§ 3.2. . Work on the movement of charge in the electrostatic field

For the charge from the electrostatic field the force acts. Therefore, when moving the charge in the electrostatic field, work is performed.

Forces electrostatic field are conservative, i.e. The work of the power of the electrostatic field to move the charge does not depend on the shape of the path, but is determined only by the position of the initial and endpoint points. Show it. Let the dot charge + Q 0 move in the stationary field point charge + Q in vacuo from point 1 to point 2. Elementary work of the Coulomb force acting on the charge from charge on Putdl is equal to Da \u003d F DL COSα. By the law of Kulon

, DL COSα \u003d DR. Then

. That is, the work is determined only by the position of points 1 and 2.

In the mechanics, we determined that:

conservatives are called forces whose work does not depend on the form of the path, but only the coordinate of the initial and final position of the material point is determined;

the field of conservative forces is potentially.

For potential fields, you can enter the concepts of potential and potential difference. Denote: the potential φ, the potential difference φ 1 -φ 2. Measured in SI in Volt (B).

The potential of this point of the electrostatic field is numerically equal to the work of the power of the electrostatic field to move a single positive charge From this point of the field to infinity.

The potential difference φ 1 -φ 2 between points of the electrostatic field (1 and 2) is numerically equal to the work performed by the field with the movement of a single positive charge along an arbitrary pathway from point 1 to point 2.

Previously, a formula was obtained for the operation of the point charge field q to move the charge Q 0 from point 1 to point 2:

. On the other hand, the work of the forces of any electrostatic field when the charge is moving 0 from point 1 to point 2 is equal to A 12 \u003d Q 0. (φ 1 -φ 2). Hence,

. From here we find an expression for the potential of the electrostatic field of the point chargeq in vacuo:

.

Principle of superposition of fields: The potential of the electrostatic field created by the charge system is equal to the algebraic amount of the potentials of fields created by each of these charges separately

.

The potential energy of charge Q 0 at the point of the electrostatic field with the potential φ: W n \u003d Q 0. φ. This means that the potential is the energy characteristics of the electrostatic field.

The electrostatic field is characterized by two values: 1) intensity (power characteristic); 2) potential (energy characteristics). It can be assumed that these values \u200b\u200bare somehow connected with each other. We show that it is so.

Work of the forces of the field to move the charge Q 0 on the cutting path :, where - Projection of the vector on the direction of movement . On the other hand, this work will be equal to the decrease. potential energy Charge:. Equating the right parts of expressions for the field work, we get that

From here

What means: The projection of the tension of the electrostatic field on some arbitrary direction is equal to the potential derivative in this direction with the opposite sign. Here - The speed of potential changes in this direction.

Due to the arbitrariness of the choice of direction, you can record





, or:

. This formula expresses the connection of the voltage of the electrostatic field with the potential: the voltage of the electrostatic field is equal to the gradient of the potential taken with the opposite sign. The minus sign means that the field strength is directed toward the decrease in the potential.

Thus, if the value of the potential φ at each point of the field is known, then the voltage can be found at each point of the field by the formula

It is possible to solve the inverse task, i.e. According to the specified values at each point to find the potential difference between two arbitrary field points by the formula

. The integral can be taken along any line connecting points 1 and 2 (because the operation of the power of the electrostatic field does not depend on the shape of the path).

For a homogeneous field

or

, whered - the distance between points 1 and 2 along the power line.

For graphic image of the electrostatic field, the surface of equal potential or equipotential surfaces also serve.

Equipotential surface is such a surface, all points of which have the same potential.

Equipotential surfaces are carried out so that the potential difference between adjacent surfaces was everywhere and the same. Thus, the thick of the equipotential surfaces are located, the greater in this place GRAD φ and, therefore, more voltage .

Power lines perpendicular to equipotential surfaces, because The operation of moving the charge along the equipotential surface is zero, and, therefore, the force acting on the charge is perpendicular to its movement.

For a homogeneous field, equipotential surfaces are parallel planes perpendicular power lines Fields.

6. Work when moving an electric charge in electric field

We calculate the work when moving the electrical charge in a uniform electric field with tension. If the charge of the charge occurred over the line on the leaps of the field at a distance \u003d D 1 -D 2 (Fig. 110), then the work is equal

where d 1 and d 2 are distances from the initial and endpoints to the plate V.

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

If you change the direction of movement 180 °, the operation of the electric field forces, like the work of gravity, changes the sign to the opposite. If, when moving the charge q from the point to a point with the power of the electric field, it was performed and, then when the charge of the charge q is the same path from the point from to the point in they make work - A. But since the work does not depend on the trajectory, And when moving along the trajectory, the SC is also made by work - A. It follows from here that when the charge is moved, first from the point to the point C, and then from the point from to the point B, i.e. on a closed trajectory, the total work of the power of the electrostatic field turns out equal to zero (RIE.111).

The work of the power of the electrostatic field when the electric charge is moved along any closed trajectory is zero.

The field, the operation of which for any closed trajectory is zero, is called a potential field. Gravitational and electrostatic fields are potential fields.

7. Concept of potential Potential of a point charge field

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

Energy characteristics of the field at this point. The potential does not depend on the size of the charge placed in this field.

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

The potential reference point is chosen depending on the problem: a) the potential of the Earth, b) the potential of an infinitely remote point of the field, c) the potential of the negative condenser plate.

The consequence of the principle of superposition of fields (the potentials are developing algebraically).

The potential of the electrostatic field at the point R is equal to the ratio of the potential energy of the test point charge Q ", placed at this point, to the magnitude of this charge Q."

φ - does not depend on q "!

8. The potentialness of the potentials. Communication between tension and potential

When the values \u200b\u200bof these two potentials are not equal to each other, the vector difference between the potentials of exposure and counteraction occurs. It determines the direction of energy movement during energy exchange: from the environment in the system or in the opposite direction. In contrast to the potential difference between the medium and the equilibrium system, there is a difference between local potentials inside the non-equilibrium system. Therefore, two different definitions should be given: 1. The potential difference in relation to the equilibrium system is the difference between the potential of the system as a whole and the potential ambient (or the potential of the neighboring system). 2. The potential difference within the non-equilibrium system is the difference between the local potential subsystems within this system. The potential difference is directed from a greater potential value to a smaller, it can be written in the form Δp 12 \u003d (p 1 - p 2) E 12, (3) where p 1 and p 2 - the potentials of the system or the surrounding medium; E 12 - ort directions from the system to the medium or in the opposite direction. In general, the lower indexes can be omitted and apply the designation ΔP. The difference of local potentials is also directed, it can be written in the form Δp 12 \u003d (p j1 - p j2) E 12, (4) where R j1 and p j2 are the local potentials of different subsystems inside the nonequilibrium system; E 12 - Ort direction from subsystem 1 to subsystem 2.

Communication between tension and potential Expresses the characteristic of the electric field. And if the tension serves it silence characteristic And allows you to determine the amount of force that acts on the charge in an arbitrarily taken point of this field, the potential is its energy characteristic. According to potentials at various points of the electro, we can determine the amount of work to move the charge using formulas: a \u003d qu, or a \u003d q (φ₁ - φ₂), where q is the charge value, U is the voltage between the points of the field and φ₁, φ₂ - the potential of the movement points . Consider the relationship between tensions and potential in an unambiguous electric field. The tension E at any point of such a field is the same, and therefore the force F that acts per unit of charge is also the same and equals E. From this it follows that the force that affects the charge Q in this field will be equal to F \u003d QE. If the distance between two points of such a field is D, then when the charge is moved, work is done: a \u003d fd \u003d ged \u003d g (φ₁-φ₂), where φ₁-φ₂ is the potentialness between the points of the field. Hence: E \u003d (φ₁-φ₂) / D, i.e. The tension of a homogeneous electric field will be equal to the potential difference, which fall per unit length, which was taken by the power line of this field. At low distances, the relationship between tensions and potential is determined similarly in the inhomogeneous field, since any field between two close points can be taken for homogeneous.

9.Electricity. Capacitor.

Electrical capacitor capacitor. The physical quantity determined by the ratio q. one of the plates of the condenser to the voltage between the capacitor plates is called capacitor electrical capacity:. With a constant location of the plates, the electrical capacity of the capacitor is a constant value at any charge on the plates. Electricity unit. Unit of electrical capacity in the international system - farad (F). The electrical capacity of 1 f possesses such a capacitor, the voltage between the plates of which is 1 V when reports to the plates of multi-dimensional charges of 1 CL.

Capacitors. The simplest ways of separation of multi-person electrical charges - Electrification in contact, electrostatic induction - allow you to get on the surface of the body only a relatively small number of free electrical charges. For the accumulation of significant quantities of multilayer electrical charges apply capacitors. Capacitor - This is a system of two conductors (plates) separated by a dielectric layer, whose thickness is small compared to the size of the conductors. For example, two flat metal plates located in parallel and separated by a dielectric layer form flat capacitor. If the plates of a flat condenser inform equal to the module charges of the opposite sign, the electric field strength between the plates will be twice as much as the field strength in one plate. Outside the plates, the electric field strength is zero, since equal charges of a different sign on two plates create outside the plates of electric fields, the tensions of which are equal to the module, but are opposite to the direction

10.Electrical dipole.

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

Distance between charges is called shoulder dipole.

The main characteristic of the dipole is the vector value called electric moment dipole (P).

