Calculate work when moving electric charge in homogeneous electric field With tension. If the charge of the charge occurred along the field strength line by distance (Fig. 134), then the work is equal

Fig. 134.

where and - distances from the initial and endpoints to the plate AT.

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 It does not depend on the trajectory of his movement.

This conclusion is confirmed by the most accurate experiments.

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 the charge is moving from the point AT exactly WITH The power of the electric field was done, then when the charge is moved along the same path from the point WITH exactly AT They do work. But since the work does not depend on the trajectory, then when moving along the trajectory SKV This is also done. From here it follows that when the charge is moving, first from the point AT exactly WITH and then from the point WITH exactly AT . on a closed trajectory, the total work of forces electrostatic field It turns out to be zero (Fig. 135).

Fig. 135.

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.

Potential charge energy in the electric field. Continue comparing the gravitational interaction of bodies and electrostatic interaction of charges. The body mass in the field of gravity of the Earth has potential energy.

Gravity work is equal to change potential energytaken with the opposite sign:

In the same way as the body mass in the field of gravity has the potential energy, proportional mass of the body, the electrical charge in the electrostatic field has potential energy proportional to the charge. The work of the power of the electrostatic field is equal to the change in the potential energy of charge in the electric field, taken with the opposite sign:

(40.1)

Potential. At one point of the electrostatic field, different charges may have different potential energy, but the ratio of potential energy to the charge for this point of the field is constant. This value is taken for the energy characteristics of this point of the field.

The physical value equal to the ratio of the potential energy of an electric charge in the electric field to the charging is called the potential of the electric field:

Hence the potential charge energy in the electrostatic field is equal to the charge of the electric field potential at this point:

The value of the potential electric charge energy at this point of the electric field is determined not only by the characteristics of the electric field, but also the charge sign placed at this point point, and the choice of zero reference level of potential energy.

The potential is scalar value. If at a certain point of space, two charges simultaneously created electrical fields with potentials and, the potential of two electric fields is equal to the algebraic amount of potentials and:

In the same way, you can find the potential of an electric field created by any number of electrical charges.

Potential difference. The measure of energy change in the interactions of bodies is the work. We found out that when moving an electric charge, the work of the power of the electrostatic field is equal to a change in the potential charge energy taken with the opposite sign, therefore, from expressions (40.1) and (40.3) we obtain

When moving the electric charge in the electrostatic field, the operation of the forces of the field is equal to the product of the charge on the difference in the potentials of the initial and end points of the charge trajectory.

Since the work of the power of the electrostatic field when the charge is moved from one point of space to another does not depend on the trajectory of the charge between these points, the difference in the potentials of two points of the electric field is the magnitude that does not depend on the trajectory of the charge traffic. The difference of potentials, therefore, can serve energy characteristic electrostatic field.

If the field potential at an infinitely large distance from the point electrical charge in vacuo is taken equal to zero, then it is determined at a distance from charge by the formula

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 silence characteristic electric field  tenseand its energy characteristic  potentialconsider elementary work electric field forces on infinitely small movement 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.

Capacity is determined by the geometric sizes and the form of the conductor and the electrical properties ambient (her 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 quantityhaving the meaning of the strength of the current 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 contour, it is equal to the work of these forces to move the unit positive charge Along the contour. 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 of electrostatic (or stationary) fields cannot 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.

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 sizedetermined by the potential energy of a single positive charge placed at this point.

1.7 Communication between tensions and potential.

For any charge, which is in the electric field, there is a force that can move this charge. We define the work and the movement of a point positive charge from the point of about the point performed by the electric field negative charge (Fig. 158). According to the law of the coulon, the force moving the charge is variable and equal

where the variable distance between charges. Note that according to the same law (inverse proportionality, the square of the distance) changes the force moving the mass in the gravitational field of mass (see § 17).

Therefore, the operation of the charge of charge in the electric field (performed by electrical forces) will be expressed by a formula similar to the formula for the operation of the mass in the gravitational field (performed by gravitational forces):

Formula (19) is excreted in exactly the same way, which formula (8) was derived in § 17.

It is even easier to derive the formula (19) by integrating:

The minus sign in front of the integral is delivered due to the fact that for ripple charges, the magnitude is negative, while the work should be positive, since the movement of the charge occurs in the direction of force.

Comparing formula (19) with a general formula (4) from § 17, we will come to the conclusion that the value is the potential charge energy at this point of the electric field:

The minus sign shows that as the charge is moved by the field, its potential energy decreases, turning into the work of moving. Value

equal to the potential energy of a single positive charge is called the electric field potential or electrical potential. Electric potential It does not depend on the magnitude of the flow of charge and therefore can serve as the characteristic of the electric field, just as the gravitational potential serves as the characteristic of the gravitational field.

Substituting the expression of the potential (21) in the work formula (19), we obtain

Personal reception

Thus, the difference in the potentials of two field points is equal to the operation of the field forces to move a single positive charge from one point to another.

Move now the charge (acting against the field strength) from some point to infinity then, according to formulas (21) and (23), and

When we get there, the potential of the point of the electric field equal to work Displacement of a single positive charge from this point to infinity.

From formula (24), we establish a potential measurement unit called Volt (B):

i.e. Volt is the potential of such a point of the field, when moving from which the charge "And the infinity work is performed in the potential dimension

Now, given the formula (25), it can be shown that installed in § 75 unit of the electric field strength is really equal

If the charge creates the field is negative, then the field force prevents the movement of a single positive charge to infinity, thereby making negative work. Therefore, the potential of any point of the field created by a negative charge is negative (just as the gravitational potential of any point of the field is negative). If the charge that creates the field is positive, then the field forces themselves move a single positive charge to infinity, making a positive work. Therefore, the potential of any point of the Positive charge field is positive. Based on these considerations, you can record expression (21) in a more general form:

where the sign minus refers to the case of a negative charge, and the sign plus - to the case of a positive charge

If the field is created by several charges, its potential is equal to the algebraic sum of the potentials of the fields of all these charges (the potential is the scalar value: the ratio of work to the charge). Therefore, the potential of the field of any charged system can be calculated based on the previously presented formulas, pre-breaking the system to a large number of point charges.

The work of charging the charge in the electric field, as well as the operation of the mass movement in the gravitational field, does not depend on the shape of the path, and depends only on the difference in the potentials of the initial and endpoint paths. Hence, electric power are potential forces (see § 17). The surface, in all points of which the potential is the same, called the equipotential. From formula (22) it follows that the operation of the charge along the equipotential surface is zero (as it means that the power of the electric field is sent perpendicular to equipotential surfaces, i.e. power lines Fields are perpendicular to the equipotential surfaces (Fig. 159).