the main Electrodynamics Charge energy in an electrostatic field. Potential of electrostatic field. Potential energy of interaction of an electric charge with an electric field

Charge energy in an electrostatic field. Potential of electrostatic field. Potential energy of interaction of an electric charge with an electric field

The energy of a system of point charges as the sum of the energy of pair interaction of charges (and the principle of superposition)

Potential charge energy in an electric field.The work done by the forces of an electric field when moving a positive point charge q from position 1 to position 2, imagine as a change in the potential energy of this charge:

where W n1 and W n2 - potential charge energies q in positions 1 and 2. With a small charge movement q in the field created by a positive point charge Q, the change in potential energy is

With the final charge movement q from position 1 to position 2 at distances r 1 and r 2 on charge Q,

If the field is created by a system of point charges Q 1 , Q 2, ¼, Q n, then the change in the potential energy of the charge q in this field:

The above formulas allow you to find only change potential energy of a point charge q, and not the potential energy itself. To determine the potential energy, it is necessary to agree at what point in the field to consider it equal to zero. For potential energy of a point charge qlocated in an electric field created by another point charge Qwe get

where C Is an arbitrary constant. Let the potential energy be equal to zero at an infinitely large distance from the charge Q (at r ® ¥), then the constant C \u003d 0 and the previous expression takes the form

In this case, the potential energy is defined as the work of moving the charge by the forces of the field from a given point to an infinitely remote. In the case of an electric field created by a system of point charges, the potential energy of the charge q:

Potential energy of a system of point charges.In the case of an electrostatic field, potential energy serves as a measure of the interaction of charges. Let there be a system of point charges in space Q i (i = 1, 2, ... ,n) Energy interactions of all n charges is determined by the ratio

where r ij - the distance between the respective charges, and the summation is performed so that the interaction between each pair of charges is taken into account once.

Potential of electrostatic field.A field of conservative force can be described not only by a vector function, but an equivalent description of this field can be obtained by determining a suitable scalar value at each of its points. For an electrostatic field, this quantity is electrostatic field potentialdefined as the ratio of the potential energy of the test charge q to the magnitude of this charge, j \u003d W P / q, whence it follows that the potential is numerically equal to the potential energy that a unit positive charge possesses at a given point in the field. The unit for measuring potential is Volt (1 V).

Point charge field potential Q in a homogeneous isotropic medium with a dielectric constant e:

The principle of superposition.Potential is a scalar function; the principle of superposition is valid for it. So for the field potential of a system of point charges Q 1, Q 2 ¼, Q n we have

where r i is the distance from the point of the field with potential j to the charge Q i . If the charge is arbitrarily distributed in space, then

where r - distance from elementary volume d x, d y, d z to the point ( x, y, z), where the potential is determined; V - the amount of space in which the charge is distributed.

Potential and work of electric field forces.Based on the determination of the potential, it can be shown that the work of the electric field forces when moving a point charge q from one point of the field to another is equal to the product of the magnitude of this charge by the potential difference at the initial and final points of the path, A = q (j 1 - j 2). If, by analogy with potential energy, we assume that at points infinitely remote from electric charges - field sources, the potential is zero, then the work of the electric field forces when the charge moves q from point 1 to infinity can be represented as A ¥ = q j 1. Thus, the potential â at a given point of the electrostatic field is physical quantity numerically equal to the work performed by the forces of an electric field when moving a unit positive point charge from a given point of the field to an infinitely distant: j \u003d A ¥ / q. In some cases, the electric field potential is more clearly defined as physical quantity numerically equal to the work of external forces against electric field forces when a unit positive point charge moves from infinity to a given point. The last definition is conveniently written as follows:

In modern science and technology, especially when describing the phenomena occurring in the microworld, a unit of work and energy is often used, called electron volt (eV). This is the work done when moving a charge equal to the charge of an electron between two points with a potential difference of 1 V: 1 eV \u003d 1.60 × 10 -19 C × 1 V \u003d 1.60 × 10 -19 J.