When moving the charge in the electrostatic field, acting on the charge of Coulomb forces, make work. Let the charge Q 0 0 move to the charge field Q0 from the point C to the point in along the arbitrary trajectory (Fig. 1.12). On Q 0 there is a Coulomb force

With elementary charging D l., this force makes the work of DA

Where  is the angle between the vectors and. Validud. l.cos \u003d DR is the projection of the vector to the direction of force. Thus, Da \u003d FDR ,. Complete work on the movement of charge from point C in B is determined by the integral , wherer 1 and R 2 - charge distances Q to the points C and B. From the resulting formula, it follows that the work performed when the electric charge Q 0 is moved in the field of the point charge Q, does not depend on the form of the movement trajectory, and depends only on the initial and end point of the movement .

The speakers section show that the field satisfying this condition is potential. Consequently, the electrostatic field of a point charge - potential, and the power in it - conservative.

If the charges Q and Q 0 of one sign, the work of the repulsion forces will be positive when they are removed and negative when they are converging (in the latter case, the work is performed by external forces). If the charges q and q 0 are different, then the work of attraction forces will be positive when they are rapprocheted and negative when removing each other (the latter case is also performed by external forces).

Let the electrostatic field in which the charge Q 0 moves, created by the charge system Q 1, Q 2, ..., q n. Consequently, independent forces apply on Q 0 , The resultant which is equal to the vector sum. Work and the equal force is equal to the algebraic amount of the work of the components, , wherer i 1 and R i 2 are the initial and final distances between the charges Q i and Q 0.

Circulation of tension vector.

When the charge is moved along an arbitrary closed path, L, the operation of the power of the electrostatic field is zero. Since, the final position of the charge is equal to the initial R 1 \u003d R 2, then the circle at the integral sign indicates that the integration is made by a closed path). So kaki, then. From here we get. Reducing both parts of equality byQ 0, we get or where l. \u003d Ecos - Projection of vector E in the direction of elementary movement. Integral circulation of tension vector. So circulation of the tension of the electrostatic field along any closed circuit is zero . This conclusion is a condition polation of the field.

Potential charge energy.

In a potential field of body, the potential energy and the work of the conservative forces are performed due to the decrease of potential energy.

Therefore, work A. 12 can be represented as the difference in potential charge energies q. 0 In the initial and endpoints of the charge field q. :

Potential energy charge q. 0, located in the charge field q. on distance r. From him is equal

Considering that when removing the charge on infinity, the potential energy appeals to zero, we get: const. = 0 .

For of the same name charges potential energy of their interaction ( repulsion) positivefor variemen charges potential energy from interaction ( attraction) negative.

If the field is created by the system n. Point charges, then the potential energy of charge q. 0, located in this field, is equal to the sum of its potential energies created by each of the charges separately:

The potential of the electrostatic field.

The attitude does not depend on the test charge Q0 and is energy characteristics of the field calledpotential :

Potential φ at any point of the electrostatic field there scalar physical quantity determined by the potential energy of a single positive charge placed at this point.

1.7 Communication between tensions and potential.

Electrostatic field - Al. Field of still charge.
Fel, acting on the charge, moves it, making a slave.
In a homogeneous electric field FEL \u003d QE - constant value

Work field (email) does not depend From the form of the trajectory and on a closed trajectory \u003d zero.

Electrostatics (from electric ... and static) , The section of the theory of electricity, in which the interaction of fixed electrical charges is studied. It is carried out by means of an electrostatic field. The main law of E. - Kulona law, which determines the strength of the interaction of fixed point charges, depending on their magnitude and the distance between them.

Electrical charges are sources of an electrostatic field. This fact expresses Gauss theorem. The electrostatic field is potentially, i.e. the work of the forces acting on the charge from the electrostatic field does not depend on the form of the path.

The electrostatic field satisfies the equations:

div D. \u003d 4Pr, Rot E. = 0,

where D - vector electric induction (see Induction Electrical and Magnetic), E - The voltage of the electrostatic field, R is the density of the electrical charge. The first equation is the differential form of the Gauss Theorem, and the second expresses the potential nature of the electrostatic field. These equations can be obtained as a special case of Maxwell equations.

Typical problems E. - Finding the distribution of charges on the surfaces of conductors according to the well-known complete charges or potentials of each of them, as well as the calculation of the energy of the conductor system by their charges and potentials.

To establish communication between the power characteristic of the electric field  tenseand it energy characteristic  potentialconsider elementary work Electric field forces on an infinitely small movement of point charge q.: D. A \u003d Q.E.d. l., the same work is equal to the decrease of potential charge energy q.: D. A \u003d d. W. P \u003d  Q.d, where D is the change in the potential of the electric field at the length of the displacement D l.. Equating the right parts of expressions, we get: E.d. l. D or in the Cartesian coordinate system

E X.d. x + E Yd. y + E zd. z \u003d.D, (1.8)

where E X., E Y., E Z.- Projections of the vector of tension on the axis of the coordinate system. Since the expression (1.8) is a complete differential, then for the projections of the tension vector we have

Equipotential surface - the concept applicable to any potential vector field, for example, to a static-electrical field or to Newtonium gravitational field (gravity). Equipotential surface is a surface to which the potential of this potential field Takes a constant value. Another equivalent, definition - surface, at any point orthogonal field lines.

