reach the end of the solenoid? Apparently, they somehow diverge and return to the solenoid from the other end (Fig. 13.9). Exactly the same field is observed outside a magnetic rod. well and what is it magnet? Our equations say that field B arises from the presence of currents. And we know that ordinary iron bars (not batteries or generators) also create magnetic fields. You might expect that there would be other terms on the right-hand side of (13.12) or (13.13) representing the "density of magnetized iron" or some similar quantity. But there is no such member. Our theory says that the magnetic effects of iron arise from some internal currents already taken into account by the j term.
Matter is very complex when viewed from a deep point of view; We already saw this when we tried to understand dielectrics. In order not to interrupt our presentation, we will postpone a detailed discussion of the internal mechanism magnetic materials type of iron. For now we have to accept that any magnetism arises due to currents and that permanent magnet there are constant internal currents. In the case of iron, these currents are created by electrons orbiting around own axes. Each electron has a spin that corresponds to a tiny circulating current. One electron, of course, does not produce a large magnetic field, but an ordinary piece of matter contains billions and billions of electrons. Usually they rotate in any way so that the overall effect disappears. What is surprising is that in a few substances like iron, most of electrons rotate around axes directed in one direction - in iron, two electrons from each atom take part in this joint movement. A magnet contains a large number of electrons spinning in the same direction, and, as we will see, their combined effect is equivalent to the current circulating across the surface of the magnet. (This is very similar to what we find in dielectrics - a uniformly polarized dielectric is equivalent to a distribution of charges on its surface.) It is therefore no coincidence that a bar magnet is equivalent to a solenoid.
§ 6. Relativity of magnetic And electric fields
When we said that the magnetic force on a charge is proportional to its speed, you probably thought, “What speed? In relation to what frame of reference? From the definition of B given at the beginning of this chapter, it is in fact clear that this vector will be different depending on the choice of frame of reference in which we define the speed of the charges. But we did not say anything about which system is suitable for determining the magnetic field.
It turns out that it's good any inertial system. We will also see that magnetism and electricity are not independent things, they must always be taken together as one total electromagnetic field. Although in the static case Maxwell's equations are divided into two separate pairs, one pair for electricity and one for magnetism, with no apparent connection between the two fields, yet in nature itself there is a very deep relationship between them, arising from the principle of relativity. Historically, the principle of relativity was discovered after Maxwell's equations. In fact, it was the study of electricity and magnetism that led Einstein to the discovery of the principle of relativity. But let's see what our knowledge of the principle of relativity tells us about magnetic forces ah, if we assume that the principle of relativity applies (and in fact it does) to electromagnetism.
Let's think about what will happen to a negative charge moving at speed v 0 parallel to a wire through which current flows (Fig. 13.10).
Fig. 13.10. Interaction of a wire with a current and a particle with a charge q,
considered in two coordinate systems.
a - a wire is at rest in the system S; b - a charge is at rest in the system S".
Let's try to understand what is happening using two reference systems: one associated with the wire, as in Fig. 13.10, A, and the other with a particle, as in Fig. 13.10, b. We will call the first frame of reference S, and the second S".
In system S there is clearly a magnetic force acting on the particle. The force is directed towards the wire, therefore, if nothing interferes with the charge, its trajectory will bend towards the wire. But in the system S" There cannot be a magnetic force on the particle because the particle's speed is zero. So why will she continue to stand still? Will we see different things in different systems? The principle of relativity states that in a system S" we would also see how the particle approaches the wire. We must try to understand why this might happen.
Let's return to our atomic description wire through which current flows. In a common conductor such as copper, electric currents are generated by the movement of a portion of the negative electrons (called conduction electrons), while the positive nuclear charges and the remaining electrons remain anchored within the material. Let the density of conduction electrons be r, and their speed in the system S there is v. Density of stationary charges in the system S there is r +, which must be equal to r - with the opposite sign, because we are taking an uncharged wire. Therefore, there is no electric field outside the wire, and the force on a moving particle is simply
Using the result we found in equation (13.18) for the magnetic field at a distance r from the axis of the wire, we conclude that the force acting on the particle is directed towards the wire and is equal in magnitude
Using equations (13.4) and (13.5), the current I can be written as r + vA, where A is the area cross section wire. Then
We could continue to consider general case arbitrary speeds v And v 0 , but it wouldn't be any worse to take special case when the speed v 0 particles coincides with the speed v conduction electrons. Therefore we will write v=v 0 , and equation (13.20) takes the form
Now let's look at what's happening in the system S", where the particle is at rest and the wire runs past it (to the left in Fig. 13.10, b) with speed v. Positive charges moving along with the wire will create some magnetic field near the particle IN". But the particle is now rests So magnetic no force acts on her! If any force arises, it must appear due to the electric field. It turns out that a moving wire creates an electric field. But she can only do it if she seems charged; it should be so that a neutral wire carrying current appears charged if it is set in motion.