Integral representation of the energy of a continuous charge distribution, c alignment with the case of energy of a system of point charges

Let there be a charge in the volume element. To determine the interaction energy of all elements in volume V, one can use formula (12.4). going in it from the sum to the integral:

Where is the potential created by all charges at the point where the charge is located.

For the first charge, formulas (12.4) and (12.5) seem similar, especially since (12.5) is “derived” from (12.4). However, there is a fundamental difference between them. Formula (12.4) takes into account only the interaction energy between charged balls, but does not take into account the interaction energy between the elements of charges located on each ball. And (12.5) takes into account the first and second.

Given the foregoing, the energy of interaction of charges can be written in the form:

Value is the energy of charged balls, taking into account the interaction of charges with each other on each ball. Own energy depends on the laws of distribution of the charges of the ball and the values \u200b\u200bof the charges. If there is a solitary ball, then.

Then (12.7)

This means that the self-energy of a point charge is infinity.

But with. This leads to serious difficulties when using the model of point charges.

Electric energy of a charged solitary conductor and capacitor

If a solitary conductor has a charge q, then around it there is an electric field whose potential on the surface of the conductor is equal, and the capacitance is C. We increase the charge by dq. When transferring the charge dq from infinity, work equal to . But the potential of the electrostatic field of a given conductor at infinity is zero. Then

When the charge dq is transferred from the conductor to infinity, the electrostatic field forces do the same work. Therefore, as the conductor charge increases by dq, the potential field energy increases, i.e.

Integrating this expression, we find the potential energy of the electrostatic field of a charged conductor with an increase in its charge from zero to q:

Applying the relation, we can obtain the following expressions for the potential energy W:

The energy of the electrostatic field, expressed as the volume integral of the vectors of intensity E and electric displacementD .

The expression of the force acting on a conductor immersed in a liquid or gaseous dielectric through the bulk energy density of the electric field near the conductor.

In the presence of a medium, the calculation of the forces acting on conductors and dielectrics is complicated.

First of all, the expression for bulk force becomes unjust, even if we mean the molecular charge density. This is due to the fact that there is an average macroscopic density that does not take into account the polarization of individual molecules. Meanwhile, in a nonuniform electric field, a force acts on a polarized molecule. One could try to average this force over the volume, but such a procedure runs into significant difficulties. We will use the energy method of calculating forces.

Let's consider some typical tasks. We find the force acting on a dielectric ball placed in a weakly inhomogeneous field. The last condition means that the field should change little on the size of the ball. Then the dipole moment of the ball will be approximately the same as in a uniform field: where E is the external field (in the absence of the ball). Since the moment of the ball is proportional to the field, it behaves as a quasielastic dipole and, therefore, its energy in the field. Now we will make a virtual movement of the ball in an external inhomogeneous field and write down the energy balance: where is the force acting on the ball from the field side:

(19.1)

i.e., the dielectric is drawn into a strong field. If (weak insulator), then expression (19.1) is valid for an insulator of arbitrary shape, since in this case the interaction of individual parts of the insulator, which are polarized independently of each other, can be neglected. Then the volumetric force acting on the dielectric,

(19.2)

i.e., it is determined by a change in the energy density of the electric field when a dielectric is introduced.

In addition to the force acting in the inhomogeneous electric field on the dielectric as a whole, internal stresses also arise in it, called striction forces. Consider a dielectric plate placed in a flat capacitor (Fig. 11.5). It is clear that under the action of striction forces, the plate

Fig. 11.5. To the calculation of striction forces.

stretches along the field. Let's try to calculate the striction forces in this example. We will use the energy method. With a slight stretching of the plate, the change in the field energy consists of two parts. First, in the layer, the field energy in vacuum is replaced by the field energy in the medium. Here, the field in the vacuum gap, which does not change when the dielectric is deformed, since we take the charge on the capacitor unchanged (see above). Secondly, it is necessary to take into account the change in energy in the entire volume of a substance due to a change in its density, on which the dielectric constant depends: where .The partial derivative is taken here at a constant temperature in order to exclude a temperature dependence. The total energy balance per unit area of \u200b\u200bthe dielectric has the form

Hence the tension acting on the dielectric,

(19.4)

can be considered as the difference in tension from the outside and from the inside dielectric, where E is the electric field inside the latter.

