1. The electric field in vacuum

History has it that the Greek philosopher Thales (Fig. 1.1) observed the action of electric forces more than twenty-five centuries ago: he discovered that amber, when rubbed with woolen flannel, is able to attract light objects. Amber — in Greek «elektron» — gave us the term «electricity».

Fig. 1.1. Thales (625–547 BC) — ancient Greek philosopher and scientist

The ancient Greeks were also familiar with the special properties of iron ore (magnetite, or lodestone), which is a natural magnet. The word «magnet» comes from the Greek «stone from Magnesia» — after the name of a city in Asia Minor, near which a deposit of this ore was located.

The systematic study of electricity and magnetism began during the Renaissance, but it was only toward the end of the last century that physicists achieved a clear understanding of the foundations of the classical theory of these phenomena.

1.1. Electric Charge. Law of Conservation of Charge

All bodies in nature are made up of atoms or molecules, which, in turn, consist of nuclei and electrons possessing electric charge.

Electric charge is a phenomenological characteristic of the properties of elementary particles and their interactions.

Between charged elementary particles there exist special interaction forces called electric forces. It has been established experimentally that these forces can be either attractive or repulsive; therefore, to describe electric interaction, two types of electric charges are introduced, conventionally called negative and positive: like charges repel each other, while unlike charges attract (Fig. 1.2).

Fig. 1.2. Two negative charges repel each other, a negative and a positive charge attract each other, two positive charges repel each other

The charge of electrons is considered negative, and the charge of protons — positive. Neutrons, which are part of nuclei, carry no electric charge. Electric interaction forces bind the nucleus and electrons into a single stable system — the atom.

The smallest electric charge in magnitude experimentally found in nature is the charge of the electron. The charge of the proton is exactly equal to it in magnitude and opposite in sign:

Note that the value of the elementary charge given above is usually used for approximate calculations and for solving school and university problems. In fact, experimental physicists have determined it with much greater precision. Currently, the tables (Journal of Physics G. Nuclear and Particle Physics. Vol. 37, No. 7A, July 2010, Article 075021) give the following value of the elementary charge

Owing to the exact equality of the magnitudes of the proton and electron charges, the total positive and negative charges in each atom are equal in magnitude, and therefore bodies are usually electrically neutral. However, by applying some effort, electrons can be torn away from some bodies, which then become positively charged, and transferred to other bodies, which become negatively charged. Such bodies are macroscopically charged. The electric charge of any body is a multiple of the elementary charge e, that is, it varies discretely:

where N — is an integer. The discreteness of the possible values of electric charge is customarily referred to as the quantization of electric charge.

Numerous experiments have proved that the law of conservation of electric charge holds:

A physical system whose boundary surface cannot be crossed by charged particles is customarily called electrically isolated. Therefore, in many cases, in particular when deriving equations that are the integral or differential form of the law of conservation of charge, the following formulation of it is quite useful:

Obviously, «introducing» charged particles into the system or «removing» them from the system is, in an algebraic sense, one and the same thing. Note also that a similar approach proves productive when writing other conservation laws as well: of energy, momentum, angular momentum, etc.

At the microscopic level, the law of conservation of charge follows from the analysis of reactions between elementary particles and, of course, nuclear reactions. Let us take, for example, the alpha decay of a uranium isotope (Fig. 1.3):

Fig. 1.3. Alpha decay of uranium-238 isotope

The atomic number Z of the uranium nucleus is 92, which means that the nucleus contains 92 protons, that is, its charge is q_U = 92e. For thorium Z = 90, that is, the charge of its nucleus is q_Th=90e, and for helium Z = 2 and q_He = 2e. The fulfillment of the equality

means that electric charge is conserved in this reaction. Never have reactions been observed involving nuclei or elementary particles in which the law of conservation of electric charge is violated. This does not mean that particles with electric charge cannot disappear or be created, but in that case particles with the same but opposite-sign charge must also disappear or be created. The main point is that the total charge remains unchanged before and after the reaction. As an example, let us consider the so-called annihilation reaction: an electron e^– with charge –e collides with its antiparticle — the positron e⁺, whose charge is positive and equal to +e. As a result, two photons g are produced (Fig. 1.4).

Fig. 1.4. Annihilation reaction of an electron and a positron

It is easy to verify that the reaction

satisfies the law of conservation of electric charge: the total charge before and after the reaction equals zero. At the same time, a reaction such as, for example,

in which charge is not conserved.

The electron is the lightest of the charged particles, and owing to the law of conservation of charge (and the law of conservation of energy), it simply has nothing to decay into. Therefore the electron is stable, and this is a necessary prerequisite for the stability of atoms, molecules, matter, and of ourselves.

1.2. Coulomb's Law. The Superposition Principle

Let there be two charged macroscopic bodies whose dimensions are negligibly small compared with the distance between them. In this case each body can be regarded as a material point, or «point charge».

The French physicist C. Coulomb (1736–1806) experimentally established the law that bears his name (Coulomb's law) (Fig. 1.5):

Fig. 1.5. C. Coulomb (1736–1806) — French engineer and physicist

Fig. 1.6 shows the electric repulsive forces arising between two like point charges.

Fig. 1.6. Electric repulsive forces between two like point charges

Recall that , where and — are the position vectors of the first and second charges, so the force acting on the second charge as a result of its electrostatic — «Coulomb» interaction with the first charge can be rewritten in the following «expanded» form

Let us note the following rule, convenient when solving problems: if the first index of the force is taken to be the number of the charge on which this force acts, and the second – the number of the charge that creates this force, then keeping the same order of indices on the right-hand side of the formula automatically ensures the correct direction of the force — corresponding to the sign of the product of the charges: — repulsion and — attraction, with the coefficient always.

To measure the forces acting between point charges, an instrument created by Coulomb, called a torsion balance, was used (Figs. 1.7, 1.8).

Fig. 1.7. C. Coulomb's torsion balance (drawing from a 1785 work). The force acting between charged balls a and b was measured

Fig. 1.8. C. Coulomb's torsion balance (suspension point)

A light beam is suspended on a thin elastic thread, with a metal ball fixed at one end and a counterweight at the other. Next to the first ball, another identical stationary ball can be placed. A glass cylinder protects the sensitive parts of the instrument from air movement.

To establish the dependence of the electrostatic interaction force on the distance between the charges, arbitrary charges are imparted to the balls by touching them with a third charged ball mounted on a dielectric handle. From the angle of twist of the elastic thread, the repulsive force of the like-charged balls can be measured, and from the instrument's scale, the distance between them.

It should be said that Coulomb was not the first scientist to establish the law of charge interaction that now bears his name: 30 years before him, B. Franklin arrived at the same conclusion. Moreover, the precision of Coulomb's measurements was inferior to the precision of experiments carried out earlier (H. Cavendish).

To introduce a quantitative measure for determining the precision of the measurements, let us assume that in reality the interaction force of the charges is inversely proportional not to the square of the distance between them, but to some other power:

.

No scientist would undertake to assert that d = 0 exactly. The correct conclusion should read as follows: experiments have shown that d does not exceed...

The results of some of these experiments are given in Table 1.

Table 1.

Results of direct experiments testing Coulomb's law

Charles Coulomb himself verified the inverse-square law to within an accuracy of a few percent. The table gives the results of direct laboratory experiments. Indirect data, based on observations of magnetic fields in outer space, lead to even stronger constraints on the value of d. Thus, Coulomb's law can be considered a reliably established fact.

In SI, the unit of current (the ampere) is a base unit, and consequently the unit of charge q turns out to be a derived one. As we shall see later, the current I is defined as the ratio of the charge flowing through the cross-section of a conductor during a time to this time:

From this it is evident that the magnitude of a steady current is numerically equal to the charge flowing through the cross-section of a conductor per unit time, and accordingly:

The proportionality coefficient in Coulomb's law is written as:

With this form of writing, the experiment yields the value of the quantity , which is customarily called the electric constant. The approximate numerical value of the electric constant is as follows:

Since most often enters equations in the combination

let us give the numerical value of the coefficient itself

As in the case of the elementary charge, the numerical value of the electric constant has been determined experimentally with high precision:

The coulomb is too large a unit for practical use. For example, two charges of 1 C each, placed in vacuum at a distance of 100 m from each other, repel each other with a force

For comparison: a body of the following mass presses on the ground with the same force

This is approximately the mass of a freight railroad car, for example, loaded with coal.

The Superposition Principle for Fields

The superposition principle is a statement according to which the resulting effect of a complex process of action is the sum of the effects caused by each action separately, provided that the latter do not mutually influence one another (Physical Encyclopedic Dictionary, Moscow, «Soviet Encyclopedia», 1983, p. 731). It has been established experimentally that the superposition principle holds for the electromagnetic interaction considered here.

In the case of the interaction of charged bodies, the superposition principle manifests itself as follows: the force with which a given system of charges acts on some point charge is equal to the vector sum of the forces with which each charge of the system acts on it.

Let us illustrate this with a simple example. Suppose there are two charged bodies acting on a third with forces and respectively. Then the system consisting of these two bodies — the first and the second — acts on the third body with a force

This rule is valid for any charged bodies, not only for point charges. The interaction forces of two arbitrary systems of point charges are calculated in Appendix 1 at the end of this chapter.

It follows that the electric field of a system of charges is determined by the vector sum of the field strengths created by the individual charges of the system, i.e.

Adding the electric field strengths according to the rule of vector addition expresses the so-called superposition principle (independent superposition) of electric fields. The physical meaning of this property is that an electrostatic field is created only by charges at rest. Consequently, the fields of different charges «do not interfere» with one another, and therefore the total field of a system of charges can be calculated as the vector sum of the fields from each of them separately.

Since the elementary charge is very small, while macroscopic bodies contain a very large number of elementary charges, the distribution of charge over such bodies can in most cases be regarded as continuous. In order to describe exactly how the charge is distributed over a body (uniformly, non-uniformly, where there is more charge, where there is less, etc.), let us introduce charge densities of the following three kinds:

· volume charge density :

where dV — is a physically infinitesimal element of volume;

· surface charge density :

where dS — is a physically infinitesimal element of surface;

· linear charge density :

where — is a physically infinitesimal element of line length.

Here and throughout, is the charge of the physically infinitesimal element under consideration (of volume, of a surface patch, of a line segment). By a physically infinitesimal portion of a body, here and below, is meant such a portion of it which, on the one hand, is so small that under the conditions of the given problem it can be regarded as a material point, while, on the other hand, it is so large that the discreteness of the charge (see relation ) of this portion can be neglected.

General expressions for the interaction forces of systems of continuously distributed charges are given in Appendix 2 at the end of the chapter.

Example 1. An electric charge of 50 nC is uniformly distributed over a thin rod 15 cm long. On the extension of the rod's axis, at a distance of 10 cm from its nearest end, there is a point charge of 100 nC (Fig. 1.9). Determine the force of interaction between the charged rod and the point charge.

Fig. 1.9. Interaction of a charged rod with a point charge

Solution. In this problem the force F cannot be determined by writing Coulomb's law in the form or (1.3). Indeed, what is the distance between the rod and the charge: r, r + a/2, r + a? Since, under the conditions of the problem, we have no right to assume that a << r, applying Coulomb's law in its original formulation, valid only for point charges, is impossible; it is necessary to use the standard technique for such situations, which is as follows.

It is always useful, if not to say necessary, before proceeding to the concretization and performance of the calculation, to analyze the symmetry of the problem. From a practical point of view, such an analysis is useful in that, as a rule, given a sufficiently high symmetry of the problem, it sharply reduces the number of quantities that need to be calculated, since it turns out that many of them are equal to zero.

Let us divide the rod into infinitesimal segments of length , the distance from the left end of such a segment to the point charge being equal to .

