The successful fulfillment of the tasks facing the Armed Forces largely depends on the effectiveness of the system of command and control of the military organization of the state. The bodies of state leadership include the President of the Russian Federation, the Federal Assembly, consisting of the State Duma and the Federation Council, the Government of the Russian Federation, and the Security Council. Military command and control bodies include specially created military bodies that manage the formation and functioning of the Armed Forces.

General leadership of the Armed Forces (and other military formations and bodies) of the Russian Federation is carried out by the Supreme Commander-in-Chief.

According to the Constitution and the Law "On Defense" it is the President of Russia. The president (from Latin - sitting in front) is the elected head of state in modern states with a republican form of government. In the Russian Federation, the post of President of the Republic was established in 1991. This marked a new stage in the development of Russian statehood.

The president received power from the hands of the people - he was elected on the basis of a universal, equal, direct secret ballot of the citizens of Russia. In accordance with his position, the President of the Russian Federation, like the presidents (heads) of other states, is the Supreme Commander-in-Chief of the Armed Forces. His powers as the Supreme Commander-in-Chief are determined by the Federal Law

"On Defense" (Articles 4 and 13).

In accordance with the Constitution of the Russian Federation (Article 80), the President of the Russian Federation is the head of state. By virtue of his position and the tasks assigned to him, the President of the Russian Federation acts as a guarantor of the security of the Russian Federation. Only the President of the Russian Federation is tasked with protecting the stability of the state as a whole, its sovereignty and state integrity, which creates conditions for the exercise of their powers in a normal constitutional regime by all other state authorities and officials.

Upon taking office in the oath taken by the President of the Russian

The Federation swears to "...protect the sovereignty and independence, security and integrity of the state...".

The performance of key functions in the system of state authorities and the defense of the country is ensured by the vesting of the President with extensive powers in the field of armed defense of the state.

In exercising his powers, the President determines the main directions of the military policy of the Russian Federation, among which the most important are the problems of creating, strengthening and improving the military organization, technical equipment of the Armed Forces, determining the prospects for the development of military equipment, and the mobilization capabilities of the state.

Among the powers of the President, there are many that most directly affect the combat readiness of the Armed Forces. It approves the military doctrine of the Russian Federation, concepts and plans for the construction and development of the Armed Forces, other troops and military formations. Only the President of the State, as the Supreme Commander-in-Chief, has been given the right to approve such a fundamental document as the Plan for the Use of the Armed Forces of the Russian Federation. No less important is the consideration and approval by the President of the Mobilization Plan of the Armed Forces, which in its essence is a directive document for the transfer of the entire state mechanism, and not just the Armed Forces, to functioning in wartime conditions.

The plan determines the procedure for the work of state authorities of Russia, constituent entities of the Russian Federation, local self-government and the country's economy in wartime. In the conditions of peace, the Federal State Program for the operational equipment of the territory of the Russian Federation is being prepared and approved by the President, it is planned to create stocks of material values ​​of the state and mobilization reserves. In addition, the President approves the Regulations on Territorial Defense and the Civil Defense Plan.

As head of state in his foreign policy activities, he negotiates and signs international treaties of the Russian Federation in the field of defense, including treaties on joint defense, collective security, reduction and limitation of armed forces and weapons, on the participation of the Armed Forces of the Russian Federation in peacekeeping operations and international security.

The President of the Russian Federation approves federal state programs for armaments and the development of the defense industrial complex. The President of the country also approves plans for the deployment on the territory of the Russian Federation of facilities with nuclear charges, as well as facilities for the elimination of weapons of mass destruction and nuclear waste. It also approves all programs of nuclear and other special tests. No such test can be carried out without the approval of the President.

All places of deployment (location) of formations and larger formations of the Armed Forces, other troops, as well as issues of their movement to other locations are considered by the President of the Russian Federation.

By exercising direct control over the Armed Forces, he approves the structure and composition of the Armed Forces, other troops, military formations up to and including unification, as well as the staffing of military personnel of the Armed Forces of the Russian Federation, other troops, military formations and bodies. The list of military posts filled by senior officers in the Armed Forces and other troops is approved by the President. He also has the right to appoint military personnel to these positions and to award them the highest officer ranks.

Most important documents , such as general military charters, provisions on the Battle Banner of a military unit, the Naval flag, the procedure for military service, military councils, military commissariats are approved by the President of the Russian Federation and are the laws of army and navy life.