Strictive pressure is usually called the quantity

(19.5)

This pressure does not contribute to the force acting on the dielectric as a whole, provided that it is surrounded by vacuum.

Finally, we consider an arbitrary system of charged bodies immersed in a homogeneous liquid dielectric. As we already know, such an environment weakens the field by a factor of time without changing its configuration. From this, in particular, it follows that the field energy is also several times smaller than in vacuum. This means that the work of moving charges and the forces between bodies also decrease by a factor of one. At first glance, this conclusion seems trivial: once the field decreases by a factor of 1, then the force of its effect on the charge should decrease by the same amount. However, the field in the medium is understood as the average field, whereas the local field acting on the charge depends on the shape of the cavity, i.e., on the shape of the charged body. To understand what is the matter here, let us return to the previous example. Let the dielectric is now liquid and fill the entire capacitor. Nevertheless, we can imagine that between the dielectric and the capacitor plate there is a very

a thin gap in which the field is equal so that all the previous consideration remains valid. In this case, the field pressure directly on the plate is equal to the same as in vacuum, instead of the expected attenuation by a factor of one. This example confirms that the force exerted by the field on the charged body really depends on the shape of the body.

However, a liquid dielectric, as a rule, has mechanical contact with the body and also acts on it with some force, which is given by expression (19.4) in the considered example. Finally, it is necessary to take into account the additional pressure in the liquid, which arises due to the electric field and is equal to the striction pressure (19.5). Thus, the total pressure on the plate

(19.6)

in accordance with energy considerations.

We emphasize once again that such a simple result is obtained only for a liquid homogeneous dielectric. The mechanical contact of conductors with a solid dielectric is, as a rule, uncertain. In addition, internal elastic stresses now depend not on local strictive pressure, but on the forces acting on the entire dielectric.

We calculate the potential energy of electric charges for the most simple, but very important special cases.
Potential charge energy in a uniform field Let the charge q move in a uniform electric field with intensity E from point 1 to point 2. The position of point 1 is determined by the radius vector and point 2 by the radius vector r2. The force acting on the charge F \u003d qE is constant. The work of the force F does not depend on the shape of the trajectory connecting points 1 and 2. This follows from the general proof of the potentiality of the electrostatic field. It is possible to carry out a proof using direct calculation of the work when the charge moves along different paths in the same way as it was done in the Mechanics for gravitational forces. Now we will not do it.
The easiest way to calculate the work if the charge moves along a straight line connecting point 1 and point 2 (Fig. 1.78). The displacement vector Дг \u003d r2 - rv Work is equal to the scalar product of force and displacement:
A \u003d F Ar ^ qE (r2-r1) \u003d qE r2-qE yy (1.18.1)
On the other hand, according to (1.17.1), A \u003d ~ (W 2 ~ Comparing expressions (1.18.1) and (1.17.1), we obtain the expression for the potential charge energy in a uniform field:
Wp ~ -qE g. (1.18.2)
A uniform field is created, in particular, in the space between parallel plates carrying charges of opposite signs (Fig. 1.79). It is natural to choose a coordinate system so that the X axis is perpendicular to the plates. Then the projections E „and E are equal to zero and
at z
neither (1.18.2) takes the form:
Wp \u003d -q (Exx + Eyu + Ezz) \u003d ~ qExx. (1.18.3)

