Energy of the electric field of the system of charges. Electric energy of the system of charges

An energetic approach to interaction. The energetic approach to the interaction of electric charges is, as we will see, very fruitful in its practical applications, and in addition, it opens up the possibility of looking differently at the electric field itself as a physical reality.

First of all, we will find out how you can come to the concept of the interaction energy of a system of charges.

1. First, consider a system of two point charges 1 and 2. Let us find the algebraic sum of the elementary workings of the forces F, and F2, with which these charges interact. Suppose that in some K-frame in time cU the charges have made displacements dl, and dl 2. Then the corresponding work of these forces

6Л, 2 \u003d F, dl, + F2 dl2.

Considering that F2 \u003d - F, (according to Newton's third law), we rewrite the previous expression: Mlj, \u003d F, (dl1-dy.

The value in brackets is the movement of charge 1 relative to charge 2. More precisely, it is the movement of charge / in / ("- a frame of reference rigidly connected with charge 2 and moving along with it translationally relative to the original / (- system. Indeed, the movement dl, charge 1 in / (- the system can be represented as displacement dl2 / ("- system plus displacement dl, charge / relative to this / (" - system: dl, \u003d dl2 + dl,. Hence dl, - dl2 \u003d dl " and

So, it turns out that the sum of elementary work in an arbitrary / (- frame of reference is always equal to the elementary work performed by the force acting on one charge, in the frame of reference, where the other charge is at rest. In other words, the work 6L12 does not depend on the choice of the initial / ( - reference systems.

Force F „acting on the charge / from the side of charge 2, conservative (as a central force). Therefore, the work of this force on displacement dl can be represented as a decrease in the potential energy of charge 1 in the field of charge 2 or as a decrease in the potential energy of interaction of the considered pair of charges:

where 2 is a value that depends only on the distance between these charges.

2. Now let us turn to a system of three point charges (the result obtained for this case can be easily generalized to a system of an arbitrary number of charges). The work done by all forces of interaction during elementary displacements of all charges can be represented as the sum of the work of all three pairs of interactions, that is, 6L \u003d 6L (2 + 6L, 3 + 6L 2 3. But for each pair of interactions, as soon as which was shown, ik \u003d - d Wik, therefore

where W is the interaction energy of a given system of charges,

W «\u003d wa + Wt3 + w23.

Each term of this sum depends on the distance between the corresponding charges, so the energy W

a given system of charges is a function of its configuration.

Similar reasoning is obviously valid for a system of any number of charges. Hence, it can be argued that each configuration of an arbitrary system of charges has its own energy value W and the work of all interaction forces when this configuration changes is equal to the loss of energy W:

bl \u003d -ag. (4.1)

Energy of interaction. Let us find an expression for the energy W. First, consider again a system of three point charges, for which we have shown that W \u003d - W12 + ^ 13 + ^ 23- We transform this sum as follows. We represent each term Wik in symmetric form: Wik \u003d] / 2 (Wlk + Wk), since Wik \u003d Wk, Then

Let's group members with the same first indices:

Each sum in parentheses is the energy Wt of interaction of the i-th charge with the rest of the charges. Therefore, the last expression can be rewritten as follows:

Generalization of an arbitrary

the obtained expression for a system of charges is obvious, for it is clear that the above reasoning is completely independent of the number of charges that make up the system. So, the interaction energy of the system of point charges

Bearing in mind that Wt \u003d<7,9, где qt - i-й заряд системы; ф,- потенциал, создаваемый в месте нахождения г-го заряда всеми остальными зарядами системы, получим окончательное выражение для энергии взаимодействия системы точечных зарядов:

Example. Four identical point charges q are located at the vertices of a tetrahedron with an edge a (Fig. 4.1). Find the interaction energy of the charges of this system.

The interaction energy of each pair of charges is here the same and equal to \u003d q2 / Ale0a. There are six such interacting pairs in total, as can be seen from the figure, therefore the interaction energy of all point charges of this system is

W \u003d 6 #, \u003d 6<72/4яе0а.