Twice a year, the President issues decrees on the conscription of citizens for military service, as well as on the dismissal of conscripted military personnel from military service.

As the Supreme Commander-in-Chief of the Armed Forces, the President of the country has the right to promptly make the most important decisions related to the defense of the country and ensuring the security of citizens. He, in accordance with the Law of the Russian Federation on martial law, enacts normative legal acts of wartime and terminates them, forms and abolishes executive authorities for the period of wartime in accordance with the federal constitutional law on martial law. In the event of aggression against Russia or an immediate threat of aggression, the President of the Russian Federation issues an order to introduce martial law. It can be introduced throughout the entire country or in certain areas that have been attacked, threatened by attack, or that are of particular importance for the defense of the country.

Introducing martial law, the President gives special powers to public authorities, local governments and organizations. When martial law is introduced, special bodies of military administration may be created, the power of which extends to civilians. All bodies and officials are ordered to assist the military command in the use of the forces and means of the given territory for defense, security and order. Certain constitutional rights of citizens may be restricted (for example: freedom of assembly, demonstration, freedom of the press).

When martial law is introduced, the President of the Russian Federation immediately informs the Federation Council and the State Duma about this. The presidential decree on the introduction of martial law must be approved by the Federation Council.

The President of the Russian Federation, in accordance with federal laws, has the right to decide on the involvement of the Armed Forces, other troops and military formations in the performance of tasks using weapons not for their intended purpose.

Fulfilling his constitutional duties and the tasks assigned to him by the Federal Law "On Defense", the President of the Russian Federation - Supreme Commander of the Armed Forces of the Russian Federation ensures the preparation of the country to repel possible aggression, manages all aspects of the process of maintaining the army and navy of Russia in a combat-ready state corresponding to the level threats to the national security of the country.

The President of Russia forms and heads the Security Council of the Russian Federation. Its main functions are the development of proposals to ensure the protection of the constitutional order, state sovereignty, the territorial integrity of the country, participation, together with other bodies, in the development of the military policy of the Russian Federation. Thus, the President of the Russian Federation occupies an independent and extremely important place in the system of ensuring the security of the state and the armed protection of its citizens. Its powers are aimed at ensuring the coordinated interaction of all branches of power in protecting the state sovereignty of Russia, strengthening the defense capability of the Armed Forces. The President of the Russian Federation, as the Supreme Commander-in-Chief of the Armed Forces, has a special responsibility to the people for the state of the Russian Armed Forces and their readiness to defend their state and their people.

In the Russian Federation, in accordance with the Constitution of the Russian Federation, the representative and legislative body is the Federal Assembly, which consists of two chambers - the Federation Council and the State Duma. The Constitution and the Law "On Defense" clearly define the powers of the Federal Assembly in the field of defense.

Council of the Federation is the upper house of the Federal Assembly and

acts as a representative body of the subjects of the federation. To his charge

includes the approval of decrees of the President of the Russian Federation on the introduction of martial law and a state of emergency; as well as on the involvement of the Armed Forces, other troops, military formations and bodies with the use of weapons to perform tasks not for their intended purpose; resolving the issue of the possibility of using the Armed Forces of the Russian Federation outside the territory of the Russian Federation. The Federation Council considers defense spending established by federal laws on the federal budget adopted by the State Duma, as well as federal laws in the field of defense adopted by the State Duma.

The State Duma is a representative body of the entire population of the Russian Federation and consists of deputies elected by the citizens of the Russian Federation on the basis of universal, equal and direct suffrage by secret ballot.

The State Duma: considers defense spending established by federal laws on the federal budget; adopts federal laws in the field of defense, thereby regulating various aspects of the organization of defense and military development.

In addition to these powers, the Federation Council and the State Duma exercise parliamentary control in this area through their security and defense committees.

Government of the Russian Federation one of the main organs

exercise of state power in the Russian Federation. It heads the system of federal executive bodies.

1. In accordance with Article 114 of the Constitution of the Russian Federation, it takes measures to ensure the defense of the country and its security. The content of the Government's activities in this area is formulated in more detail in the Law of the Russian Federation "On Defense". According to this law, the Government: develops and submits to the State Duma proposals on defense spending in the federal budget;

2. organizes the provision of the Armed Forces of the Russian Federation, bodies with material means, energy and other resources and services on their orders;

3. organizes the development and implementation of state programs for armaments and the development of the defense industrial complex;

4. determines the conditions for the financial and economic activities of organizations of the Armed Forces;

5. organizes the development of the Federal State Program for the operational equipment of the country's territory for the purposes of defense and takes measures to implement this program;

6. determines the organization, tasks and carries out general planning of civil and territorial defense;

7. organizes control over the export of weapons and military equipment, strategic materials, technologies and dual-use products, etc.