Formula (1.18.3) is similar to the formula Wp \u003d mgh for the potential energy of the body above the surface of the Earth. The role of the mass is played by the charge, the acceleration of gravity is the field strength, and instead of the height h is the x coordinate. But the sign of energy is different: minus instead of plus. Here is the point. Mass is always positive, and gravity is necessarily directed vertically downward. Given these circumstances, the formula Wp \u003d mgh was written. It contains the free fall acceleration modulus, and the height h is measured from the Earth’s surface. Formula (1.18.3) is more general. The charge q can be both positive and negative; the field strength can be directed anywhere, and its projection can have either a positive value or a negative one depending on the choice of the coordinate system.
In particular, if the field strength E is directed vertically down and the X axis up, then
Wp \u003d qE \\ x \\ (1.18.4)
in exact accordance with the expression Wp \u003d mgh.
If the electric field does a positive job, then the energy of the charged body in the field decreases: AW 0. This movement of a charged particle is similar to the movement of a stone thrown up. In this case, the potential energy of the particle grows, and the kinetic energy decreases: the particle slows down.
Potential Energy Zero
The potential energy in electrodynamics is determined, as in mechanics, up to an arbitrary constant. Instead of the expression (1.18.2) we could write:
W \u003d -qE-r + C, (1.18.5)
where C is an arbitrary constant. In this case, the change in potential energy remains the same, and the work determines precisely the change in potential energy, and not the energy itself. Writing down the formula (1.18.2), we actually equated the constant C to zero. This corresponds to a certain choice of zero level of potential energy. For example, for the case shown in Figure 1.79, the potential energy is considered equal to zero on the surface of the plate B. But, as with the action of gravitational forces, the zero level of potential energy is chosen arbitrarily. We can assume that W is O at a distance from plate B. Then
Wp \u003d -qExx-qExx y
The physical meaning is not the potential energy itself, but the difference in its values, determined by the work of the field when the charge moves from the initial position to the final.
The energy of interaction of point charges
In the course of mechanics, an expression was obtained for the interaction energy of point bodies:
ty and »W \u003d -G ---.
R g
If instead of point masses we take two charges q1 and q2 of opposite sign (the charges are attracted), we can obtain a similar expression for the potential energy of their interaction:
w (1.18.6)
R g y
For charges of the same sign (charges repel), the sign of potential energy will be opposite:
w (1.18.7)
R g y
Formulas (1.18.6) and (1.18.7) can be combined into one if, instead of charge modules, we take their algebraic values:
W. (1.18.8)
P r v "
The sign of potential energy will automatically turn out to be correct.
If the charges ql and q2 have the same signs, then the potential energy of their interaction is positive (Fig. 1.80, a). It is the larger, the smaller the distance between the charges, since the work that the Coulomb forces can do when pushing the charges away from each other will be more. If the charges have opposite signs, then the energy is negative and its maximum value, equal to zero, is achieved when r -\u003e oo (Fig. 1.80, b). The larger r, the more work the attractive forces will perform when the charges approach each other.

Fig. 1.80
When writing potential energy in the form (1.18.8), a certain choice of the zero level of potential energy has already been made. It is believed that the potential energy of the infinitely distant charges is zero: Wp - »0 as r -» oo. This choice of zero level is convenient, but not required. Instead of the expression (1.18.8), one could write with the same success that
(1.18.9)
p r y
where C is an arbitrary constant. This shows that the positive or negative value of the potential energy has no special physical meaning. The sign of potential energy will be determined when fixing an arbitrary constant C. By changing the value of C, we can change the sign of Wp at a given distance r between charges.
Potential energy of a system of point charges
The potential energy of the system of point charges qv q2, ..., qN is equal to the sum of the potential energies of all pairs of interacting charges. For three charges
w kbSi + hbS * + hw3 l
R G1.2 G1.3 G2.3
Prove it yourself using the following trick. Initially, the charges q2 and qz are located at an infinitely large distance from the charge qv. Then, the charge q2 moves to a point located at a distance of gl 2 from the first charge. After that, the charge qz moves to a point at a distance r1 3 from the first charge and r2 3 from the second. It is necessary to calculate the work of the Coulomb forces performed during these movements, and equate it to the change in potential energy, taken with the opposite sign.
In the general case, N charges
N N
Wp \u003d I llk7rh\u003e (1.18.11)
i \u003d lfc \u003d l (i * k)
where r; k is the distance between the charges of ink numbers. Coeffi- 1
2 is obtained due to the fact that when summing the potential energy is taken into account twice in the form of identical
ri, k rk, i
It is advisable to remember the formulas for the potential energy of an electric charge in a uniform field (1.18.2) and for two point charges (1.18.8). They will be found quite often.
? 1. Is it possible to create an electrostatic
le, whose tension lines are parallel- ^^^^^^^
are lax, and the module of tension of age -
in the direction perpendicular to
niyam (Fig. 1.81)? Fig. 1.81
Draw a graph of the dependence of the potential energy of the oppositely charged particles on the distance, provided that the arbitrary constant C in the formula (1.18.9) is positive.
What will the formula (1.18.8) look like if the charges are in a medium with a dielectric constant є?

We begin by discussing the potential energy that a charge has in an electrostatic field. First of all, it is necessary to recall the conditions under which the concept of potential energy can be introduced.

4.1 Conservative forces

A force is called conservative (or potential) if the work of this force does not depend on the shape of the trajectory and is determined only by the initial and final position of the body.

Let, for example, a body under the action of a conservative force ~ move from the initial

point 1 to end point 2 (Fig. 16). Then the work of force ~ depends only on the position

points 1 and 2 themselves, but not from the trajectory of the body. For example, for trajectories 1! a! 2 and 1! b! 2, the value of A will be the same.

Fig. 16. The concept of conservative power

Note that the work of a conservative force along any closed path is zero. Indeed, let's exit point 1 along trajectory 1! a! 2 and go back along trajectory 2! b! 1. On the first path, the force will do the work A, and on the second path the work will be A. As a result, the total work will be zero.

So, the concept of potential energy can be introduced only in the case of conservative force. Potential energy W is a mathematical expression that depends on the coordinates of the body, such that the work of force is equal to a change in this expression with a minus sign:

Or, the same thing:

A \u003d (W2 W1) \u003d W1 W2:

As we see, the work of a conservative force is the difference between the potential energy values \u200b\u200bcalculated respectively for the initial and final positions of the body.

Examples of conservative forces are well known to you. For example, gravity is conservative. The spring force is also conservative. That is why we can talk about the potential energy of a body raised above the earth, or about the potential energy of a deformed spring.

But the friction force is not conservative: the work of the friction force depends on the shape of the trajectory and is not equal to zero in a closed path. Therefore, there is no “potential energy of the body in the field of friction force”.

4.2 Electrostatic field potential

It turns out that the force with which an electrostatic field acts on a charged body is also conservative. The work of this force, performed when the charge moves, is called the work of the electrostatic field. Thus, we have the most important fact:

The work of the electrostatic field does not depend on the shape of the trajectory along which the charge moves, and is determined only by the initial and final positions of the charge. The work of the field in a closed path is zero.

This fact is also called the potentiality of the electrostatic field. Like the gravitational field, an electrostatic field is potential. The work of the electrostatic field is the same for all paths along which the charge can move from one fixed point in space to another.

Strict mathematical proof of the potentiality of the electrostatic field is beyond the scope of the school curriculum. However, “at the physical level of rigor”, we can verify the validity of this fact using the following simple argument.

It is easy to see that if the electrostatic field were not potential, then it would be possible to build a perpetual motion machine! In fact, then there would be a closed trajectory, when the charge moved along which the field would do a positive job (and no changes in the surrounding bodies would occur). We spin a charge along this trajectory, draw an unlimited amount of energy from nowhere and all the energy problems of mankind have been resolved :-) But this, alas, is not observed, this is flagrantly contrary to the law of energy conservation.

Since the electrostatic field is potential, we can talk about the potential energy of the charge in this field. Let's start with a simple and important case.