Another approach to solving this problem is based on the use of formula (4.3). The potential φ at the location of one of the charges, due to the field of all other charges, is equal to φ \u003d 3<7/4яе0а. Поэтому

Total energy of interaction. If the charges are distributed continuously, then, expanding the system of charges into a set of elementary charges dq \u003d р dV and passing from summation in (4.3) to integration, we obtain

where φ is the potential created by all charges of the system in an element of volume dV. A similar expression can be written for the distribution of charges, for example, over a surface; for this it is sufficient to replace p by o and dV by dS in formula (4.4).

One may mistakenly think (and this often leads to misunderstandings) that expression (4.4) is only a modified expression (4.3), corresponding to the replacement of the concept of point charges by the concept of a continuously distributed charge. In reality, this is not the case - both expressions differ in their content. The origin of this difference is in a different sense of the potential φ included in both expressions, which is best explained by the following example.

Let the system consist of two balls with charges q, and q2 "The distance between the balls is much larger than their sizes, so the charges ql and q2 can be considered pointwise. Let us find the energy W of this system using both formulas.

According to formula (4.3)

W \u003d "AUitPi +2\u003e where, φ [is the potential created by the charge q2 at the place

finding a charge has a similar meaning

and potential f2.

According to formula (4.4), we must break the charge of each ball into infinitesimal elements p AV and multiply each of them by the potential φ, created not only by the charges of another ball, but also by the elements of the charge of this ball. It is clear that the result will be completely different, namely:

W \u003d Wt + W2 + Wt2, (4.5)

where Wt is the energy of interaction with each other of the charge elements of the first ball; W2 - the same, but for the second ball; Wi2 is the interaction energy of the charge elements of the first ball with the charge elements of the second ball. The energies W, and W2 are called the intrinsic energies of charges qx and q2, and W12 is the energy of interaction of the charge with the charge q2.

Thus, we see that the calculation of the energy W according to the formula (4.3) gives only Wl2, and the calculation according to the formula (4.4) gives the total interaction energy: in addition to W (2, also the intrinsic energies IF, and W2. Ignoring this circumstance is often a source gross mistakes.

We will return to this question in § 4.4, and now we will obtain several important results using formula (4.4).

Electric energy of the system of charges.

Field work with dielectric polarization.

Electric field energy.

Like all matter, an electric field has energy. Energy is a function of the state, and the state of the field is given by the intensity. Whence it follows that the energy of the electric field is an unambiguous function of the strength. Since, it is extremely important to introduce the concept of energy concentration in the field. A measure of the concentration of the field energy is its density:

Let's find an expression for. Consider for this the field of a flat capacitor, assuming it to be uniform everywhere. The electric field in any capacitor arises during its charging, which can be represented as the transfer of charges from one plate to another (see figure). Elementary work͵ spent on charge transfer is equal to:

where, and the complete work:

which goes to increase the field energy:

Taking into account that (there was no electric field), for the energy of the electric field of the capacitor we obtain:

In the case of a flat capacitor:

since, is the volume of the capacitor equal to the volume of the field. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the energy density of the electric field is:

This formula is valid only in the case of an isotropic dielectric.

The energy density of an electric field is proportional to the square of the strength. This formula, although obtained for a uniform field, is valid for any electric field. In general, the field energy can be calculated by the formula:

The expression includes the dielectric constant. This means that the energy density in a dielectric is greater than in a vacuum. This is due to the fact that when creating a field in the dielectric, additional work is performed, associated with the polarization of the dielectric. Let us substitute the value of the electric induction vector into the expression for the energy density:

The first term is associated with the field energy in vacuum, the second - with the work expended on the polarization of a unit volume of the dielectric.

The elementary work spent by the field on the increment of the polarization vector is equal to.

Work on the polarization of a unit volume of a dielectric is:

since, as required.

Consider a system of two point charges (see figure) according to the principle of superposition at any point in space:

Electric field energy density

The first and third terms are associated with the electric fields of charges and, respectively, and the second term reflects the electrical energy associated with the interaction of charges:

The intrinsic energy of the charges is positive, and the interaction energy can be both positive and negative.