We need to figure this out. Let's try to calculate the charge density in the wire in the system S", using what we know about it in the system S. At first glance, one might think that the densities are the same, but from Ch. 15 (issue 2) we know that when moving from one system to another, the lengths change, therefore, the volumes will also change. Because the density charges depend on the volume occupied by the charges, the densities will also change.
Before determining the charge densities in the system S", need to know what's going on with the electrical charge groups of electrons when charges move. We know that the apparent mass of a particle acquires a factor of 1/C(1-v 2 /c 2). Does something similar happen to its charge? No! Charges never don't change regardless of whether they are moving or not. Otherwise, we could not observe experimentally the conservation of the full charge.
Let us take a piece of matter, such as a conductor, and let it be initially uncharged. Now let's heat it up. Since electrons have a different mass than protons, the speeds of electrons and protons will change differently. If the charge of a particle depended on the speed of the particle that carries it, then in the heated piece the charges of electrons and protons would not be compensated. A piece of material would become charged when heated.
Fig. 13.11 . If the distribution of charged particles has a charge density p 0, then from the point of view of a system moving with relative speed v, the charge density will be equal to r=r 0 /C (1 - v 2 /s 2).
We saw earlier that a very small change in charge on each of the electrons in the piece would result in huge electric fields. Nothing like this has ever been observed.
Moreover, it can be noted that average speed electrons in a substance depends on its chemical composition. If the charge of an electron were to change with speed, the net charge in a piece of matter would change during the course of the chemical reaction. As before, direct calculation shows that even a very small dependence of charge on speed would lead in the simplest chemical reactions to huge fields. Nothing similar has been observed, and we come to the conclusion that the electric charge individual particle does not depend on the state of motion or rest.
So, the charge of the particle q is an invariant scalar quantity independent of the reference frame. This means that in any system the charge density of a certain distribution of electrons is simply proportional to the number of electrons per unit volume. We only need to take into account the fact that the volume Maybe change due to relativistic contraction distances
Let's now apply these ideas to our moving wire. If we take a wire of length L 0, in which the density stationary charges is r 0 , then it will contain the total charge Q- r 0 L 0 A 0 . If the same charges move in another system with a speed v, then they will all be in a piece of material
less length
but the same section A 0, since the dimensions in the direction perpendicular to the movement do not change (Fig. 13.11).
If r denotes the density of charges in the system where they move, then the total charge Q will be r L.A. 0 . But it must also be equal to r 0 L 0 A, because the charge in any system is the same, therefore, rL=r 0 L 0, or using (13.22)
Density moving charges totality charges changes in the same way as the relativistic mass of the particle. Let us now apply this result to the density of positive charges r + in our wire. These charges are at rest in the system S. However, in the S system, where the wire moves at a speed v, the density of positive charges becomes equal
Negative charges in the system S" are at rest, therefore their density in this system is the “rest density” r 0 . In equation (13.23) r 0 =r - because their charge density is equal to r - if wire is at rest, i.e. in the system S, where the speed of negative charges is equal to v. Then for conduction electrons we get
Now we can understand why the system S" electric fields arise: because in this system in the wire there is a resulting charge density r", given by the formula
Using (13.24) and (13.26) we have
Since the wire at rest is neutral, r - = -r + , we get
Our moving wire is positively charged and should create a field E" at the point where the external particle at rest is located. We have already solved the electrostatic problem of a uniformly charged cylinder. The electric field at a distance r from the cylinder axis is
The force acting on a negatively charged particle is directed towards the wire. We have a force directed equally in both systems; electrical force in the system S" directed in the same way as the magnetic force in the system S. The magnitude of the force in the system S" is equal to
Comparing this result for F" with our result for F in equation (13.21), we see that the magnitudes of the forces from the point of view of two observers are almost the same. More precisely,
therefore, for the low speeds we are considering, both forces are the same. We can say that, at least for low speeds, magnetism and electricity are simply “two different sides the same thing."
