The principle of superposition of electric fields is written in the form. The principle of electrostatic superposition. The principle of superposition of electrostatic fields

One of the tasks that electrostatics sets for itself is the assessment of field parameters for a given stationary distribution of charges in space. And the principle of superposition is one of the options for solving such a problem.

Superposition principle

Let us assume the presence of three point charges interacting with each other. With the help of experiment it is possible to measure the forces acting on each of the charges. To find the total force with which two other charges act on one charge, you need to add the forces of each of these two according to the parallelogram rule. In this case, the logical question is: are the measured force that acts on each of the charges and the totality of forces from two other charges equal to each other, if the forces are calculated according to Coulomb’s law. The research results demonstrate a positive answer to this question: indeed, the measured force is equal to the sum of the calculated forces according to Coulomb’s law on the part of other charges. This conclusion is written in the form of a set of statements and is called the principle of superposition.

Definition 1

Superposition principle:

• the force of interaction between two point charges does not change if other charges are present;
• the force acting on a point charge from two other point charges is equal to the sum of the forces acting on it from each of the point charges in the absence of the other.

The principle of superposition of charge fields is one of the foundations for the study of such a phenomenon as electricity: its significance is comparable to the importance of Coulomb’s law.

In the case when we are talking about a set of charges N (i.e., several field sources), the total force experienced by the test charge q, can be determined by the formula:

F → = ∑ i = 1 N F i a → ,

where F i a → is the force with which it affects the charge q charge q i if there is no other N - 1 charge.

Using the principle of superposition using the law of interaction between point charges, it is possible to determine the force of interaction between charges present on a body of finite dimensions. For this purpose, each charge is divided into small charges d q (we will consider them point charges), which are then taken in pairs; the interaction force is calculated and finally the vector addition of the resulting forces is carried out.

Field interpretation of the superposition principle

Field interpretation: The field strength of two point charges is the sum of the intensities created by each of the charges in the absence of the other.

For general cases, the principle of superposition with respect to tensions has the following notation:

E → = ∑ E i → ,

where E i → = 1 4 π ε 0 q i ε r i 3 r i → is the intensity of the i-th point charge, r i → is the radius of the vector drawn from the i-th charge to a certain point in space. This formula tells us that the field strength of any number of point charges is the sum of the field strengths of each of the point charges, if there are no others.

Engineering practice confirms compliance with the superposition principle even for very high field strengths.

The fields in atoms and nuclei have a significant strength (of the order of 10 11 - 10 17 V m), but even in this case the principle of superposition was used to calculate energy levels. In this case, the results of calculations coincided with experimental data with great accuracy.

However, it should also be noted that in the case of very small distances (on the order of ~ 10 - 15 m) and extremely strong fields, the superposition principle is probably not satisfied.

Example 1

For example, on the surface of heavy nuclei at a strength of the order of ~ 10 22 V m, the superposition principle is satisfied, and at a strength of 10 20 V m, quantum mechanical nonlinearities of interaction arise.

When the charge distribution is continuous (i.e. there is no need to take into account discreteness), the total field strength is given by the formula:

E → = ∫ d E → .

In this entry, integration is carried out over the charge distribution region:

• when charges are distributed along the line (τ = d q d l - linear charge distribution density), integration is carried out along the line;
• when charges are distributed over the surface (σ = d q d S - surface distribution density), integration is carried out over the surface;
• with volumetric charge distribution (ρ = d q d V - volumetric distribution density), integration is carried out over the volume.

The principle of superposition makes it possible to find E → for any point in space for a known type of spatial charge distribution.

Identical point charges q are given, located at the vertices of a square with side a. It is necessary to determine what force is exerted on each charge by the other three charges.

Solution

In Figure 1 we illustrate the forces affecting any of the given charges at the vertices of the square. Since the condition states that the charges are identical, it is possible to choose any of them for illustration. Let's write down the summing force affecting the charge q 1:

F → = F 12 → + F 14 → + F 13 → .

The forces F 12 → and F 14 → are equal in magnitude, we define them as follows:

F 13 → = k q 2 2 a 2 .

Drawing 1

Now let’s set the direction of the O X axis (Figure 1), design the equation F → = F 12 → + F 14 → + F 13 →, substitute the force modules obtained above into it and then:

F = 2 k q 2 a 2 · 2 2 + k q 2 2 a 2 = k q 2 a 2 2 2 + 1 2 .

Answer: the force exerted on each of the given charges located at the vertices of the square is equal to F = k q 2 a 2 2 2 + 1 2.

Example 3

An electric charge is given, distributed uniformly along a thin thread (with linear density τ). It is necessary to write down an expression that determines the field strength at a distance a from the end of the thread along its continuation. Thread length – l .

Drawing 2

Solution

Our first step will be to highlight a point charge on the thread d q. Let us compose for it, in accordance with Coulomb’s law, a record expressing the strength of the electrostatic field:

d E → = k d q r 3 r → .

At a given point, all tension vectors have the same direction along the OX axis, then:

d E x = k d q r 2 = d E .

The condition of the problem is that the charge has a uniform distribution along the thread with a given density, and we write the following:

Let's substitute this entry into the previously written expression for the electrostatic field strength, integrate and get:

E = k ∫ a l + a τ d r r 2 = k τ - 1 r a l + a = k τ l a (l + a) .

Answer: The field strength at the indicated point will be determined by the formula E = k τ l a (l + a) .

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Let there be two charged macroscopic bodies, the sizes of which are negligible compared to the distance between them. In this case, each body can be considered a material point or a “point charge”.

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

Rice. 1.5. C. Coulon (1736–1806) - French engineer and physicist

In Fig. Figure 1.6 shows the electrical repulsive forces that arise between two point charges of the same name.

Rice. 1.6. Electrical repulsive forces between two like point charges

Let us recall that , where and are the radius vectors of the first and second charges, therefore 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 for solving problems: if the first index of the force is the number of that charge, on which this force acts, and the second is the number of that charge, which creates this force, then compliance with the same order of indices on the right side of the formula automatically ensures the correct direction of the force - corresponding to the sign of the product of charges: - repulsion and - attraction, while the coefficient is always.

To measure the forces acting between point charges, a device created by Coulomb, called torsion scales(Fig. 1.7, 1.8).

Rice. 1.7. Torsion scales by Ch. Coulomb (drawing from the work of 1785). The force acting between charged balls a and b was measured

Rice. 1.8. Torsion scales Sh. Coulomb (suspension point)

A light rocker arm is suspended from a thin elastic thread, with a metal ball attached at one end and a counterweight at the other. Next to the first ball, you can place another identical motionless ball. The glass cylinder protects sensitive parts of the device from air movement.

To establish the dependence of the strength of electrostatic interaction on the distance between charges, the balls are given arbitrary charges by touching them with a third charged ball mounted on a dielectric handle. Using the angle of twist of the elastic thread, you can measure the repulsive force of similarly charged balls, and using the scale of the device, you can measure the distance between them.

It must be said that Coulomb was not the first scientist to establish the law of interaction of charges, which now bears his name: 30 years before him, B. Franklin came to the same conclusion. Moreover, the accuracy of Coulomb's measurements was inferior to the accuracy of previously conducted experiments (G. Cavendish).

To introduce a quantitative measure to determine the accuracy of measurements, let us assume that in fact the force of interaction between charges is not the inverse of the square of the distance between them, but some other power:

None of the scientists will undertake to claim that d= 0 exactly. The correct conclusion should be: experiments have shown that d does not exceed...

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

Table 1.

Results of direct experiments to test Coulomb's law

Charles Coulomb himself tested the inverse square law to within a few percent. The table shows the results of direct laboratory experiments. Indirect evidence based on observations of magnetic fields in space leads to even stronger restrictions on the magnitude d. Thus, Coulomb's law can be considered a reliably established fact.

The SI unit of current is ( ampere) is basic, hence the unit of charge q turns out to be a derivative. As we will see later, the current strength I is defined as the ratio of the charge flowing through the cross-section of the conductor in time to this time:

From this it can be seen that the strength of the direct current is numerically equal to the charge flowing through the cross section of the conductor per unit time, according to this:

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

With this form of recording, the value of the quantity follows from the experiment, which is usually called electrical constant. The approximate numerical value of the electrical constant is as follows:

Since it most often appears in equations as a combination

Let's give the numerical value of the coefficient itself

As in the case of an elementary charge, the numerical value of the electric constant is determined experimentally with high accuracy:

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

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

This is approximately the weight of a freight railway car, for example, with coal.

Principle of field superposition

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

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

Let's explain this with a simple example. Let there be two charged bodies acting on a third body with forces and respectively. Then the system of these two bodies - the first and the second - acts on the third body with a force

This rule is true for any charged bodies, not only for point charges. The interaction forces between 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.

The addition of 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 the electrostatic field is created only by charges at rest. This means that the fields of different charges “do not interfere” with each other, and therefore the total field of a system of charges can be calculated as a vector sum of the fields from each of them separately.

Since the elementary charge is very small, and macroscopic bodies contain a very large number of elementary charges, the distribution of charges over such bodies in most cases can be considered continuous. In order to describe exactly how the charge is distributed (uniformly, non-uniformly, where there are more charges, where there are fewer, etc.) the charge throughout the body, we introduce charge densities of the following three types:

· bulk densitycharge:

Where dV- physically infinitesimal volume element;

· surface charge density:

Where dS- physically infinitesimal surface element;

· linear charge density:

where is a physically infinitesimal element of the line length.

Here everywhere is the charge of the physically infinitesimal element under consideration (volume, surface area, line segment). By a physically infinitesimal section of a body, here and below we mean a section of it that, on the one hand, is so small that under the conditions of this problem, it can be considered a material point, and, on the other hand, it is so large that it is a discrete charge (see . ratio) of this area can be neglected.

General expressions for the interaction forces between 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 continuation of the axis of the rod 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.

Rice. 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). In fact, what is the distance between the rod and the charge: r, r + a/2, r + a? Since, according to the conditions of the problem, we do not have the right to assume that a << r, application of Coulomb's law in its original It is impossible to formulate a formulation that is valid only for point charges; it is necessary to use a standard technique for such situations, which consists of the following.

It is always useful, if not necessary, to analyze the symmetry of the problem before starting to specify and perform calculations. From a practical point of view, such an analysis is useful in that, as a rule, with 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 is equal to .

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

Therefore, the charge of the segment is equal to , from where, in accordance with Coulomb’s law, the force acting on spot charge q as a result of its interaction with point charge is equal to

As a result of interaction spot charge q at all rod, a force will act on it

Substituting numerical values ​​here, for the force modulus we obtain:

From (1.5) it is clear that when , when the rod can be considered a material point, the expression for the force of interaction between the charge and the rod, as it should be, takes the usual form of Coulomb’s law for the force of interaction between 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. According to the condition, the charge is uniformly distributed on a ring of radius . Dividing by the circumference, we obtain the linear charge density on the ring Select an element on the ring with length . Its charge is .

Rice. 1.10. Interactions of a ring with a point charge

At the point q this element creates an electric field

We are only interested in the longitudinal component of the field, because 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 here we find the required force of interaction between the ring and the charge q:

Let's discuss the result obtained. At large distances to the ring, the value of the radius of the ring 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 usual Coulomb law. At short distances the situation changes dramatically. 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 is mutually compensated.

Since at and at the electric field is zero, somewhere at an intermediate value the electric field of the ring is maximum. Let's find this point by differentiating the expression for the tension E by distance

Equating the derivative to zero, we find the point where the field is maximum. It is equal at this point

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

Solution. First, we will discuss the solution to this problem using the dimensional analysis method. The strength of interaction between the threads can depend on the charge densities on them, the distance between the threads and the electrical constant, that is, the required formula has the form:

where is a dimensionless constant (number). Note that due to the symmetrical arrangement of the threads, charge densities can only enter them in a symmetrical manner, in the same degrees. The dimensions of the quantities included here in SI are known:

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

Compared to mechanics, a new quantity has appeared here - the dimension of the electric charge. Combining the two previous formulas, we obtain the equation for dimensions:

Electrostatic field- a field created by electric charges that are motionless in space and constant in time (in the absence of electric currents).

An electric field is a special type of matter associated with electric charges and transmitting the effects of charges on each other.

If there is a system of charged bodies in space, then at every point of this space there is a force electric field. It is determined through the force acting on a test charge placed in this field. The test charge must be small so as not to affect the characteristics of the electrostatic field.

Electric field strength- vector physical quantity that characterizes the electric field at a given point and is numerically equal to the ratio of the force acting on a stationary test charge placed at a given point in the field to the magnitude of this charge:

From this definition it is clear why the electric field strength is sometimes called the force characteristic of the electric field (indeed, the entire difference from the force vector acting on a charged particle is only in a constant factor).

At each point in space at a given moment in time there is its own vector value (generally speaking, it is different at different points in space), thus, this is a vector field. Formally, this is expressed in the notation

representing the electric field strength as a function of spatial coordinates (and time, since it can change with time). This field, together with the field of the magnetic induction vector, is an electromagnetic field, and the laws to which it obeys are the subject of electrodynamics.

Electric field strength in SI is measured in volts per meter [V/m] or newtons per coulomb [N/C].

The number of lines of the vector E penetrating some surface S is called the flux of the intensity vector N E .

To calculate the flux of vector E, it is necessary to divide the area S into elementary areas dS, within which the field will be uniform (Fig. 13.4).

The tension flow through such an elementary area will be equal by definition (Fig. 13.5).

where is the angle between the field line and the normal to the site dS; - projection of the area dS onto a plane perpendicular to the lines of force. Then the field strength flux through the entire surface of the site S will be equal to

Since then

where is the projection of the vector onto the normal and to the surface dS.

Superposition principle- one of the most general laws in many branches of physics. In its simplest formulation, the principle of superposition states:

the result of the influence of several external forces on a particle is the vector sum of the influence of these forces.

The most famous principle of superposition is in electrostatics, in which it states that the strength of the electrostatic field created at a given point by a system of charges is the sum of the field strengths of individual charges.

The principle of superposition can also take other formulations, which completely equivalent above:

The interaction between two particles does not change when a third particle is introduced, which also interacts with the first two.

The interaction energy of all particles in a many-particle system is simply the sum of the energies pair interactions between all possible pairs of particles. Not in the system many-particle interactions.

The equations describing the behavior of a many-particle system are linear by the number of particles.

It is the linearity of the fundamental theory in the field of physics under consideration that is the reason for the emergence of the superposition principle in it.

Let's consider a method for determining the value and direction of the tension vector E at each point of the electrostatic field created by a system of stationary charges q 1 , q 2 , ..., Q n .

Experience shows that the principle of independence of the action of forces discussed in mechanics (see §6) is applicable to Coulomb forces, i.e. resultant force F, acting from the field on the test charge Q 0, equal to the vector sum of forces F i applied to it from the side of each of the charges Q i:

According to (79.1), F=Q 0 E And F i ,=Q 0 E i, where E is the strength of the resulting field, and E i is the field strength created by the charge Q i. Substituting the last expressions into (80.1), we get

Formula (80.2) expresses the principle of superposition (imposition) of electrostatic fields, according to which tension E the resulting field created by the system of charges is equal to geometric sum field strengths created at a given point by each of the charges separately.

The principle of superposition is applicable to calculate the electrostatic field of an electric dipole. Electric dipole- a system of two equal in modulus opposite point charges (+ Q, - Q), distance l between which there is significantly less distance to the considered points of the field. A vector directed along the dipole axis (a straight line passing through both charges) from a negative charge to a positive charge and equal to the distance between them is called dipole arml . Vector

coinciding in direction with the dipole arm and equal to the product of the charge

| Q| on the shoulder l , called electric dipole moment p or dipole moment(Fig. 122).

According to the superposition principle (80.2), the tension E dipole fields at an arbitrary point

E=E + + E - ,

Where E+ and E- - field strengths created by positive and negative charges, respectively. Using this formula, we calculate the field strength along the extension of the dipole axis and at the perpendicular to the middle of its axis.

1. Field strength along the extension of the dipole axis at the point A(Fig. 123). As can be seen from the figure, the dipole field strength at the point A is directed along the dipole axis and is equal in magnitude

E A =E + -E - .

Marking the distance from the point A to the middle of the dipole axis through l, based on formula (79.2) for vacuum we can write

According to the definition of a dipole, l/2<

2. Field strength at a perpendicular raised to the axis from its middle, at the point IN(Fig. 123). Dot IN equidistant from the charges, therefore

Where r" - distance from point IN to the middle of the dipole arm. From the similarity of isosceles-

of the given triangles based on the dipole arm and vector еv, we obtain

E B =E + l/ r". (80.5)

Substituting the value (80.4) into expression (80.5), we obtain

Vector E B has the direction opposite to the electric moment of the dipole (vector R directed from negative to positive charge).