In order to put into practice the decisions taken by state authorities relating to the Armed Forces, constant organizational work is necessary. This activity is carried out by special military bodies united in the system of the Ministry of Defense of the Russian Federation. The activity of these bodies is an integral part of the exercise of the federal executive power.

The system of military bodies exercising state power in the Armed Forces, i.e. military command, consists of:

– central authorities;

- management bodies of associations, military formations and units;

- military commissariats (local military authorities);

- chiefs of garrisons (senior naval commanders);

- military commanders.

As a specific type of governing bodies, it is customary to single out headquarters.

Headquarters- this is the main body in the hands of the corresponding commander (commander) for the operational control of the troops and fleet forces subordinate to him.

Direct leadership of the Russian Armed Forces is carried out by Minister of Defense through the Ministry of Defense and the General Staff.

The Minister of Defense is the direct head of all personnel of the Armed Forces of the Russian Federation. He is personally responsible for the fulfillment of the tasks assigned to the ministry. On the most important issues of life and activities of the Armed Forces of the Russian Federation, he issues orders and directives, and also puts into effect regulations, instructions, and other legal acts regulating various issues of life, life and activities of the troops.

The Ministry of Defense of the Russian Federation is involved in the development

proposals on issues of military policy and on the military doctrine of the Russian Federation, develops a concept for the development of the Armed Forces of the Russian Federation. It develops a federal state program for armaments and the development of military equipment, as well as proposals for a state defense order, for defense spending in the draft federal budget. Of great importance is the coordination and financing of work carried out for defense purposes; organizing scientific research, ordering and financing the production and purchase of weapons and military equipment, food, clothing and other property, material and other resources for the Armed Forces. The Ministry cooperates with the military departments of foreign states, and also exercises a number of other powers.

The main body of operational command and control of the troops and forces of the fleet of the Armed Forces of the Russian Federation is General base . It develops proposals on the military doctrine of Russia, the plan for the development of the Armed Forces of the Russian Federation and coordinates the development of proposals on the size of the Armed Forces of the Russian Federation, other troops, military formations and bodies.

The General Staff is also developing a plan for the employment and mobilization plan of the Armed Forces and a federal state program for the operational equipment of the country's territory for defense purposes. It establishes quantitative standards for conscription for military service, military training, analyzes and coordinates the conduct of military registration in the country, prepares citizens for military service and their conscription for military service and military training. For the purposes of defense and security, the General Staff organizes intelligence activities, measures to maintain the combat and mobilization readiness of the Armed Forces of the Russian Federation, etc.

AT the structure of the central office of the Ministry of Defense The Russian Federation includes a number of main and central departments that are in charge of certain functions and are subordinate to certain deputies of the Minister of Defense or directly to the Minister of Defense. In addition, the central bodies of the Ministry of Defense of the Russian Federation include the directorates of the commanders-in-chief of the branches of the Armed Forces of the Russian Federation. Structurally, the Directorate of the Commander-in-Chief branch of the Armed Forces of the Russian Federation consists of the Command, the General Staff, main directorates, directorates, departments. At the head of the branch of the Armed Forces, the Office of the Commander-in-Chief of the branch is the Commander-in-Chief. He is appointed by the President of the Russian Federation and reports directly to the Minister of Defense.

Part military district administration includes: command, headquarters of the military district, departments, services and other structural units. The military district is headed by the commander of the troops of the military district.

Management structure of a separate military unit and the main duties of its officials are determined by the Charter of the Internal Service of the Armed Forces of the Russian Federation.

Each military command and control body functions in accordance with the competence assigned to it within the limits of the powers granted to it, with strict and unswerving observance and execution of laws and other legal acts based on them.

executive power , carried out by the military bodies of the Armed Forces of the Russian Federation, acts both on the basis of general principles of executive power, and specific ones, the most important of which are the principles of consistent centralization, unity of command and strict discipline.

Centralization is expressed in:

Implementation of leadership of all the Armed Forces of the state from a single

Subordination of all types of the Armed Forces of the Russian Federation to a single command;

Giving the central authorities full rights to manage subordinate

military bodies and subordinate troops;

Mandatory acts and instructions of higher bodies and officials for

downstream.

unity of command- the fundamental principle of the construction of the Armed Forces of the Russian Federation, their leadership and the relationship between servicemen. The essence of unity of command lies in vesting the commander (chief) with full administrative power in relation to subordinates and placing on him personal responsibility for all aspects of the life and activities of a military unit, unit and each serviceman.

Unity of command in the Armed Forces of the Russian Federation is carried out on a solid legal basis. This principle is enshrined legally at the legislative level. The norms of military legislation and the Statutes of the Armed Forces of the Russian Federation specifically define the duties and rights of the respective commanders (chiefs) and endow them with the necessary powers of state power.

military discipline - the most important principle of exercising executive power in military administration. However, military discipline is only a part (kind) of state discipline operating in the military field. Therefore, military command and control bodies and their officials are required to comply with the requirements of other types of state discipline.

Thus, all of the above allows us to conclude that the Armed Forces of the Russian Federation have a coherent system of command and control, which ensures their reliable controllability in various conditions of the situation.

"Military training" - The contract provides for the obligation of a citizen to undergo military service by conscription in the manner prescribed by the Federal Law "On military duty and military service." Voluntary preparation of citizens for military service. Young men studying at military departments undergo training camps or internships, which are equated to military training camps.

"Fundamentals of military service" - What is military duty? What law establishes conscription? The main burden of training in military specialties lies with the Russian Defense Sports and Technical Organization (ROSTO). On May 31, 1996, the Federal Law "On Defense" establishes military duty for citizens of the Russian Federation.

"Protection of the Fatherland" - You can even learn to shoot without an army. But the army is not a kindergarten. But few people are happy to join the army now. But why do people who have served in the army always enjoy special respect among the people? "God is not in power, but in truth" Alexander Nevsky. You can get into one of the "hot" spots. And here they are preparing not for picking flowers in the meadow, but for overcoming obstacles on the way to victory.

"Armed Forces of the Russian Federation" - Theme of the lesson ORGANIZATIONAL STRUCTURE OF THE ARMED FORCES. 3. Status. 2 Component (Connections and parts of the reduced composition and frame). The structure of the Ground Forces of the Armed Forces of the Russian Federation. Composition of the Ground Forces of the Armed Forces of the Russian Federation. Lesson 1: Ground Forces. 1 Component (Connections and parts of permanent readiness). TYPES OF ARMED FORCES, TROOPS.

"AF RF" - Combined Arms Regulations. Mobilization plans for the economy. Task No. 1 for groups Group I - What are the Ground Forces of the Armed Forces for? Seriously simplify the interaction of various types of aircraft. Ensuring the economic and political interests of the Russian Federation. Fizkultminutka. Summarizing. Group II - What is the purpose of the Navy Armed Forces?

The armed forces in any state are a key element in ensuring the country's defense capability. Their proper management depends on their proper organization. The structure of the Armed Forces of the Russian Federation ensures the rapid and correct performance of the functions assigned by law to the state military organization of the country.

Structure of the RF Armed Forces

The Armed Forces is a military organization of the Russian Federation, the main function of which is to repulse military aggression in order to ensure territorial integrity, as well as to fulfill tasks in accordance with Russia's international obligations. The Armed Forces of the Russian Federation were created on May 7, 1992. The President of the Russian Federation is the Supreme Commander. In accordance with the Decree of the President of 2008, the strength of the Russian Armed Forces is set at 2,019,629 people, of which 1.3 million are military personnel.

Organizationally, the Armed Forces consist of three branches, three separate branches of the armed forces, Logistics, as well as the Cantonment Service, which is not a branch of the Armed Forces. In addition, the Structure of the Armed Forces of the Russian Federation was also created according to the territorial principle: the territory of the Russian Federation is divided into 4 military districts.

Territorial structure

Today in the Russian Federation there are four military districts, which are assumed by the territorial structure of the Armed Forces:

1. Western military district. The command and headquarters are in St. Petersburg.
2. Eastern military district. The command and headquarters are in Khabarovsk.
3. Central military district. The command and headquarters are in Yekaterinburg.
4. Southern military district. The command and headquarters are in Rostov-on-Don.

The structure of the armed forces of the Russian Federation on the diagram:

Aircraft types

The main element of the Armed Forces are types of the Armed Forces. In the military department of Russia, the law establishes the existence of three types of armed forces: the Air Force, the Ground Forces and the Navy.

To date, the Ground Forces are the most numerous branch of the Russian armed forces. Their main function is to conduct offensive operations, the purpose of which is to defeat the enemy, capture and hold his territory, individual regions and lines, repulse the invasion of the enemy’s country and his large landing forces, and deliver artillery and missile strikes to great depths. In turn, the Ground Forces organizationally consist of military branches. These types of troops can perform tasks independently or jointly.

Motor Rifle Troops (MSV)- the most numerous branch of the army in the Ground Forces. They are also the most numerous branch of the military. To date, the motorized rifle troops are armed with armored personnel carriers and infantry fighting vehicles, which should ensure the mobility of the infantry. MSV organizationally consist of motorized rifle subunits, units and formations.

Motorized rifle, tank, artillery and other subunits and units may be part of the MSV.

Tank Troops (TV)- the main strike force, characterized by high mobility, maneuverability and resistance to the effects of weapons of mass destruction, including nuclear ones. The main tasks, based on the technical equipment of TV: the implementation of a breakthrough, the development of operational success. Artillery, motorized rifle, missile, tank units and subunits can operate as part of the TV.

Missile Forces and Artillery (RViA): nuclear and fire defeat of the enemy is the main task. It is armed with rocket and cannon artillery. The MFA includes units, units and formations of howitzer, rocket, cannon, anti-tank artillery, as well as structural elements of support, control, mortars and artillery reconnaissance.

Air Defense Forces of the Ground Forces (Air Defense SV)- this type of troops must ensure the protection of the Ground Forces from air strikes, as well as counteraction to enemy air reconnaissance. Towed, mobile, portable anti-aircraft gun systems and anti-aircraft missile systems are in service with the SV air defense.

Also, the organizational structure of the Armed Forces implies the presence in the Armed Forces of special troops and services that perform highly specialized tasks in order to ensure the daily and combat activities of the ground forces.

• Signal Corps,
• Electronic Warfare Troops,
• engineering troops,
• automobile troops,
• Railway troops, etc.

are special forces.

Air Force

Air Force similarly, the Ground Forces consist of aviation branches that ensure the fulfillment of the tasks assigned to the Air Force.

Long-range aviation is designed to strike and strike at the strategic and operational depths of enemy military groupings, its important areas in economic and strategic terms, including with the help of nuclear weapons.

Front-line aviation operates at operational depth. It can perform tasks both independently and during joint operations on land and at sea.

Army Aviation provides support to ground forces by destroying enemy armored and mobile objects. Also, the forces of the Army Aviation provide the mobility of the Ground Forces.

Military transport aviation carries out the transportation of goods, troops and equipment, and is also involved in military air operations. In peacetime, the main function is to ensure the vital activity of the Armed Forces, and in wartime, the mobility of the Armed Forces.

The structure of the armed forces of the Russian Federation assumes the presence in the composition Special Air Force, Anti-aircraft missile troops and Radio engineering troops which significantly expand the range of tasks assigned to the Air Force.

Navy

Navy- the main force of the Armed Forces of the Russian Federation to protect the interests of Russia in the Exclusive Maritime (Economic) Zone, conduct search and rescue operations, as well as conduct military operations at sea.

The Navy is composed of:

• submarine force,
• surface forces,
• coastal troops,
• naval aviation,
• parts and connections for special purposes.

The Navy is also organizationally divided into:

• Baltic Fleet,
• Black Sea Fleet,
• northern fleet,
• pacific fleet,
• Caspian flotilla.

Independent branches of the military

Some tasks require special equipment and trained personnel. The structure of the Armed Forces assumes the existence of independent branches of service:

1. Airborne troops;
2. Strategic Rocket Forces;
3. Aerospace Defense Troops.

Aerospace Defense Troops

The youngest branch of the military. Although our country began space exploration back in the 1960s, it was only in the 21st century that the Aerospace Defense Forces were separated into a separate branch of the Strategic Missile Forces.

The most important tasks are:

• detection of a missile attack;
• spacecraft constellation management;
• missile defense of the capital of Russia.

Strategic Rocket Forces

Today they are the main land component of Russia's nuclear forces. The main function is considered to be deterrence of possible aggression. But if necessary, they can deliver a preemptive strike on important economic and military targets of the enemy, as well as the destruction of his military groupings.