The surface of the conductor in electrostatics is an equipotential surface. In addition, the conductor of the conductor on the equipotential surface does not cause changes in the configuration of the electrostatic field. This fact is used in the image method that allows you to calculate the electrostatic field for complex configurations.

In the gravitational field, the level of the fixed fluid is installed on the equipotential surface. In particular, the level of oceans passes along the equipotential surface of the gravitational field of the Earth. The equipotential surface of the ocean level, continued on the surface of the Earth, is called geoid and plays an important role in geodesy.

5.Electrical Capacity - Characteristics of the conductor, the measure of its ability to accumulate an electrical charge. In the theory of electrical circuits, the container is called the mutual capacity between the two conductors; The parameter of the capacitive element electrical circuitrepresented in the form of a two-pole. Such a container is defined as the ratio of the amount of electric charge to the potential difference between these conductors.

In the system, the container is measured in the Farades. In the SGS system in centimeters.

For a single conductor, the capacity is equal to the ratio of the charge of the conductor to its potential under the assumption that all other conductors are infinite-minded and that the potential of an infinitely remote point is taken equal to zero. In mathematical form, this definition has a view

Where Q. - Charge, U. - Conductor potential.

The capacity is determined by the geometric dimensions and the form of the conductor and the electrical properties of the environment (its dielectric permeability) And does not depend on the material of the conductor. For example, the capacity of the conducting bowl of the radius R. equal (in system SI):

C. \u003d 4πε 0 ε R..

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

where S. - the area of \u200b\u200bone plated (it is understood that they are equal), d. - distance between the plates, ε - relative dielectric permeability between the plates, ε 0 \u003d 8.854 × 10 -12 F / M - electrical constant.

With parallel compound K capacitors Full capacity is equal to the amount of containers of individual capacitors:

C \u003d C 1+ C 2.+ ... + C k.

With a sequential connection K capacitors folded inverse gas tanks:

1 / C \u003d 1 / C 1+ 1 / C 2+ ... + 1 / C k.

The energy of the electrical field of the charged condenser is:

W \u003d QU / 2 \u003d Cu 2 /2 \u003d Q 2/ (2C).

6. Electric current is calledconstant If the current and its direction does not change over time.

Tok Power (often just " current") In the conductor, the scalar value is numerically equal to the charge flowing per unit of time through the seren. Denoted by the letter (in some courses -. Do not be confused with current vector density):

The main formula used to solve problems is the law of Ohm:

§ for the site electrical chain:

The current strength is equal to the ratio of the resistance.

§ For full electrical circuit:

Where E is EMF, R is an external resistance, R - internal resistance.

Unit of measurement in C - 1 amp (a) \u003d 1 pendant / second.

To measure the current force, a special device is used - an ammeter (for devices intended for measuring small currents, the names of the milliammeter, micro ammeter, galvanometer) are also used. It is included in the chain gap in the place where the current is measured. The main methods for measuring the current force: magnetoelectric, electromagnetic and indirect (by measuring voltage voltmeter on a known resistance).

When alternating current There are instant current strength, amplitude (peak) current strength and efficient current strength ( equal power DC, which highlights the same power).

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

Current in cross section of the conductor.

Among the conditions necessary for existence electric current distinguish:

· Presence in medium of free electrical charges

· Creating an electric field in an environment

Thirdness - The forces of non-electrical nature, causing the movement of electrical charges within the DC source.
Third-party are considered all the forces other than the Coulomb forces.

Electromotive force (EMF), the physical quantity characterizing the effect of third-party (non-optical) forces in the sources of direct or alternating current; In a closed conductive circuit, the operation of these forces on the movement of a single positive charge along the contour is equal. If E. page designate the intensity of the side of the third-party strength, then the EMF in the closed circuit ( L.) Equal , where dL - Element of contour length.

Potential forces electrostatic (or stationary) fields can not support d.C. In the chain, since the work of these forces on the closed path is zero. The passage of current on the conductors is accompanied by the release of energy - heating the conductors. Third-party forces lead charged particles within the current sources: generators, galvanic elements, batteries, etc. The origin of third-party forces may be different. In the generators, third-party strength is the forces by the vortex electric field arising from change magnetic field Over time, or Lorentz, force acting from the magnetic field to electrons in a moving conductor; In electroplating elements and batteries, this is chemical forces, etc. EMF determines the current strength in the chain with a given resistance (see Ohma law) . EMF is measured, as well as voltage, edges.