But it turns out that everything is even better than we said. If we take into account the fact that strength are also transformed during the transition from one system to another, it turns out that both methods of observing what is happening actually give the same physical results at any speed.
To see this, you can, for example, ask the question: what transverse momentum will a particle acquire when a force has been acting on it for some time? We know from issue. 2, ch. 16 that the transverse momentum of the particle must be the same as in the system S, so in the system S". Let us denote the transverse coordinate at and compare Dр y And Dр y ’ . Using the relativistically correct equation of motion F-dp/dt, we expect that in time Dt our particle will acquire transverse momentum Dр y in system S, given by the expression
In the S" system, the transverse momentum will be equal to
Fig. 13.12. In system S, the charge density is zero, and the current density is equal to j. There is only a magnetic field. In the system S" the charge density is equal to R", and current density j". The magnetic field here is equal to IN" and there is an electric field E".
We must compare Dр y and Dр y ", of course, for the corresponding time intervals Dt and Dt". In ch. 15 (issue 2) we saw that the time intervals related to a moving particle seem longer intervals in the particle's rest frame. Since our particle was initially at rest in the system S", That
we expect that for small D t
and everything turns out great. According to (13.31) and (13.32),
and if we combine (13.30) and (13.33), then this ratio is equal to one.
So it turns out that we get the same result, regardless of whether we analyze the motion of a particle flying next to the wire in the rest frame of the wire or in the rest frame of the particle. In the first case the force was purely “magnetic”, in the second it was purely “electric”. Both observation methods are shown in Fig. 13.12 (although in the second system there is also a magnetic field B, it does not affect the stationary particle).
If we had chosen another coordinate system, we would have found some different mixture of the E and B fields. Electric and magnetic forces are parts onephysical phenomenon- electromagnetic interaction of particles. The division of this interaction into electric and magnetic parts in to a large extent depends on the frame of reference in which we describe the interaction. But complete electromagnetic description invariant; electricity and magnetism taken together are consistent with the principle of relativity discovered by Einstein.
Once electric and magnetic fields appear in different ratios when changing the frame of reference, we must be careful in handling the E and B fields. If, for example, we are talking about the “lines” E or B, then we should not exaggerate the reality of their existence. The lines may disappear if we want to see them in a different coordinate system. For example, in the system S" there are electric field lines, but we we don't see them “moving past us with speed v in the system S 1”. In system S There are no electric field lines at all! So it makes no sense to say something like: “When I move a magnet, it carries its field with it, so the B field lines move too.” There is no way to make the concept of "speed of moving field lines" meaningful at all.
Fields are a way of describing what happens at some point in space. In particular, E and B tell us about the forces that will act on the moving particle. The question “what is the force acting on the charge from the side moving magnetic field? does not have any precise content. The force is given by the quantities E and B at the point of charge, and formula (13.1) will not change if source fields E or B are moving (the values of E and B will change as a result of the movement). Is our mathematical description only applies to fields as functions x, y, z And t, taken in some inertial frame of reference.
Later we will talk about "wave electric and magnetic fields propagating in space,” for example, about a light wave. But it's the same as talking about wave, running on a rope. We do not mean by this that any part ropes moves in the direction of the wave, and we mean that bias rope appears first in one place and then in another. Likewise for electromagnetic wave- herself wave spreads, and the magnitude of the fields changes.
So in the future, when we - or anyone else - talk about a "moving" field, you should understand that we're talking about just a short and convenient way descriptions of a changing zero under certain conditions.
§ 7. Conversion of currents and charges
You were probably concerned about the simplification we made when we took the same velocity v for the particle and the conduction electrons in the wire. We could go back and do the analysis again at two different rates, but it is easier to just notice that the charge and current densities are components of the four-vector (see issue 2, chapter 17).
We have already seen that if r 0 is the density of charges in their rest frame, then in a system where they have a speed v, the density is equal to
In this system their current density is
where m 0 - its rest mass. We also know that U and p form a relativistic four-vector. Since r and j depend on the speed v exactly as U and p, then we can conclude that r and j Also components of a relativistic four-vector. This property is the key to general analysis field of a wire moving at any speed, and we could use it if we wanted to solve the problem again with the particle speed v 0, not equal speed conduction electrons.
If we need to convert r and j into a coordinate system moving at speed And in the direction X, then we know that they are converted exactly as t And (x, y, z); therefore we have (see issue 2, chapter 15)
Using these equations, you can relate charges and currents in one system to charges and currents in another. Taking charges and currents in some system, you can solve the electromagnetic problem in this system using Maxwell's equations. The result we will get for particle movement, will be the same, regardless of the chosen reference system. Later we will return to relativistic transformations of electromagnetic fields.
