The area of economics that covers resources, extraction, transformation and use various types energy.
Energy can be represented by the following interconnected blocks:
1. Natural energy resources and mining enterprises;
2. Processing plants and transportation of finished fuel;
3. Generation and transmission of electrical and thermal energy;
4. Consumers of energy, raw materials and products.
Brief content of the blocks:
1) Natural resources are divided into:
renewable (sun, biomass, hydro resources);
non-renewable (coal, oil);
2) Extractive enterprises (mines, mines, gas rigs);
3) Fuel processing enterprises (enrichment, distillation, fuel purification);
4) Transportation of fuel ( railway, tankers);
5) Generation of electrical and thermal energy (CHP, nuclear power plant, hydroelectric power station);
6) Transfer of electrical and thermal energy ( electrical networks, pipelines);
7) Consumers of energy and heat (power and industrial processes, heating).
The part of the energy sector that deals with the problems of obtaining large quantities of electricity, transmitting it over a distance and distributing it between consumers, its development is carried out at the expense of electric power systems.
This is a set of interconnected power stations, electrical and thermal systems, as well as consumers of electrical and thermal energy, united by the unity of the process of production, transmission and consumption of electricity.
Electric power system: CHPP - combined heat and power plant, NPP - nuclear power plant, IES - condensing power plant, 1-6 - consumers of electricity CHPP
Scheme of a thermal condensing power plant
Electrical system (electrical system, ES)- the electrical part of the electrical power system.
The diagram is shown in a single-line diagram, i.e. by one line we mean three phases.
Technological process in the energy system
A technological process is the process of converting a primary energy resource (fossil fuel, hydropower, nuclear fuel) into final products (electric energy, thermal energy). Parameters and indicators technological process determine production efficiency.
The technological process is shown schematically in the figure, from which it can be seen that there are several stages of energy conversion.
Scheme of the technological process in the power system: K - boiler, T - turbine, G - generator, T - transformer, power line - power lines
In boiler K, the fuel combustion energy is converted into heat. A boiler is a steam generator. In the turbine thermal energy transforms into mechanical. In a generator, mechanical energy is converted into electrical energy. Voltage electrical energy in the process of its transmission along power lines from the station to the consumer, it is transformed, which ensures economical transmission.
The efficiency of the technological process depends on all these links. Consequently, there is a complex of operational tasks associated with the operation of boilers, thermal power plant turbines, hydroelectric power plant turbines, nuclear reactors, electrical equipment (generators, transformers, power lines, etc.). It is necessary to select the composition of the operating equipment, the mode of its loading and use, and comply with all restrictions.
Electrical installation- installation in which electricity is produced, generated or consumed, distributed. Can be: open or closed (indoors).
Electric station- a complex technological complex in which the energy of a natural source is converted into energy electric current or heat.
It should be noted that power plants (especially thermal, coal-fired ones) are the main sources of pollution environment energy.
Electrical substation- an electrical installation designed to convert electricity from one voltage to another at the same frequency.
Power transmission (power lines)- the structure consists of elevated power transmission line substations and step-down substations (a system of wires, cables, supports) designed to transmit electricity from source to consumer.
Electrical networks- a set of power lines and substations, i.e. devices connecting the power supply to the .
Consider a system of two point charges(see figure) according to the principle of superposition at any point in space:
.
Energy Density electric field
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 self-energy of the charges is a positive value 
, and the interaction energy can be either positive or negative 
.
Unlike the vector 
Electric field energy is a non-additive quantity. The interaction energy can be represented by a simpler relationship. For two point charges, the interaction energy is equal to:
,
which can be represented as the sum:
Where 
- charge field potential 
at the location of the charge 
, A 
- charge field potential 
at the location of the charge 
.
Generalizing the result obtained to a system of an arbitrary number of charges, we obtain:
,
Where 
-
system charge, 
- potential created at the location 
charge, everyone else system charges.
If the charges are distributed continuously with the volume density 
, the sum should be replaced by the volume integral:
,
Where 
- potential created by all charges of the system in an element with volume 
. The resulting expression corresponds to total electrical energy systems.
Examples.
Charged metal ball in a homogeneous dielectric.
Using this example, we will find out why the electrical forces in a dielectric are less than in a vacuum and calculate the electrical energy of such a ball.
N 
The field strength in a dielectric is less than the strength in a vacuum in 
once 
.
This is due to the polarization of the dielectric and the appearance of a bound charge at the surface of the conductor 
opposite charge of the conductor 
(see picture). Associated charges 
screen the field of free charges 
, reducing it everywhere. The electric field strength in a 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 the conductor, we find the relationship between the surface density of bound charges 
and surface density of free charges 
:
.
The resulting relationship 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. 
– 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 is described by various functions:
The calculation of energy is reduced to the sum of two integrals:
.