Airborne Troops

They were created in the 1930s. Today, they are entrusted with the function of conducting landing operations and conducting combat operations behind enemy lines.

To calculate the fields created by charges that are uniformly distributed over spherical, cylindrical or flat surfaces, the Ostrogradsky-Gauss theorem (section 2.2) is used.

Method for calculating fields using the theorem

Ostrogradsky - Gauss.

1) We choose an arbitrary closed surface enclosing a charged body.

2) We calculate the flow of the tension vector through this surface.

3) We calculate the total charge covered by this surface.

4) We substitute the calculated quantities into the Gauss theorem and express the strength of the electrostatic field.

Examples of calculation of some fields

The field of a uniformly charged infinite cylinder (thread).

Let an infinite cylinder with radius R uniformly charged with linear charge density + τ (Fig. 16).

It follows from symmetry considerations that the lines of field strength at any point will be directed along radial lines perpendicular to the axis of the cylinder.

As a closed surface, we choose a coaxial cylinder with a given (with a common axis of symmetry) cylinder of radius r and height ℓ .

Calculate the flow of the vector through this surface

,

where S main , S side are the areas of the bases and the lateral surface.

The flow of the stress vector through the areas of the bases is equal to zero, therefore

The total charge covered by the selected surface:

.

Substituting everything into the Gauss theorem, taking into account the fact that ε = 1, we get:

.

The intensity of the electrostatic field created by an infinitely long uniformly charged cylinder or an infinitely long uniformly charged thread at points located outside it:

, (2.5)

where r - distance off axis cylinder to a given point ( r ≥ R );

τ - linear charge density .

If a r < R , then the closed surface under consideration does not contain charges inside, therefore, in this region E = 0, i.e. inside the cylinder, no field .

Field of a uniformly charged infinite plane

P an infinite plane is charged with a constant surface density + σ .

As a closed surface, we choose a cylinder, the bases of which are parallel to the charged plane, and the axis is perpendicular to it (Fig. 17). Since the lines forming the side surface of the cylinder are parallel to the lines of tension, the flow of the tension vector through the side surface is zero. The flow of the tension vector through two areas of the base

.

The total charge covered by the selected surface:

.

Substituting everything into the Gauss theorem, we get:

Electrostatic field strength of an infinite uniformly charged plane

. (2.6)

It follows from this formula that E does not depend on the length of the cylinder, that is, the field strength is the same at all points. In other words, the field of a uniformly charged plane homogeneous.

Field of two infinite parallel

oppositely charged planes

P the mouth of the plane are uniformly charged with the same surface densities + σ and - σ (Fig. 18).

According to the principle of superposition,

.

It can be seen from the figure that in the area between the planes, the lines of force are codirectional, so the resulting tension

. (2.7)

Outside the volume bounded by the planes, the added fields have opposite directions, so that the resulting strength is zero.

Thus, the field is concentrated between the planes. The result obtained is approximately valid for planes of finite dimensions, if the distance between the planes is much less than their area (flat capacitor).

If charges of the same sign with the same surface density are distributed on the planes, then the field is absent between the plates, and outside the plates it is calculated by formula (2.7).

Field strength

uniformly charged sphere

Field generated by a spherical surface of radius R , charged with surface charge density σ , will be centrally symmetrical, so the lines of tension are directed along the radii of the sphere (Fig. 19, a).

As a closed surface, we choose a sphere of radius r having a common center with a charged sphere.

If a r > R , then the entire charge enters the surface Q .

The flow of the intensity vector through the surface of the sphere

Substituting this expression into the Gauss theorem, we get:

.

The strength of the electrostatic field outside a uniformly charged sphere:

, (2.8)

where r - distance from the center spheres.

This shows that the field is identical with the field of a point charge of the same magnitude, placed in the center of the sphere.

If a r < R , then the closed surface does not contain charges inside, therefore there is no field inside the charged sphere (Fig. 19, b).

Field strength volumetric

charged ball

P mouth ball radius R charged with a constant volumetric charge density ρ .

The field in this case has central symmetry. For the field strength outside the sphere, the same result is obtained as in the case of a surface-charged sphere (2.8).

For points inside the ball, the tension will be different (Fig. 20). The spherical surface covers the charge

Therefore, according to the Gauss theorem

Given that
, we get:

The intensity of the electrostatic field, inside a volumetrically charged ball

(r ≤ R ). (2.9)

.

