electrostatic field- email stationary charge field.
Fel, acting on the charge, moves it, doing work.
In a uniform electric field, Fel = qE is a constant value

Field work (electronic force) does not depend on the shape of the trajectory and on a closed trajectory = zero.

Electrostatics(from electro... and static) , a branch of the theory of electricity that studies the interaction of fixed electric charges. It is carried out through electrostatic field. The basic law of E. - Coulomb is the law that determines the strength of the interaction of motionless point charges depending on their size and distance between them.

Electric charges are sources of an electrostatic field. This fact is expressed by the Gauss theorem. The electrostatic field is potential, i.e., the work of the forces acting on the charge from the electrostatic field does not depend on the shape of the path.

The electrostatic field satisfies the equations:

div D= 4pr, rot E = 0,

where D- electric induction vector (see Electric and magnetic induction), E - electrostatic field strength, r - density electric charge. The first equation is differential form the Gauss theorem, and the second expresses the potential nature of the electrostatic field. These equations can be obtained as special case Maxwell equations.

Typical problems of electrical engineering are finding the distribution of charges on the surfaces of conductors according to the known total charges or potentials of each of them, as well as calculating the energy of a system of conductors from their charges and potentials.

To establish a connection between power characteristic electric field tension and its energy characteristics potential consider elementary work electric field forces on an infinitesimal displacement of a point charge q:d A=qE d l, the same work is equal to the loss potential energy charge q:d A = - d W P = - q d , where d is the change in the potential of the electric field over the length of travel d l. Equating the right parts of the expressions, we get: E d l d or in Cartesian coordinate system

E x d x + Ey d y+Ez d z=d , (1.8)

where E x,E y,Ez- projections of the tension vector on the axes of the coordinate system. Since expression (1.8) is total differential, then for the projections of the intensity vector we have

Equipotential surface- a concept applicable to any potential vector field, for example, to a static electric field or to a Newtonian gravitational field (Gravity). An equipotential surface is a surface on which the scalar potential of a given potential field takes on a constant value. Another, equivalent, definition is a surface that is orthogonal at any of its points. lines of force fields.

The surface of a conductor in electrostatics is an equipotential surface. In addition, placing a conductor on an equipotential surface does not cause a change in the configuration of the electrostatic field. This fact is used in the imaging method, which allows calculating the electrostatic field for complex configurations.

In a gravitational field, the level of an immobile liquid is established by the equipotential surface. In particular, the level of the oceans passes along the equipotential surface of the Earth's gravitational field. The equipotential surface of the level of the oceans, extended to the surface of the Earth, is called the geoid and plays an important role in geodesy.

5.Electric capacity- a characteristic of a conductor, a measure of its ability to accumulate an electric charge. In the theory of electrical circuits, capacitance is the mutual capacitance between two conductors; parameter of the capacitive element of the electrical circuit, presented in the form of a two-terminal network. Such capacity is defined as the ratio of the magnitude of the electric charge to the potential difference between these conductors.

In the SI system, capacitance is measured in farads. In the cgs system in centimeters.

For a single conductor, the capacitance is equal to the ratio of the conductor's charge to its potential, assuming that all other conductors are at infinity and that the potential of the point at infinity is assumed to be zero. In mathematical form this definition has the form

Where Q- charge, U- conductor potential.

The capacitance is determined by the geometric dimensions and shape of the conductor and electrical properties environment(her permittivity) and does not depend on the material of the conductor. For example, the capacitance of a conducting ball of radius R is equal to (in the SI system):

C= 4πε 0 ε R.

The concept of capacitance also refers to a system of conductors, in particular, to a system of two conductors separated by a dielectric - a capacitor. In this case mutual capacitance these conductors (capacitor plates) will be equal to the ratio of the charge accumulated by the capacitor to the potential difference between the plates. For a flat capacitor, the capacitance is:

where S- the area of ​​​​one lining (it is assumed that they are equal), d- the distance between the plates, ε - relative permittivity of the medium between the plates, ε 0 = 8.854×10 −12 F/m - electrical constant.

At parallel connection k capacitors, the total capacitance is equal to the sum of the capacitances of the individual capacitors:

C=C1+C2+ … + C k .

When connected in series k capacitors add up the reciprocals of the capacitances:

1/C = 1/C 1+ 1/C2+ … + 1/C k .

The energy of the electric field of a charged capacitor is:

W = qU / 2 = CU 2 /2 = q2/ (2C).

6.Electric current is calledpermanent , if the current strength and its direction do not change over time.

Current strength (often just " current”) in the conductor - a scalar value, numerically equal to the charge flowing per unit time through the cross section of the conductor. Denoted by a letter (in some courses - . Not to be confused with vector current density):

The basic formula used to solve problems is Ohm's Law:

§ for the plot electrical circuit:

Current is equal to the ratio of voltage to resistance.

§ for a complete electrical circuit:

Where E is EMF, R is external resistance, r is internal resistance.

The SI unit is 1 Ampere (A) = 1 Coulomb / second.

To measure the current strength, a special device is used - an ammeter (for devices designed to measure low currents, the names milliammeter, microammeter, galvanometer are also used). It is included in the open circuit in the place where you need to measure the current strength. The main methods for measuring current strength are: magnetoelectric, electromagnetic and indirect (by measuring voltage with a voltmeter at a known resistance).

When alternating current distinguish between instantaneous current strength, amplitude (peak) current strength and effective current strength ( equal to strength direct current, which produces the same power).

current density - vector physical quantity, meaning the strength of the current flowing through a unit area. For example, with a uniform density distribution:

Current over the cross section of the conductor.

Among the conditions necessary for the existence electric current distinguish:

The presence of free electric charges in the environment

creating an electric field in the environment

Third party forces - forces of a non-electric nature, causing the movement of electric charges inside a direct current source.
All forces other than the Coulomb forces are considered external.

Electromotive force (emf), a physical quantity that characterizes the action of external (non-potential) forces in sources of direct or alternating current; in a closed conducting circuit is equal to the work of these forces to move a unit positive charge along the contour. If through E pp denote the strength of the field of external forces, then the emf in a closed loop ( L) is equal to , where dl- contour length element.

Potential Forces electrostatic (or stationary) fields cannot support D.C. in the chain, since the work of these forces on a closed path is zero. The passage of current through the conductors is accompanied by the release of energy - heating of the conductors. External forces set in motion charged particles inside current sources: generators, galvanic cells, batteries, etc. The origin of external forces can be different. In generators, external forces are forces from the vortex electric field that occurs when the magnetic field over time, or the Lorentz force acting from the magnetic field on electrons in a moving conductor; in galvanic cells and batteries, these are chemical forces, etc. Eds determines the strength of the current in the circuit for a given resistance (see Ohm's law) . EMF is measured, as well as voltage, in volts.

6. Work when moving an electric charge in an electric field

Let us calculate the work when moving an electric charge in a uniform electric field with intensity . If the charge moved along the field strength line at a distance Ad \u003d d 1 -d 2 (Fig. 110), then the work is equal to

where d 1 and d 2 are the distances from the start and end points to plate B.

In mechanics, it was shown that when moving between two points in a gravitational field, the work of gravity does not depend on the trajectory of the body. The forces of gravitational and electrostatic interaction have the same dependence on distance, the force vectors are directed along the straight line connecting the interacting point bodies. It follows from this that when a charge moves in an electric field from one point to another, the work of the forces of the electric field does not depend on the trajectory of its movement.

When the direction of movement changes by 180°, the work of the electric field forces, as well as the work of gravity, changes sign to the opposite. If, when moving charge q from point B to point C, the electric field forces did work A, then when moving charge q along the same path from point C to point B, they do work - A. But since the work does not depend on the trajectory, then and when moving along the SLE trajectory, work is also done - A. It follows that when the charge moves first from point B to point C, and then from point C to point B, i.e., along a closed trajectory, the total work of the forces of the electrostatic field turns out to be equal to zero (Rie.111).

The work of the forces of the electrostatic field during the movement of an electric charge along any closed trajectory is equal to zero.

A field whose work of forces along any closed trajectory is equal to zero is called a potential field. Gravitational and electrostatic fields are potential fields.

7. The concept of potential field potential of a point charge

The potential of an electrostatic field is a scalar value equal to the ratio of the potential energy of a charge in the field to this charge:

Energy characteristic of the field at a given point. The potential does not depend on the magnitude of the charge placed in this field.

because If the potential energy depends on the choice of the coordinate system, then the potential is determined up to a constant.

For the reference point of the potential is chosen depending on the task: a) the potential of the Earth, b) the potential of the infinitely distant point of the field, c) the potential of the negative plate of the capacitor.

A consequence of the principle of superposition of fields (potentials add up algebraically).

Electrostatic field potential at a point r is equal to the ratio of the potential energy of a test point charge q" placed at a given point to the value of this charge q".

φ - does not depend on q"!

8. Potential difference. Relationship between tension and potential

When the values ​​of these two potentials are not equal to each other, there is a vector difference in the potentials of action and reaction. It determines the direction of movement of energy carriers during energy exchange: from the environment to the system or to reverse direction. In contrast to the potential difference between the medium and the equilibrium system, there is a local potential difference inside a non-equilibrium system. Therefore, two different definitions should be given: 1. The potential difference with respect to an equilibrium system is the difference between the potential of the system as a whole and the potential of the environment (or the potential of a neighboring system). 2. The potential difference inside a non-equilibrium system is the difference between the local potentials of subsystems inside this system. The potential difference is directed from a larger potential to a smaller one, it can be written as ΔР 12 = (Р 1 - Р 2) e 12, (3) where Р 1 and Р 2 are the potentials of the system or its environment; e 12 is the unit vector of the direction from the system to the medium or in the opposite direction. In the general case, the subscripts can be omitted and the notation ΔP can be used. The difference of local potentials is also directed, it can be written as ΔР 12 = (Р j1 − Р j2) e 12 , (4) where Р j1 and Р j2 are local potentials of different subsystems inside the nonequilibrium system; e 12 is the unit vector of the direction from subsystem 1 to subsystem 2.

Relationship between tension and potential expresses the characteristic of the electric field. Moreover, if the tension serves as its power characteristic and allows you to determine the magnitude of the force that acts on the charge at an arbitrarily taken point of this field, then the potential is its energy characteristic. Based on the potentials at various points of the electric field, we can determine the amount of work to move the charge using the formulas: A \u003d qU, or A \u003d q (φ₁ - φ₂), where q is the charge value, U is the voltage between the field points and φ₁, φ₂ is the potential of the points of movement . Consider the relationship between strength and potential in a single-valued electric field. The intensity E at any point of such a field is the same, and therefore the force F, which acts on a unit of charge, is also the same and equals E. It follows that the force that acts on the charge q in this field will be equal to F = qE. If the distance between two points of such a field is equal to d, then when the charge moves, work will be done: A = Fd = gEd = g(φ₁-φ₂), where φ₁-φ₂ is the potential difference between the points of the field. Hence: E= (φ₁-φ₂)/d, i.e. the intensity of a uniform electric field will be equal to the potential difference that per unit length, which was taken along the line of force of this field. At short distances, the relationship between strength and potential is determined similarly in an inhomogeneous field, since any field between two closely spaced points can be taken as homogeneous.

9.Electrocapacity. Capacitor.

The electrical capacitance of the capacitor. Physical quantity determined by the charge ratio q one of the capacitor plates to the voltage between the capacitor plates is called Capacitor capacitance:. With a constant arrangement of the plates, the capacitance of the capacitor is a constant value for any charge on the plates. Unit of electrical capacity. The unit of electric capacity in the international system is farad(F). Such a capacitor has an electric capacity of 1 F, the voltage between the plates of which is 1 V when the plates are given opposite charges of 1 C each.

Capacitors. The simplest methods of separating dissimilar electric charges - electrification by contact, electrostatic induction - make it possible to obtain only a relatively small number of free electric charges on the surface of bodies. To accumulate significant amounts of opposite electric charges, capacitors. Capacitor- this is a system of two conductors (plates), separated by a dielectric layer, the thickness of which is small compared to the dimensions of the conductors. So, for example, two flat metal plates, located in parallel and separated by a dielectric layer, form flat capacitor. If the plates of a flat capacitor are given equal charges of the opposite sign, then the electric field strength between the plates will be twice as large as the field strength of one plate. Outside the plates, the electric field strength is zero, since equal charges of different signs on two plates create electric fields outside the plates, the strengths of which are equal in magnitude, but opposite in direction.

10. Electric dipole

electric dipole - a system of two equal in magnitude, but opposite in sign, point electric charges located at some distance from each other.

The distance between charges is called dipole arm.

The main characteristic of a dipole is a vector quantity called electric moment dipole(P).

When moving the test charge q In an electric field, electric forces do work. This work with a small displacement is (Fig. 1.4.1):

Consider the work of forces in an electric field created by a time-invariant distributed charge, i.e. electrostatic field

An electrostatic field has an important property:

The work of the forces of the electrostatic field when moving the charge from one point of the field to another does not depend on the shape of the trajectory, but is determined only by the position of the starting and ending points and the magnitude of the charge.

The gravitational field has a similar property, and there is nothing surprising in this, since the gravitational and Coulomb forces are described by the same ratios.

A consequence of the independence of work from the shape of the trajectory is the following statement:

The work of the forces of the electrostatic field when moving the charge along any closed trajectory is equal to zero.

Force fields with this property are called potential or conservative .

On fig. 1.4.2 shows the lines of force of the Coulomb field of a point charge Q and two different test charge trajectories q from start point (1) to end point (2). On one of the trajectories, a small displacement is distinguished Work Δ A Coulomb forces on this displacement is equal to

The result obtained does not depend on the shape of the trajectory. On trajectories I and II shown in Figs. 1.4.2, the work of the Coulomb forces is the same. If on one of the trajectories we change the direction of charge movement q to the opposite, then the work will change sign. This implies that the work of the Coulomb forces on a closed trajectory is equal to zero.

If the electrostatic field is created by a set of point charges, then when moving the test charge q Work A the resulting field, in accordance with the principle of superposition, will consist of the work of the Coulomb fields of point charges: Since each term of the sum does not depend on the shape of the trajectory, then the total work A The resulting field is independent of the path and is determined only by the position of the start and end points.

The potentiality property of the electrostatic field allows us to introduce the concept potential energy charge in an electric field. To do this, a certain point (0) is selected in space, and the potential energy of the charge q placed at this point is taken equal to zero.

Potential charge energyq , placed at any point (1) of space, with respect to a fixed point (0) is equal to the workA 10 , which will make an electrostatic field when moving a chargeq from point (1) to point (0):

(In electrostatics, energy is usually denoted by the letter W, since the letter E indicate the field strength.)

Just as in mechanics, the potential energy is defined up to a constant value, depending on the choice of the reference point (0). Such ambiguity in the definition of potential energy does not lead to any misunderstanding, since physical meaning has not the potential energy itself, but the difference of its values ​​at two points in space.

Work done by an electrostatic field when moving a point chargeq from point (1) to point (2), is equal to the difference between the values ​​of potential energy at these points and does not depend on the path of charge movement and on the choice of point (0).

The potential φ is the energy characteristic of the electrostatic field.

Work A 12 on the movement of electric charge q from the starting point (1) to the end point (2) is equal to the product of the charge and potential difference (φ 1 - φ 2) start and end points:

In many problems of electrostatics, when calculating potentials, it is convenient to take the point at infinity as the reference point (0). In this case, the concept of potential can be defined as follows:

The field potential at a given point in space is equal to the work that electric forces do when a unit positive charge is removed from a given point to infinity.

As follows from the Gauss theorem, the same formula expresses the field potential of a uniformly charged ball (or sphere) at r ≥ R, where R is the radius of the ball.

For a visual representation of the electrostatic field, along with lines of force, use equipotential surfaces.

A surface in which the potential of the electric field has the same value at all points is calledequipotential surface orequal potential surface .

The lines of force of an electrostatic field are always perpendicular to equipotential surfaces.

The equipotential surfaces of the Coulomb field of a point charge are concentric spheres. On fig. 1.4.3 shows pictures of lines of force and equipotential surfaces of some simple electrostatic fields.

When homogeneous field equipotential surfaces are a system of parallel planes.

If a test charge q committed small movement along the field line from point (1) to point (2), then we can write:

This relation in scalar form expresses the relationship between field strength and potential. Here l is the coordinate measured along the field line.

From the principle of superposition of field strengths created by electric charges, the principle of superposition for potentials follows:

Equipotential surfaces- a concept applicable to any potential vector field, for example, to a static electric field or to a Newtonian gravitational field. An equipotential surface is a surface on which the scalar potential of a given potential field takes a constant value (potential level surface). Another, equivalent, definition is a surface, at any point orthogonal to the field lines of force.

The surface of a conductor in electrostatics is an equipotential surface. In addition, placing a conductor on an equipotential surface does not cause a change in the configuration of the electrostatic field. This fact is used in the imaging method, which allows calculating the electrostatic field for complex configurations.

In a (stationary) gravitational field, the level of a stationary fluid is established by an equipotential surface. In particular, it can be approximately stated that the level of the oceans passes along the equipotential surface of the Earth's gravitational field. The shape of the surface of the oceans, extended to the surface of the Earth, is called the geoid and plays an important role in geodesy. The geoid is thus an equipotential surface of gravity, consisting of a gravitational and centrifugal component.

EQUIPOTENTIAL LINES

Lines of equal potential values ​​of the studied electric field.