4.3 Potential charge energy in a uniform field

The potential energy of a body raised above the earth is mgh. The case of a charge in a uniform field turns out to be very similar to this mechanical situation.

Consider a uniform electrostatic field E, the lines of tension of which are directed along the X axis (Fig. 17). Let the positive charge q move along the field line from point 1 (with coordinate x1) to point 2 (with coordinate x2).


0 x 1

Fig. 17. Charge movement in a uniform field

The field acts on the charge with a force ~, which is directed along the lines of tension. Job

this power, as is easy to see, will be equal to:

A \u003d F (x2 x1) \u003d qE (x2 x1):

What will change if points 1 and 2 do not lie on the same line of tension? It turns out nothing! The formula for the field will remain the same. We will verify this using Fig. 18 .


0 x 1

Fig. 18. Charge movement in a uniform field

Moving from point 1 to point 2, let's choose path 1!3! 2, where point 3 lies on the same line of force with point 1. Then the work of A32 in section 32 is zero, because we are moving perpendicular to the force. As a result, we get:

A \u003d A13 + A32 \u003d A13 \u003d qE (x2 x1):

We see that the work of the field depends only on the abscissa of the initial and final positions of the charge. We write the resulting formula as follows:

A \u003d qEx2 qEx1 \u003d ((qEx2) (qEx1)) \u003d (W2 W1) \u003d W:

Here W1 \u003d qEx1, W2 \u003d qEx2. The field work, in accordance with formula (8), is equal to the change with a minus sign

This value is the potential energy of the charge in a uniform electrostatic field. The x indicates the abscissa of the point at which the potential energy is sought. In this case, the zero level of potential energy corresponds to the origin x \u003d 0 and is shown in the figures by a dashed line perpendicular to the tension lines4.

Recall that while q\u003e 0 is considered. It follows from formula (9) that when the charge moves along the field line, the potential energy decreases with increasing x. This is natural: after all, the field does a positive job of dispersing the charge, and the kinetic energy of the charge increases due to a decrease in its potential energy.

It is easy to show that formula (9) remains valid for q< 0. В этом случае потенциальная энергия возрастает с ростом x. Это тоже понятно: ведь сила, с которой поле действует на заряд, теперь будет направлена влево, так что движение заряда вправо будет осуществляться против действия поля. Заряд тормозится полем, кинетическая энергия заряда уменьшается, а потенциальная энергия увеличивается.

So, an important conclusion: in the formula for potential energy, q denotes the algebraic value of the charge (taking into account the sign), and not its module.

4 In fact, zero potential energy can be chosen anywhere. In other words, the potential energy is determined only up to an arbitrary additive constant C, i.e., W \u003d qEx + C. There is nothing wrong with such uncertainty: the potential energy itself has a physical meaning, and the difference of potential energies is equal to the work of the field. In this difference, the constant C decreases.

4.4 Potential energy of interaction of point charges

Let two point charges q1 and q2 be in vacuum at a distance r from each other. It can be shown that the potential energy of their interaction is given by the formula:

		kq1 q2

We accept formula (10) without proof. Two features of this formula should be discussed.

First, where is the zero level of potential energy? After all, potential energy, as can be seen from formula (10), cannot vanish. But in fact, the zero level exists, and it is at infinity. In other words, when the charges are located infinitely far from each other, the potential energy of their interaction is assumed to be zero (which is logical in this case, the charges already “do not interact”).

Secondly, q1 and q2 are again the algebraic values \u200b\u200bof the charges, i.e., charges taking into account their sign.

For example, the potential energy of interaction of two charges of the same name will be positive. Why? If we let them go, they will start to accelerate and move away from each other. Their kinetic energy increases, so the potential energy decreases. But at infinity, the potential energy vanishes, and since it decreases to zero, then it is positive.