§ 8. Superposition; right hand rule
We will end this chapter with two more remarks on magnetostatics. First: our basic equations for the magnetic field
linear up to B and j. This means that the principle of superposition (superposition) also applies to the magnetic field. A field created by two different direct currents, is the sum of the intrinsic fields from each current acting separately. Our second point relates to the right-hand rules that we have already encountered (the right-hand rule for the magnetic field created by a current). We also pointed out that magnetization iron magnet explained by the rotation of electrons in the material. The direction of the magnetic field of a spinning electron is related to its axis of rotation by the same right-hand rule. Since B is determined by the rule of a particular hand (using either vector product, or rotor), it is called axial vector. (Vectors whose direction in space is independent of references to the left or right hand are called polar vectors. For example, displacement, velocity, force and E are polar vectors.)
Physically observable quantities in electromagnetism, however, not connected with right or left hand. From Ch. 52 (issue 4) we know that electromagnetic interactions symmetrical with respect to reflection. When calculating magnetic forces between two sets of currents, the result is always invariant with respect to changes of hands. Our equations, regardless of the right-hand condition, lead to the final result, What parallel currents attract, and opposite ones repel. (Try calculating force using the left-hand rule.) Attraction or repulsion is a polar vector. This is because in describing any complete interaction we use the right-hand rule twice - once to find B of the currents, and then to find the force exerted by the field B on the second current. Using the right hand rule twice is the same as using the left hand rule twice. If we agreed to move to the left-hand system, all our B fields would change sign, but all the forces or (perhaps more clearly) the observed accelerations of objects would not change.
When we said that the magnetic force on a charge is proportional to its speed, you probably thought, “What speed? In relation to what frame of reference? From the definition given at the beginning of this chapter, it is in fact clear that this vector will be different depending on the choice of the frame of reference in which we define the speed of the charges. But we did not say anything about which system is suitable for determining the magnetic field.
It turns out that any inertial system is suitable. We will also see that magnetism and electricity are not independent things, they must always be taken together as one complete electromagnetic field. Although in the static case Maxwell's equations are divided into two separate pairs: one pair for electricity and one for magnetism, without any visible connection between both fields, yet in nature itself there is a very deep relationship between Them, arising from the principle of relativity. Historically, the principle of relativity was discovered after Maxwell's equations. In fact, it was the study of electricity and magnetism that led Einstein to the discovery of the principle of relativity. But let's see what our knowledge of the principle of relativity tells us about magnetic forces, assuming that the principle of relativity applies (and in fact it does) to electromagnetism.
Let's think about what will happen to a negative charge moving at a speed parallel to a wire through which current flows (Fig. 13.10). Let's try to understand what is happening using two reference systems: one associated with the wire, as in Fig. 13.10, a, and the other with a particle, as in Fig. 13.10, b. We will call the first reference system , and the second .
Figure 13.10. The interaction of a wire with a current and a particle with a charge, considered in two coordinate systems.
a - a wire is at rest in the system; b - a charge is at rest in the system.
In the system, the particle is clearly affected by a magnetic force. The force is directed towards the wire, therefore, if nothing interferes with the charge, its trajectory will bend towards the wire. But in the system there cannot be a magnetic force on the particle, because the particle speed is zero. So why will she continue to stand still? Will we see different things in different systems? The principle of relativity states that in the system we would also see how the particle approaches the wire. We must try to understand why this might happen.
Let us return to our atomic description of a wire through which current flows. In an ordinary conductor such as copper, electric currents arise due to the movement of part of the negative electrons (called conduction electrons), while the positive nuclear charges and the remaining electrons remain fixed within the material. Let the density of conduction electrons be , and their speed in the system be . The density of stationary charges in the system is , which should be equal with the opposite sign, because we take an uncharged wire. Therefore, there is no electric field outside the wire, and the force on a moving particle is simply
Using the result we found in equation (13.18) for the magnetic field at a distance from the axis of the wire, we conclude that the force acting on the particle is directed towards the wire and is equal in magnitude
.
Using equations (13.4) and (13.5), the current can be written as , where is the cross-sectional area of the wire. Then
(13.20)
We could continue to consider the general case of arbitrary velocities and , but it would be no worse to take the special case when the particle velocity coincides with the velocity of conduction electrons. Therefore, we will write , and equation (13.20) will take the form
(13.21)
Now let's turn to what happens in the system, where the particle is at rest and the wire runs past it (to the left in Fig. 13.10, b) with a speed. Positive charges moving along with the wire will create some magnetic field near the particle. But the particle is now at rest, so the magnetic force has no effect on it! If any force arises, it must appear due to the electric field. It turns out that a moving wire creates an electric field. But she can only do this if she seems charged; it should be so that a neutral wire carrying current appears charged if it is set in motion.
We need to figure this out. Let's try to calculate the charge density in the wire in the system, using what we know about it in the system. At first glance one might think that the densities are the same, but from Ch. 15 (issue 2) we know that when moving from one system to another, the lengths change, therefore, the volumes will also change. Since charge densities depend on the volume occupied by the charges, the densities will also change.
Before determining the charge densities in a system, you need to know what happens to the electric charge of a group of electrons when the charges move. We know that the apparent mass of a particle acquires a multiplier. Does something similar happen to its charge? No! Charges never change whether they are moving or not. Otherwise, we could not observe experimentally the conservation of the full charge.
Let us take a piece of matter, such as a conductor, and let it be initially uncharged. Now let's heat it up. Since electrons have a different mass than protons, the speeds of electrons and protons will change differently. If the charge of a particle depended on the speed of the particle that carries it, then in the heated piece the charges of electrons and protons would not be compensated. A piece of material would become charged when heated.
Figure 13.11. If the distribution of charged particles has a charge density , then from the point of view of a system moving with a relative speed , the charge density will be equal to 
.
We saw earlier that a very small change in charge on each of the electrons in the piece would result in huge electric fields. Nothing like this has ever been observed.
In addition, it can be noted that the average speed of electrons in a substance depends on its chemical composition. If the charge of an electron were to change with speed, the net charge in a piece of matter would change during the course of the chemical reaction. As before, direct calculations show that even a very small dependence of charge on speed would lead to huge fields in the simplest chemical reactions. Nothing similar has been observed, and we come to the conclusion that the electric charge of an individual particle does not depend on the state of motion or rest.
So, the particle charge is invariant scalar quantity, independent of the reference system. This means that in any system the charge density of a certain distribution of electrons is simply proportional to the number of electrons per unit volume. We only need to take into account the fact that the volume can change due to the relativistic reduction of distances.
Let's now apply these ideas to our moving wire. If you take a wire of length in which there is a density of stationary charges, then it will contain a full charge. If the same charges move in another system with a speed, then they will all be in a piece of material of shorter length
but the same section, since the dimensions in the direction perpendicular to the movement do not change (Fig. 13.11).
If we denote the density of charges in the system where they move, then the total charge will be , But this must also be equal to , because the charge in any system is the same, therefore, , or using (13.22)
The charge density of a moving set of charges changes in the same way as the relativistic mass of the particle. Let us now apply this result to the density of positive charges in our wire. These charges are at rest in the system. However, in a system where the wire moves at speed, the density of positive charges becomes equal to
Negative charges in the system are at rest, therefore their density in this system is the “rest density”. In equation (13.23), because their charge density is equal to , if the wire is at rest, that is, in a system where the speed of negative charges is equal to . Then for conduction electrons we get
. (13.26)
Now we can understand why electric fields arise in the system: because in this system in the wire there is a resultant charge density given by the formula
Using (13.24) and (13.26) we have
.
Since the wire at rest is neutral, we obtain
, (13.27)
Our moving wire is positively charged and should create a field at the point where the external particle at rest is located. We have already solved the electrostatic problem of a uniformly charged cylinder. The electric field at a distance from the cylinder axis is
. (13.28)
The force acting on a negatively charged particle is directed towards the wire. We have a force directed equally in both systems; The electric force in the system is directed in the same way as the magnetic force in the system. The magnitude of the force in the system is equal to
. (13.29)
Comparing this result for with our result for in equation (13.21), we see that the magnitudes of the forces from the point of view of the two observers are almost the same. More precisely,
therefore, for the low speeds we are considering, both forces are the same. We can say that, at least for low speeds, magnetism and electricity are simply “two different sides of the same thing.”
But it turns out that everything is even better than we said. If we take into account the fact that forces are also transformed during the transition from one system to another, then it turns out that both methods of observing what is happening actually give the same physical results at any speed.