Note that bound charges arise on the surface and in the volume of the dielectric ball:
,
,
Where 
- volumetric density of free charges in the ball.
Conduct the proof yourself using connections 
,
and Gauss's theorem 
.
The self-energy of each shell is equal respectively (see example 1.):
,
,
and the interaction energy of the shells:
.
The total energy of the system is:
.
If the shells are charged with equal charges of opposite sign 
(spherical capacitor), the total energy will be equal to:
Where 
- capacity of the spherical capacitor.
The voltage applied to the capacitor is:
,
Where 
And 
- electric field strength in layers.
Electrical induction in layers:
- surface density free charges on the capacitor plates.
Considering the connection 
from the definition of capacity, we get:
.
The resulting formula is easily generalized to the case of a multilayer dielectric:
.
Within electrostatics, it is impossible to answer the question of where the energy of a capacitor is concentrated. The fields and the charges that formed them cannot exist separately. They cannot be separated. However, alternating fields can exist regardless of the charges that excite them (solar radiation, radio waves, ...), and they transfer energy. These facts force us to admit that the carrier of energy is the electrostatic field .
When moving electric charges Coulomb interaction forces do a certain amount of work d A. The work done by the system is determined by the decrease in interaction energy -d W charges
| . | (5.5.1) |
Interaction energy of two point charges q 1 and q 2 located at a distance r 12, is numerically equal to the work of moving the charge q 1 in the field of a stationary charge q 2 from point with potential to point with potential:
| . | (5.5.2) |
It is convenient to write down the interaction energy of two charges in a symmetric form
 .
|
(5.5.3) |
For a system from n point charges (Fig. 5.14) due to the principle of superposition for the potential, at the point where k-th charge, we can write:
Here φ k , i- potential i-th charge at the location point k-th charge. In total, the potential φ is excluded k , k, i.e. The effect of the charge on itself, which is equal to infinity for a point charge, is not taken into account.
Then the mutual energy of the system n charges is equal to:
 |
(5.5.4) |
This formula is valid only if the distance between the charges significantly exceeds the size of the charges themselves.
Let's calculate the energy of a charged capacitor. The capacitor consists of two, initially uncharged, plates. We will gradually remove charge d from the bottom plate q and transfer it to the top plate (Fig. 5.15).
As a result, a potential difference will arise between the plates. When transferring each portion of charge, elementary work is performed
Using the definition of capacity we get
General work, spent on increasing the charge of the capacitor plates from 0 to q, is equal to:
This energy can also be written as
Field work during dielectric polarization.
Electric field energy.
Like any matter, electric field has energy. Energy is a function of state, and the state of the field is given by strength. Whence it follows that the energy of the electric field is an unambiguous function of intensity. Since, it is necessary to introduce the idea of energy concentration in the field. A measure of the field energy concentration is its density:
Let's find an expression for. For this purpose, let us consider the field of a flat capacitor, considering it uniform everywhere. An electric field in any capacitor arises during the process of charging, which can be represented as the transfer of charges from one plate to another (see figure). The elementary work spent on charge transfer is:
where and the complete work:
which goes to increase the field energy:
Considering that (there was no electric field), for the energy of the electric field of the capacitor we obtain:
In the case of a parallel plate capacitor:
since, - the volume of the capacitor is equal to the volume of the field. Thus, the energy density of the electric field is equal to:
This formula is valid only in the case of an isotropic dielectric.
The energy density of the electric field is proportional to the square of the intensity. This formula, although obtained for a uniform field, is true for any electric field. In general, the field energy can be calculated using the formula:
The expression includes dielectric constant. This means that in a dielectric the energy density is greater than in a vacuum. This is due to the fact that when a field is created in a dielectric, additional work is performed associated with the polarization of the dielectric. Let us substitute the value of the electrical induction vector into the expression for 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.
The work of polarization per unit volume of a dielectric is equal to:
since that is what needed to be proven.
Let's 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 self-energy of charges is positive, and the interaction energy can be either positive or negative.
Unlike a vector, the energy of an electric field is not an additive quantity. The interaction energy can be represented by a simpler relationship. For two point charges, the interaction energy is equal to:
which can be represented as the sum:
where is the charge field potential at the location of the charge, and is the charge field potential at the location of the charge.
Generalizing the result obtained to a system of an arbitrary number of charges, we obtain:
where is the charge of the system, is the potential created at the location of the charge, everyone else system charges.
If the charges are distributed continuously with volume density, the sum should be replaced by the volume integral:
where is the potential created by all charges of the system in an element with volume. The resulting expression corresponds to total electrical energy systems.
Energy approach to interaction. The energy 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 opportunity to take a different look at the electric field itself as a physical reality.
First of all, we will find out how we 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’s find the algebraic sum of the elementary works of forces F and F2 with which these charges interact. Let in some K-frame of reference during the time cU the charges have made movements dl, and dl 2. Then the corresponding work of these forces
6L, 2 = F, dl, + F2 dl2.
Considering that F2 = - F, (according to Newton’s third law), we rewrite the previous expression: Mlj, = F,(dl1-dy.
The value in parentheses is the movement of charge 1 relative to charge 2. More precisely, this is the movement of charge / in the /("-frame of reference, rigidly connected with charge 2 and moving with it translationally with respect to the original /(-system. Indeed, movement dl, charge 1 in the /(-system can be represented as the displacement dl2 of the /("-system plus the displacement dl, charge / relative to this /("-system: dl, = dl2+dl,. Hence dl, - dl2 = dl" , And
So, it turns out that the sum of elementary work in an arbitrary /(-reference frame is always equal to the elementary work performed by the force acting on one charge in a reference frame where the other charge is at rest. In other words, the work 6L12 does not depend on the choice of the initial /( -reference systems.
The force F„ acting on the charge / from the side of charge 2 is conservative (as the 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 pair of charges under consideration:
where 2 is a value that depends only on the distance between these charges.
2. Now let's move on 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 that all interaction forces do during elementary movements of all charges can be represented as the sum of the work of all three pairs of interactions, i.e. 6A = 6A (2 + 6A, 3 + 6A 2 3. But for each pair of interactions, as soon as what was shown is 6L ik = - d Wik, therefore
where W is the interaction energy of a given system of charges,
W «= wa + Wtз + w23.
Each term of this sum depends on the distance between the corresponding charges, so the energy W
of a given system of charges is a function of its configuration.
Similar reasoning is obviously valid for a system of any number of charges. This means that we can say that each configuration of an arbitrary system of charges has its own energy value W and the work of all interaction forces when changing this configuration is equal to the decrease in energy W:
bl = -ag. (4.1)
Energy of interaction. Let's find an expression for the energy W. First, consider again a system of three point charges, for which we showed that W = - W12+ ^13+ ^23- Let's transform this sum as follows. Let us represent each term Wik in a symmetric form: Wik= ]/2(Wlk+ Wk), since Wik=Wk, Then
Let's group members with the same first indices:
Each sum in parentheses is the energy Wt of interaction of the ith charge with the remaining charges. Therefore, the last expression can be rewritten as follows:
Generalization of arbitrary
The resulting expression for the system from the number of charges is obvious, because it is clear that the arguments carried out are completely independent of the number of charges making up the system. So, the interaction energy of a system of point charges
Keeping in mind that Wt =<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 edge a (Fig. 4.1). Find the interaction energy of the charges of this system.
The interaction energy of each pair of charges here is the same and equals = 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 a given system
W = 6№, = 6<72/4яе0а.
Another approach to solving this issue 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 = 3<7/4яе0а. Поэтому
Total energy of interaction. If the charges are distributed continuously, then, decomposing the system of charges into a set of elementary charges dq = p dV and passing from summation in (4.3) to integration, we obtain
where f is the potential created by all charges of the system in an element with volume dV. A similar expression can be written for the distribution of charges, for example, over a surface; To do this, it is enough to replace p by o and dV by dS in formula (4.4).
One might mistakenly think (and this often leads to misunderstandings) that expression (4.4) is only a modified expression (4.3), corresponding to replacing the idea of point charges with the idea of a continuously distributed charge. In reality this is not so - both expressions differ in their content. The origin of this difference is in the different meaning of the potential φ included in both expressions, which is best explained with the following example.
Let the system consist of two balls with charges d and q2. The distance between the balls is much larger than their sizes, so the charges ql and q2 can be considered point charges. Let us find the energy W of this system using both formulas.
According to formula (4.3)
W= "AUitPi +2> where, f[ is the potential created by the charge q2 at the location
finding a charge has a similar meaning
and potential f2.
According to formula (4.4), we must divide 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 the other ball, but also by the charge elements of this ball. It is clear that the result will be completely different, namely:
W=Wt + W2+Wt2, (4.5)
where Wt is the energy of interaction of the charge elements of the first ball with each other; W2 - the same, but for the second ball; Wi2 is the energy of interaction between the charge elements of the first ball and the charge elements of the second ball. The energies W and W2 are called the intrinsic energies of the charges qx and q2, and W12 is the energy of charge-charge interaction q2.
Thus, we see that calculating the energy W using formula (4.3) gives only Wl2, and calculating using formula (4.4) gives the total interaction energy: in addition to W(2, also the own energies IF and W2. Ignoring this circumstance is often the source gross mistakes.
We will return to this issue in § 4.4, and now we will obtain several important results using formula (4.4).
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