But the potential energy of interaction of opposite charges is negative. Indeed, let's remove them at a very great distance from each other so that the potential energy is zero and let go. The charges will begin to accelerate, drawing closer, and the potential energy decreases again. But if she was zero, then where did she go? Only towards negative values.

Formula (10) also helps to calculate the potential energy of a system of charges if the number of charges is more than two. To do this, sum the energies of each pair of charges. We will not write out a general formula; we better illustrate what has been said with a simple example, shown in Fig. 19.

Fig. 19. The interaction of the three charges

If the charges q1, q2, q3 are at the vertices of the triangle with sides a, b, c, then the potential energy of their interaction is:

kq1 q2	kq2 q3	kq1 q3

4.5 potential

From the formula W \u003d qEx, we see that the potential energy of the charge q in a homogeneous field is directly proportional to this charge.

We see the same thing from the formula W \u003d kq1 q2 \u003d r: the potential energy of a charge q1 located in the field of a point charge q2 is directly proportional to the value of charge q1.

If an electric body acts on electrically charged bodies, then it is able to do the work of moving charged bodies. The electrostatic field created by a point charge is central, that is, the force acting on a point charge in such a field is directed along a straight line connecting the charge source and the test charge. Earlier, we showed that any central force is potential, that is, the work of this force does not depend on the shape of the trajectory, but is determined only by the initial and final position of the body.

We briefly recall the proof of this most important statement. Let the point test charge q move in the central field created by the motionless charge Q (Fig. 174). The force acting on the test charge is determined by the law of Coulomb

Where is the vector drawn from the charge of the source Q to point A, at which the test charge is located. When a charge moves along arcs of circles centered on a charge Q (for example, along arcs AB, CD), the work of electric force is zero, so the force and displacement vectors are mutually perpendicular. When moving in the radial direction (for example, along the segments BC, DE), the operation depends only on the initial and final distances to the source charge. So the work of the electrostatic field when moving along the segments DE and D1E1 are obviously equal. The most beautiful proof of this statement is related to the symmetry of the field - let's turn our system around the axis passing through the source so that the segment D1E1 coincides with the segment DE - the distribution of the field does not change, why should the field work?

Since the principle of superposition is valid for electrostatic field strength, any electrostatic field is potential. Indeed, let a point charge q be in an electric field created by a system of fixed point charges Q1, Q2, ..., QN. When the charge moves to a small displacement vector, by definition, the electric field will do the job where

The resulting force acting on the moving charge q is equal to the sum of the forces acting on the side of each of the stationary point charges Qk. The work of this force can be calculated by the formula

In order to calculate the work along the final section of the trajectory, it is necessary to divide the path into small sections (Fig. 175), then using formula (1) to calculate the work in each small section, and then sum them up

. (2) In fact, this amount is double, since each resulting force is the sum of the forces in accordance with formula (1). Note that in formula (2), the resulting force changes, as it is calculated at different points on the trajectory.

As we showed earlier, the work of the electric field of a point charge does not depend on the shape of the trajectory, that is, each term from formula (1) does not depend on the shape of the trajectory, therefore, the entire sum does not depend on the shape of the trajectory. Thus, any electrostatic field is potential.

Therefore, for a point charge located in an electrostatic field, one can introduce the potential interaction energy U (x, y, z). This function has the following physical meaning: the electric field when moving a point charge from one point with coordinates (x1, y1, z1) to another, with coordinates (x2, y2, z2) is equal to the change in potential energy taken with the opposite sign:

. (3) The change of sign in this definition is quite logical: if the electric field has done a positive job (A\u003e 0), then its energy decreases (ΔU< 0). Для вычисления работы силы взаимодействия между двумя точечными заряженными телами достаточно подсчитать эту работу при движении вдоль радиального отрезка при изменении расстояния от r1 до r2 (Рис. 176). Если построить зависимость силы взаимодействия между зарядами от расстояния r между телами, тогда площадь под графиком этой зависимости в указанных пределах и будет равна искомой работе (Рис. 177). Зависимость силы электростатического взаимодействия от расстояния аналогична силе гравитационного взаимодействия, с одним существенным отличием: гравитационная сила всегда есть сила притяжения, а электрическая может быть как силой притяжения, так и силой отталкивания. В частности два положительных заряда отталкиваются. Поэтому выражение для работы электрического поля, будет аналогично формуле для работы гравитационной силы, но иметь противоположный знак

This work is equal to a decrease in the potential energy of interaction, that is, from this expression we can determine the expression for the potential energy of the electrostatic interaction of two point charges. (4) With this definition, the potential energy of interaction of two charges of the same sign is positive and tends to zero at an infinite distance between the bodies. The force of interaction of charges of opposite signs is directed in the opposite direction, so the work of this force with increasing distance between charges will be negative. However, we do not need to make any additional reservations, since formula (4) automatically takes into account the signs of charges - if the charges are opposite, then their product (respectively, energy) is negative. The sign of the potential energy of interaction of charges has a very clear meaning. The charges of the same sign repel, therefore, when they “scatter” over an infinitely large distance, the electric field will do a positive job - therefore, initially the system of these charges has the ability to do the job, therefore its energy is positive, when the charges move away from each other, their energy decreases to zero. Charges of opposite signs are attracted, in order to remove them to an infinitely large distance, external forces must do positive work. In this case, the energy of a pair of charges should increase, therefore, initially it is negative, and when the charges are removed from each other, it increases to zero. In general, the usual situation - attraction corresponds to negative energy, and repulsion - positive. We only note that such evidence is valid only when choosing a zero level of potential energy at infinity. Formula (4) determines the potential energy of interaction of two point charged bodies. The values \u200b\u200bof the charges of the bodies Q and q enter, as one would expect, in this formula symmetrically. The division of charges into a source charge and a test charge is conditional, they can be completely interchanged. Therefore, this formula is preferable to write in a symmetric form: the interaction energy of two point charges q1 and q2 is equal to, (5) and it makes sense to work done by the field with increasing distance between charges from r to infinity, regardless of whether the first charge moves, or the second , or both charges move, finally, regardless of the trajectories of motion of both charges. Further, it cannot be said which particular charge this energy “belongs” to, we will show later that the interaction energy of the charges is part of the energy of the electrostatic field itself, that is, it is “spread” throughout the space where the field created by these charges exists. If the system consists of more than two charges, then to calculate the interaction energy of these charges, it is necessary to sum the interaction energies of all pairs of charges

here Uik is the interaction energy of the charges qi and qk located at a distance rik from each other (Fig. 178).

40 Question:

Electrostatic field - el. field of motionless charge.

The fel acting on the charge moves it, making a rabo. In a uniform electric field, Fel \u003d qE is a constant

The work of the field (electric power) does not depend on the shape of the trajectory and on a closed path \u003d zero.

POTENTIAL ENERGY OF A CHARGED BODY IN A HOMOGENEOUS ELECTROSTATIC FIELD

Electrostatic energy is the potential energy of a system of charged bodies (because they interact and are able to do the job).

Since the work of the field does not depend on the shape of the trajectory, then at the same time

comparing the work formulas, we obtain the potential charge energy in a uniform electrostatic field

If the field does a positive job (along the lines of force), then the potential energy

charged body decreases (but according to the law of conservation of energy, kinetic energy increases) and vice versa.

ELECTROSTATIC FIELD POTENTIAL

Energy characteristic el. fields.

Equal to the ratio of the potential energy of a charge in the field to this charge.

The scalar quantity that determines the potential energy of the charge at any point el. fields.

The potential value is considered relative to the selected zero level.

DP (or otherwise VOLTAGE)

This is the potential difference at the initial and final points of the charge path.

The voltage between two points (U) is equal to the potential difference of these points and is equal to the work of the field in moving a unit charge.

COMMUNICATION BETWEEN FIELD TENSION AND DIFFERENCE OF POTENTIALS