Task 2.3 . In the field of an infinitely long plane with a surface charge density σ a small ball of mass suspended from a thread m , which has a charge of the same sign as the plane. Find the charge of the ball if the thread forms an angle with the vertical α

Decision. Let us return to the analysis of the solution of Problem 1.4. The difference is that in problem 1.4 the force
is calculated according to the Coulomb law (1.2), and in problem 2.3 - from the definition of the electrostatic field strength (2.1)
. The strength of the electrostatic field of an infinite uniformly charged plane is derived using the Ostrogradsky-Gauss theorem (2.4).

P The field of the plane is homogeneous and does not depend on the distance to the plane. From fig. 21:

.

 note , that to find the force acting on a charge placed in the field of a distributed charge, it is necessary to use the formula

,

and the strength of the field created by several distributed charges is found by the principle of superposition. Therefore, the following tasks are devoted to finding the strength of the electrostatic field of distributed charges using the Ostrogradsky-Gauss theorem.

Task 2.4. Get ahead of the field strength inside and outside a uniformly charged plate of thickness d , volume charge density inside the plate ρ . Plot dependency graph E (X ).

Decision. We place the origin of coordinates in the middle plane of the plate, and the axis OH direct it perpendicular to it (Fig. 22, a). We apply the Ostrogradsky-Gauss theorem to calculate the strength of the electrostatic field of a charged infinite plane, then

.

From the definition of volumetric charge density

,

then for tension we get

.

This shows that the field inside the plate depends on X . The field outside the plate is calculated similarly:

This shows that the field outside the plate is uniform. Tension dependency graph E from X in fig. 22b.

Task 2.5. The field is created by two infinitely long filaments charged with linear charge densities – τ 1 and + τ 2 . The threads are located perpendicular to each other (Fig. 23). Find the field strength at a point at a distance r 1 and r 2 from threads.

R solution. Let us show in the figure the strength of the field created by each thread separately. Vector directed to the first strand, since it is negatively charged. Vector directed from the second strand, since it is positively charged. Vectors and mutually perpendicular, so the resulting vector will be the hypotenuse of the right triangle. Vector modules and are determined by formula (2.5).

According to the principle of superposition

.

According to the Pythagorean theorem

Task 2.6 . The field is created by two charged infinitely long hollow coaxial cylinders with radii R 1 and R 2 > R 1 . The surface charge densities are – σ 1 and + σ 2 . Find the strength of the electrostatic field at the following points:

a) dot BUT located at a distance d 1 < R 1 ;

b) point AT located at a distance R 1 < d 2 < R 2 ;

c) point With located at a distance d 3 > R 1 > R 2 .

The distances are measured from the axis of the cylinders.

Decision. Coaxial cylinders are cylinders that have a common axis of symmetry. Let's make a drawing and show points on it (Fig. 24).

E BUT = 0.

dot AT located inside the larger cylinder, so at this point the field is created only by the smaller cylinder:

.

Let us express the linear charge density in terms of the surface charge density. To do this, we use formulas (1.4) and (1.5), from which we express the charge:

Equate the right sides and get:

,

where S 1 is the surface area of ​​the first cylinder.

Taking into account the fact that
, we finally get:

dot With located on the outside of both cylinders, so the field is generated by both cylinders. According to the principle of superposition:

.

Taking into account the directions and calculations obtained above, we get:

.

Task 2.7 . The field is created by two charged infinitely long parallel planes. The surface charge densities are σ 1 and σ 2 > σ 1 . Find the strength of the electrostatic field at points located between the plates and outside the plates. Solve the problem for two cases:

a) the plates are charged with the same name;

b) the plates are oppositely charged.

Decision. In vector form, the strength of the resulting field is written in the same way in any case. According to the principle of superposition:

.

Vector modules and are calculated by formula (2.6).

a) If the planes are charged with the same name, then between the planes of tension they are directed in different directions (Fig. 26, a). Modulus of resulting tension

Beyond the planes of tension and directed in one direction. Since the field of infinite charged planes is homogeneous, that is, does not depend on the distance to the planes, then at any point both to the left and to the right of the planes the field will be the same:

.

b) If the planes are differently charged, then, on the contrary, between the planes of tension they are directed in one direction (Fig. 26, b), and outside the planes - in different directions.