When moving a test charge q. in electric field electric power Make work. This work with a small movement is equal to (Fig. 1.4.1):

Consider the work of the forces in the electric field, created in the time distributed charge, i.e. Electrostatic field

The electrostatic field has an important property:

Work forces electrostatic field When the charge is moved from one point of the field to another, it does not depend on the form of the trajectory, but is determined only by the position of the initial and endpoint and the size of the charge.

A gravitational field has a similar property, and there is nothing surprising, since the gravitational and pendant forces are described by the same relations.

The consequence of the independence of the work on the form of the trajectory is the following statement:

The operation of the power of the electrostatic field when the charge is moving along any closed trajectory is zero.

Power fields possessing this property called potential or conservative .

In fig. 1.4.2 depicted power lines Coulomb Field Of Spot Charge Q. and two different trajectories of the trial charge q. From the starting point (1) to the end point (2). On one of the trajectories allocated a small movement operation Δ A. Coulomb forces on this movement is equal

The result obtained does not depend on the form of the trajectory. On the trajectories I and II shown in Fig. 1.4.2, the works of the Coulomb forces are the same. If one of the trajectories change the direction of movement of charge q. On the opposite, then the work will change the sign. From here it follows that on a closed trajectory, the work of the Coulomb forces is zero.

If the electrostatic field is created by the set spot charges then when moving a trial charge q. Work A. The resulting field in accordance with the principle of the superposition will be folded from the works of Coulomb fields of point charges: since each member of the summary depends on the form of the trajectory, then the full work A. The resulting field does not depend on the path and is determined only by the position of the initial and end points.

The properties of the potentiality of the electrostatic field allows you to enter the concept potential energy Charge in the electric field. To do this, in space, some point (0) is selected, and the potential energy of charge q.placed at this point is taken equal to zero.

Potential energy chargeq. placed at any point (1) of space relative to the fixed point (0) is equal to workA. 10 which the electrostatic field will make when the charge is movingq. From point (1) to point (0):

W. p1 \u003d. A. 10 .

(In electrostatics, energy is accepted to mark the letter W.since the letter E. Denote the field strength.)

Also, as in mechanics, the potential energy is determined with an accuracy of a constant value depending on the choice of the reference point (0). Such ambiguity in determining potential energy does not lead to any misunderstandings because physical meaning It has no potential energy itself, but the difference between its values \u200b\u200bat two points of space.

Work performed by electrostatic field when moving point chargeq. From point (1) to point (2), equal to the difference of potential energy values \u200b\u200bat these points and does not depend on the path of movement of the charge and from the point selection (0).

The potential φ is energy characteristic electrostatic field.

Work A. 12 by movement electric charge q. From the starting point (1) to the end point (2) is equal to the work of the charge on potential difference (φ 1 - φ 2) initial and end points:

In many tasks of electrostatics, when calculating the potentials for the support point (0), it is convenient to take an infinitely remote point. In this case, the concept of potential can be determined as follows:

The field potential at this point of space is equal to the work that electrical forces makes when the single positive charge is removed from this point to infinity.

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

For a visual presentation, electrostatic fields along with power lines use equipotential surfaces.

Surface, at all points of which the potential electric field has the same values \u200b\u200bcalledequipotential surface orthe surface of equal potential .

Power lines The electrostatic fields are always perpendicular to the equipotential surfaces.

Equipotential surfaces of the Coulomb field of point charge are concentric spheres. In fig. 1.4.3 Presents patterns of power lines and the equipotential surfaces of some simple electrostatic fields.

When uniform field Equipotential surfaces are a system of parallel planes.

If a trial charge q. performed small moving along the power line From point (1) to point (2), then you can write:

This ratio in the scalar form expresses the relationship between field strength and potential. Here l. - Coordinate, counted along the power line.

Of the principle of superposition of the tensions of fields generated by electric charges, the principle of superposition for potentials is followed:

Equipotential surfaces - the concept applicable to the lovo-substantial vector field, for example, to a static electric field or to the Newtonian gravitational field. The equipotential surface is a surface on which the scalar potential of this potential field takes constant value (surface level of the level). 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 (stationary), the gravitational field, the level of the fixed fluid is installed on the equipotential surface. In particular, it can be approximately argued that the level of oceans passes on the equipotential surface of the gravitational field of the Earth. The shape of the surface of the oceans, continued on the surface of the Earth, is called geoid and plays an important role in geodesy. Geoid is thus the equipotential surface of gravity consisting of gravitational and centrifugal components.

Equipotential lines

Lines of equal values \u200b\u200bof the potential of the expired electric field.

§ 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.

The forces of the 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 point charge + Q 0 moves to the stationary point charge + q field 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 work forces of the electrostatic field to move 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 potential energy of 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 to the power lines of the field.

We calculate the work when moving the electrical charge in a uniform 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 IN.

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 that when the charge in the electric field is moved, from one point to another work of the electric field forces does not depend on the trajectory of its 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 IN exactly FROM The power of the electric field was done, then when the charge is moved along the same path from the point FROM exactly IN 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 IN exactly FROM and then from the point FROM exactly IN . According to a closed trajectory, the total work of the power of the electrostatic field 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 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.

The work of gravity is equal to the change in the potential energy taken 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 potential difference, therefore, can serve as the energy characteristic of the 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

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 obtain, therefore, the potential of the point of the electric field is equal to the operation of moving 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 (potential - scalar value: The ratio of work to the charging). 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. Consequently, electric forces 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. the field lines are perpendicular to the equipotential surfaces (Fig. 159).