To see this, you can, for example, ask the question: what transverse momentum will a particle acquire when a force has been acting on it for some time? We know from issue. 2, ch. 16 that the transverse momentum of the particle must be the same both in the system and in the system. Let us denote the transverse coordinate and compare and . Using the relativistically correct equation of motion, we expect that over time our particle will acquire transverse momentum in the system, given by the expression
In the system, the transverse momentum will be equal to
Figure. 13.12. In the system, the charge density is zero, and the current density is equal to . There is only a magnetic field. In the system, the charge density is equal to , and the current density is . The magnetic field here is equal and there is an electric field.
We must compare and , of course, for the corresponding time intervals and . In ch. 15 (issue 2) we saw that the time intervals related to a moving particle seem longer than the intervals in the particle’s rest frame. Since our particle was initially at rest in the system, we expect that for small
and everything turns out great. According to (13.31) and (13.32),
and if we combine (13.30) and (13.33), then this ratio is equal to one.
So it turns out that we get the same result, regardless of whether we analyze the motion of a particle flying next to the wire in the rest frame of the wire or in the rest frame of the particle. In the first case the force was purely “magnetic”, in the second it was purely “electric”. Both observation methods are shown in Fig. 13.12 (although in the second system there is also a magnetic field, it does not affect the stationary particle).
If we had chosen another coordinate system, we would have found some different mixture of fields and . Electric and magnetic forces are parts of one physical phenomenon - the electromagnetic interaction of particles. The division of this interaction into electric and magnetic parts depends to a large extent on the frame of reference in which we describe the interaction. But the full electromagnetic description is invariant; electricity and magnetism taken together are consistent with the principle of relativity discovered by Einstein." In the system at the point of charge, and formula (13.1) will not change if the source of the fields or moves (the values of and will change as a result of the movement). Our mathematical description applies only to the fields as functions of and , taken in some inertial frame of reference.
Later we will talk about “a wave of electric and magnetic fields propagating through space,” such as a light wave. But this is like talking about a wave running along a rope. We do not mean that any part of the rope moves in the direction of the wave, but we mean that the displacement of the rope appears first in one place and then in another. Similarly for an electromagnetic wave - the wave itself propagates, and the magnitude of the fields changes.
So in the future, when we - or anyone else - talk about a "moving" field, you should understand that we are simply talking about a short and convenient way of describing a changing field under certain conditions.
Laws of transformation and relativity
An electromagnetic field differs from any system of particles in that it is a physical system with infinite a large number degrees of freedom. This property is associated with a certain state of the field. Indeed, in the region of existence of the field, the values of the independent components constitute an infinite number of quantities, since any region of space contains an infinitely large number of points.
Electric and magnetic fields are various manifestations single electromagnetic field, which also obeys the principle of superposition. The division of the electromagnetic field into an electric field and a magnetic field is relative in nature, since it depends on the choice of the reference system.
For example, a charge moves in an inertial reference frame S with a constant speed v or when moving identical charges towards each other at a constant speed v. In this reference frame, both the electric and magnetic fields of this charge are observed, but changing with time. When moving to another inertial reference frame S *, moving with the charge, only an electric field is observed, since the charge is at rest in it. If in the S - frame of reference there is a constant, inhomogeneous magnetic field (for example, a horseshoe magnet), then in the S * - frame moving relative to the S - frame, alternating electric and magnetic fields are observed.
The relationships between electric and magnetic fields are not the same in different reference systems.
Experiments show that the charge of any particle is invariant, that is, it does not depend on the speed of the particle and on the choice of inertial frame of reference. Gauss's theorem
is valid not only for charges at rest, but also for moving ones, i.e. it is invariant with respect to inertial frames of reference.
When moving from one inertial reference system to another, the electric and magnetic fields are transformed. Let there be two inertial systems reference: S and the system moving relative to it, with speed S *. If at some spatio-temporal point A of system S the values of the fields and are known, then what will be the values of these fields * and * at the same spatio-temporal point A of system S *? Spatiotemporal point A is a point whose coordinates and time in both reference systems are interconnected by Lorentz transformations, i.e.
The laws of transformation of these fields according to the special theory of relativity are expressed by the following four formulas:
Symbols || and ^ the longitudinal and transverse (relative to the vector) components of the electric and magnetic fields are marked; c is the speed of light in vacuum;
From the equations it is clear that each of the vectors * and * is expressed both through and through, which indicates the unified nature of the electric and magnetic fields.
For example, the strength modulus of vector E of a freely moving relativistic charge is described by the formula
where a is the angle between the radius vector and the velocity vector.