Unlike a vector, the energy of an electric field is not an additive quantity. The energy of interaction can be represented by a simpler relationship. For two point charges, the interaction energy is:

which can be represented as the sum:

where is the potential of the charge field at the location of the charge, and is the potential of the charge field at the location of the charge.

Generalizing the result obtained for a system of an arbitrary number of charges, we get:

where is the system charge, is the potential created at the location of the charge, all the rest system charges.

If the charges are distributed continuously with the bulk density, the sum should be replaced by the volume integral:

where is the potential created by all charges of the system in an element of volume. The resulting expression matches total electrical energy systems.

The work of an electric field to move a charge

Work concept A electric field E charge movement Q is introduced in full accordance with the definition of mechanical work:

where - potential difference (the term voltage is also used)

In many problems, continuous charge transfer is considered for some time between points with a given potential difference U(t), in this case the formula for work should be rewritten as follows:

where is the current

Electric current power in the circuit

Power W electric current for a section of the circuit is determined in the usual way, as a derivative of the work A by time, that is, by the expression:

This is the most common expression for power in an electrical circuit.

Taking into account Ohm's law:

Electrical power dissipated on resistance R can be expressed as through the current: ,

Accordingly, work (released heat) is the integral of power over time:

Energy of electric and magnetic fields

For electric and magnetic fields, their energy is proportional to the square of the field strength. It should be noted that, strictly speaking, the term electromagnetic field energy is not entirely correct. Calculation of the total energy of the electric field of even one electron leads to a value equal to infinity, since the corresponding integral (see below) diverges. The infinite energy of the field of a completely finite electron is one of the theoretical problems of classical electrodynamics. Instead, physics usually uses the concept energy density of the electromagnetic field (at a certain point in space). The total energy of the field is equal to the integral of the energy density over the entire space.

The energy density of the electromagnetic field is the sum of the energy densities of the electric and magnetic fields.

In the SI system:

where E - electric field strength, H - magnetic field strength, - electric constant, and - magnetic constant. Sometimes for constants and - the terms dielectric constant and magnetic permeability of vacuum are used - which are extremely unfortunate, and are now almost never used.

Energy flows of the electromagnetic field

For an electromagnetic wave, the energy flux density is determined by the Poynting vector S (in the Russian scientific tradition - the Umov-Poynting vector).

In the SI system, the Poynting vector is:,

The vector product of the strengths of the electric and magnetic fields, and is directed perpendicular to the vectors E and H... This naturally agrees with the transverse property of electromagnetic waves.

At the same time, the formula for the energy flux density can be generalized for the case of stationary electric and magnetic fields, and has exactly the same form:.

The very fact of the existence of energy flows in constant electric and magnetic fields, at first glance, looks very strange, but this does not lead to any paradoxes; moreover, such flows are found experimentally.

· The potential of an electric field is a value equal to the ratio of the potential energy of a point positive charge, placed at a given point of the field, to this charge

or the potential of an electric field is a value equal to the ratio of the work of the field forces to move a point positive charge from a given point of the field to infinity to this charge:

The potential of the electric field at infinity is conventionally assumed to be zero.

Note that when moving a charge in an electric field, the work A c. C external forces is equal in modulus to work A s.p field strength and is opposite to it in sign:

A c.c. \u003d - A c.p.

· Electric field potential created by a point charge Q on distance rfrom the charge,

· The potential of an electric field created by a metal carrying charge Q sphere of radius R, on distance rfrom the center of the sphere:

inside the sphere ( r<R) ;

on the surface of the sphere ( r=R) ;

out of scope (r\u003e R) .

In all the formulas given for the potential of a charged sphere, e is the dielectric constant of a homogeneous unlimited dielectric surrounding the sphere.

· The potential of the electric field generated by the system ppoint charges, at a given point in accordance with the principle of superposition of electric fields is equal to the algebraic sum of potentials j 1, j 2, ... , j ngenerated by individual point charges Q 1, Q 2, ..., Q n:

· Energy W interaction of the system of point charges Q 1, Q 2, ..., Q n is determined by the work that this system of charges can perform when they move away from each other to infinity, and is expressed by the formula

where is the potential of the field created by all p-1 charges (excluding ith) at the point where the charge is located Q i.