The uniformity of the distribution of the charge over the rod means that the linear charge density is constant and equal to

,

Consequently, the charge of the segment is equal to , whence, in accordance with Coulomb's law, the force acting on the point charge q as a result of its interaction with the point charge , is equal to

As a result of the interaction of the point charge q with the entire rod, a force will act on it

Substituting the numerical values here, for the magnitude of the force we obtain:

From (1.5) it is seen that at , when the rod can be regarded as a material point, the expression for the interaction force of the charge and the rod, as it should, takes the usual form of Coulomb's law for the interaction force of two point charges:

Example 2. A ring of radius carries a uniformly distributed charge . What is the force of interaction between the ring and a point charge q located on the axis of the ring at a distance from its center (Fig. 1.10).

Solution. By the condition, the charge is uniformly distributed over a ring of radius . Dividing by the circumference, we obtain the linear charge density on the ring Let us isolate an element of the ring of length . Its charge is equal to .

Fig. 1.10. Interaction of a ring with a point charge

At point q this element creates an electric field

We are interested only in the longitudinal component of the field, since when summing the contribution from all elements of the ring, only it is nonzero:

Integrating over , we find the electric field on the axis of the ring at a distance from its center:

From this we find the required force of interaction of the ring with the charge q:

Let us discuss the result obtained. At large distances from the ring , the quantity of the ring's radius under the radical sign can be neglected, and we obtain the approximate expression

This is not surprising, since at large distances the ring looks like a point charge and the interaction force is given by the ordinary Coulomb's law. At small distances the situation changes sharply. Thus, when a test charge q is placed at the center of the ring , the interaction force is zero. This is also not surprising: in this case the charge q is attracted with equal force by all elements of the ring, and the action of all these forces mutually cancels.

Since the electric field is zero both at and at , somewhere at an intermediate value the electric field of the ring is maximal. Let us find this point by differentiating the expression for the field strength E with respect to the distance

Setting the derivative equal to zero, we find the point where the field is maximal. At this point it is equal to

Example 3. Two mutually perpendicular infinitely long threads, carrying uniformly distributed charges with linear densities and , are located at a distance a from each other (Fig. 1.11). How does the interaction force between the threads depend on the distance a?

Solution. Let us first discuss the solution of this problem by the method of dimensional analysis. The interaction force between the threads may depend on the charge densities on them, the distance between the threads, and the electric constant, that is, the sought formula has the form:

where is a dimensionless constant (a number). Note that, owing to the symmetric arrangement of the threads, the charge densities on them can enter only in a symmetric way, to equal powers. The dimensions of the quantities entering here are known in SI:

Fig. 1.11. Interaction of two mutually perpendicular infinitely long threads

Compared with mechanics, a new quantity has appeared here — the dimension of electric charge. Combining the two preceding formulas, we obtain an equation for the dimensions:

Equating the powers of M and T on both sides of this equation, we immediately obtain There is no quantity of charge dimension on the left-hand side, whence it follows that or Finally, equating the powers of the length dimension, we obtain the equation whence it follows that Finally we have:

Thus, it turns out that the interaction force between the threads does not depend on the distance between them. Recall that the dimensionless constant C cannot be determined by the method of dimensional analysis. In essence we have already obtained the answer to the problem, but let us also give its exact solution, which will allow us to find C. In Fig. 1.11, on the right, a top view is shown of the plane containing the thread point A marks the intersection of the drawing plane with the thread . The electric field strength created by the thread at the point where the element of the second thread is located, is equal to

On the element of the thread a force acts

We are, however, interested only in the component of this force along the axis , since the longitudinal component is compensated exactly by the same force acting on the symmetric element of the thread below. Let us express all distances in terms of the angle :

We finally obtain the expression for the component of the force acting on the element

Integrating this expression over the angle from to , we find the total force acting on the thread:

We have again confirmed that the force between the threads in this problem does not depend on the distance between them. Moreover, this time we have determined the dimensionless constant:

1.3. The Electric Field. Field Strength and Potential

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Продолжение:

Часть 1 1. The electric field in vacuum
Часть 2 1.4. Flux of a vector. The Ostrogradsky–Gauss theorem for a
Часть 3 1.5. Application of Gauss's theorem for calculating the electric field
Часть 4 Appendices - 1. The electric field in vacuum
Часть 5 - 1. The electric field in vacuum