· The potential is related to the electric field strength by the relation

In the case of an electric field with spherical symmetry, this relationship is expressed by the formula

or in scalar form

and in the case of a homogeneous field, i.e., a field whose strength at each point is the same both in magnitude and in direction

where j 1 and j 2 - potentials of points of two equipotential surfaces; d -the distance between these surfaces along the electric line of force.

· Work done by an electric field when moving a point charge Q from one point of the field with potential j 1, to another one with potential j 2

A=Q ∙(j 1 - j 2), or

where E l - the projection of the tension vector on the direction of movement; dl - moving.

In the case of a uniform field, the last formula takes the form

A \u003d Q ∙ E ∙ l ∙ cosa,

where l - moving; a - the angle between the directions of the vector and displacement.

A dipole is a system of two point electric charges equal in size and opposite in sign, the distance lbetween which there is much less distance rfrom the center of the dipole to the observation points.

The vector drawn from the negative charge of the dipole to its positive charge is called the arm of the dipole.

Charge product | Q| a dipole on its shoulder is called the electric moment of the dipole:

Dipole field strength

where r- the electric moment of the dipole; r- the modulus of the radius vector drawn from the center of the dipole to the point, the field strength in which we are interested; α is the angle between the radius vector and the arm of the dipole.

Dipole field potential

Mechanical moment acting on a dipole with an electric moment, placed in a uniform electric field of strength

or M \u003d p ∙ E ∙sin,

where α is the angle between the directions of the vectors and.

In an inhomogeneous electric field, in addition to a mechanical moment (a pair of forces), a certain force acts on the dipole. In the case of a field with symmetry about the axis x, the force is expressed by the ratio

where is the partial derivative of the field strength characterizing the degree of field inhomogeneity in the direction of the axis x.

With strength F x is positive. This means that, under the action of its action, the dipole is drawn into the region of a strong field.

Potential energy of a dipole in an electric field

Consider a system of two point charges (see figure) according to the principle of superposition at any point in space:

Electric field energy density

The first and third terms are related to the electric fields of charges and accordingly, and the second term reflects the electrical energy associated with the interaction of charges:

The self-energy of charges is positive
, and the interaction energy can be both positive and negative
.

Unlike vector the energy of the electric field is not an additive quantity. The energy of interaction can be represented by a simpler relationship. For two point charges, the interaction energy is:

which can be represented as the sum:

where
- potential of the charge field at the location of the charge , and
- potential of the charge field at the location of the charge .

Generalizing the result obtained for a system of an arbitrary number of charges, we get:

where -
system charge, - potential created at the location
charge, everyone elsesystem charges.

If the charges are distributed continuously with a bulk density , the sum should be replaced by a volume integral:

where is the potential created by all charges of the system in an element of volume
... The resulting expression matches total electrical energysystems.

Examples.

Charged metal ball in a homogeneous dielectric.

Using this example, we will find out why the electric forces in a dielectric are less than in a vacuum and calculate the electric energy of such a ball.

H the field strength in the dielectric is less than the strength in vacuum in time
.

This is due to the polarization of the dielectric and the appearance at the surface of the conductor of a bound charge opposite sign of the charge of the conductor (see figure). Associated charges screen the field of free charges reducing it everywhere. The electric field strength in the dielectric is equal to the sum
where
- field strength of free charges,
- field strength of bound charges. Considering that
, we find:

→
→

→
→
→

Dividing by the surface area of \u200b\u200bthe conductor, we find the relationship between the surface density of bound charges
and surface density of free charges :

The resulting ratio is suitable for a conductor of any configuration in a homogeneous dielectric.

Let's find the energy of the electric field of the ball in the dielectric:

It is taken into account here that
, and the elementary volume, taking into account the spherical symmetry of the field, is chosen in the form of a spherical layer. - the capacity of the ball.

Since the dependence of the electric field strength inside and outside the ball on the distance to the center of the ball r is described by various functions: