The potential scalar physical value is characteristic. Potential

In addition to the difference in potentials, characterizing two field points, use the concept of potential, which is the energy characteristic of each point of the field. We introduce this concept, based on the expression (12) for spot charge:

When .(14)

Due to the arbitrariness of the point of choice, the index can be omitted.

The potential of the field at this point is called a scalar value numerically equal work Displacement of a single charge from the specified point of the field in infinity :

On the other hand, with from (11) we get ,

where W is the potential energy of the charge system and. In this regard, it is possible to give another equivalent to the first, determination of the potential.

The potential is physical quantity, numerically equal potential energy single charge placed at this field

In practice, there are no cases with infinity, but determine the potential relative to another point, relative to the ground or ground (joined) device of the device. Similarly, the potential (potential energy) of the field of gravity is often determined relative to the surface of the Earth.

The principle of superposition is performed for potential, therefore the potential of the field of charge system is equal to the amount of the potentials of the fields of each charge separately:

From expressions (15) and (16) a unit of measurement unit is established - Volt: I B \u003d I J / I CL.

5 Communication between voltage and potential difference

The electrostatic field can be specified by the values \u200b\u200bof either the voltage vector, or the potential at each point. Obviously, there is a connection between these values.

The elementary work of charging in the electrostatic field can be calculated as follows: or, loss of potential energy.

Equating the right-wing parts of both equalities, we get because

We divide the last expression on. Since, then, then

In accordance with the expression (18) the magnitude of tension electric field Depends on the rate of potential change along the radius - vector. If the charge that creates the field. It is point, then the rate of potential change along the radius - vector is the maximum largest. In mathematics, the concept of a gradient (GRAD) is introduced in the direction of the maximum rate of change in the direction of the gradient (GRAD), the vector of which is equal to the maximum rate of change of this value, directed towards the most rapid increase. Taking into account this equality (17) can be recorded in vector form: (18)

The "-" sign in the record (18) reflects the fact that the voltage vector is directed towards the most rapid decrease in the potential. For homogeneous electrostatic field \u003d Subst, i.e. ,

where and the potentials of points removed from the beginning of the reference at distances R 1 and R 2. In electrical engineering, the difference in potentials is called the voltage between the two points.:. T. about. for uniform field.

Graphically electric field can be depicted using equipotential surfacesthat is, surfaces of equal potential. For these surfaces , and since , then, that is, the voltage vector is perpendicular to the equivalent surfaces (figures 17 and 18).

Figure 17 Figure 18

From formulas (12) and (13) it follows that loss of the potential energy of a single positive charge during the transition from one point of the field to another numerically equal to the operation of the field for the movement of the charge between these points :.

When moving the charge there is a conversion electrical Energy in mechanical.

Since the electric field is potential. And the electrical forces are conservative, then with such a transformation, the law of conservation of energy is performed.

End of work -

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Electrostatic field

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If the circuit with the current is in an inhomogeneous magnetic field, then the different forces act on different parts.

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Magnetic field work
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Dia-, para- and ferromagnets and their application.
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Maxwell equations
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It is known that the forces of gravitational and electrical interaction equally depend on the distance, the vectors of gravitational and Coulomb forces in the interaction of point bodies are directed in a straight line connecting the interacting bodies. When moving the body between two points in the gravitational field, the operation of gravity does not depend on the form of the trajectory of its movement. Therefore, it can be assumed that when moving the charged particle in the electrostatic field, from one point to another operation of the electric field forces also does not depend on the form of the trajectory. When the particle is moving along a closed trajectory, this work 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.

Work forces in homogeneous electric field. We prove the independence of the work of the power of the electrostatic field on the type of the trajectory of the charge between the two points of the homogeneous field. Let in a homogeneous electric field tension electrical charge q. moves from the point IN exactly D. (Fig. 1.23). If the charge was moving in a straight line BD., the work of the power of the electric field would be equal to

where S is the movement vector module, the angle between the directions of the Coulomb strength vector and the charge travel vector.

If the charge is out of point B. First moves straight to the point C.and then straight from the point C. exactly D.The work of the power of the electric field will still be equal. Really,

Thus, the work of the forces of a uniform electric field when moving the electric charge in a straight line BD. And on scrap BCD. The same is equal to the product of an electric charge on the electric field strength and the distance to which the charge moved along the electric field strength line:

Any curve connecting points B. and D. In a homogeneous electric field, it is possible to approximately represent consisting of consecutive segments located in parallel and perpendicular to the intensity lines. Applying the same reasoning for each site of the trajectory, we obtain that the expression obtained for work is fair to calculate the work of the forces of a homogeneous electric field when the charge is moved along any trajectory.

Fig. 1.24.

Work forces in the field of point charge. We define the work that is performed when the charge is moving in an electric field, the source of which is a point positive charge Q.. Let a positive trial charge q. is first at point M.distance from charge Q,and then it turns out at the point N. Distance from charge Q.(Fig. 1.24). Suppose also that the charge was moving at first along the radius in a straight line MK.and then on the arc KN. circumference with radius. Then work on the movement of charge from the point M. exactly N. equal to the amount of work on the plots MK. and KN.. Work on the movement of charge on the arc is zero, since the vectors of the strength and movement are perpendicular to each other. It remains to find the work performed by the field when moving the test charge along the radius, that is, along the line of tension. But in this area at every point, the force acting on the charge from the side of the electric field takes a new value. However, on a very small segment of the path along the line of tension, the elementary work of the electric field forces can be taken equal to

where r. - distance from the source of the field to the dedicated segment. The work of the power of the electric field when moving the test charge q. From the point M, located at a distance from charge Q., exactly K.located at a distance, can be found as a sum elementary work On small segments of the way:

The operation of finding such a sum in the transition to infinitely small values \u200b\u200bof movement is called integration. Integration in this case leads to the following result:

(1.6)

It is easy to show that if the test charge is moved along another trajectory, for example, from the point M. exactly L. on the arc of the circle and then along the radial direct to the point N.The result will be the same. Consequently, the work of the point of the point charge on the movement of the test charge on an arbitrary path does not depend on the form of the trajectory and is determined only by the provisions of the initial and endpoints, that is, the field of point charge is potential.

But if the current forces are potential, then the work of these forces can be represented as a difference in potential energies in the initial and endpoints of the trajectory:

The potential energy of the interaction of charges is determined with an accuracy of an arbitrary constant term, the value of which can be set so as to simplify the solution of the problem. However, the work of the forces of the field does not depend on this arbitrary constant, since it is determined by the difference in potential energies. The permanent integration value is chosen in such a way that when the charge is removed to infinity (that is, the potential energy appealed to zero. Under this condition it turns out that

Thus, the electrostatic field is potentially, and any charged particle in it has potential energy. Different trial charges will have in the same field of the field with different potential energy. However, the ratio of potential energy to the magnitude of the trial charge will be the same.

Similarly, how was introduced power characteristic electric field - its tension, introduce energy characteristic Electric field - potential:

(1.9)

The potential of the electric field at a given point is the ratio of potential energy, which has a trial charge placed at this field, to this charge. From (1.9) it follows that the potential is numerically equal to the potential energy, which would have a single positive charge at a given point.

Comparing the expressions (1.8) and (1.9), we will receive an expression for the potential of a point charge field:

The potential is defined with an accuracy of an arbitrary constant, depending on the selection of the zero level of potential. Typically, it is believed to zero the field potential at the point remotely remotely far from the point source of the field; That is, with the potential. With this condition. Then the expression for the potential of the point of the point charge is:

Consider the field created by the system N. spot charges . Distance from each charge to this point of the field will be denoted. Work performed by this field above charge q.is equal to the algebraic amount of work of the forces due to each charge separately: q can be expressed through the potential difference:

It follows that the potential is numerically equal to workwhich the field forces makes a single positive charge when removing it from this field of the field into infinity.

Formula (1.13) can be used to establish capacity units. The capacity of the potential takes the potential at such a point in the field, to move into which an infinitely remote unit positive charge must be done, equal to one. In the system of the SI unit, the capacity of the potential is called Volta, is the difference in the potentials of two points of the electric field, during the transition between which the charge field does the field:

In the SGSE system, the capacity unit has no special name.

Lecture 6.Electrostatic field

Plan lectures

Electrical charges, their properties. The law of the coulon.

Electrostatic field strength. The principle of superposition of electric fields. Gaussian theorem.

The potential and difference of the potentials of the electrostatic field.

Electrical capacity of conductors and capacitors. Energy electrostatic field.

Electrical charges, their properties. The law of the coulon.

Electric charge is a physical value that characterizes the intensity of electromagnetic interaction of tel. Electric charge itself does not exist, its carrier can only be a particle of matter.

Basic properties

Duality: In nature, there are charges of two signs, the same are repelled, the variepetes are attracted. In this regard, the conditional charges are divided into positive and negative.

A positive charge named with a glass wand, loss of silk or paper.

Negative - a charge that has an amber or ebonite wand, loss of fur or wool.

Quantization: If the physical value takes only certain discrete values, it is said that it is quantized (discrete). Experience shows that any electrical charge quantum, i.e. consists of an integer number of elementary charges:

where

\u003d 1,2, ... an integer; E \u003d 1.6 · 10 -19 CB - elementary charge.

The lowest (elementary) negative charge has an electron, a positive - proton.

1 pendant - charge passing through the cross section of the conductor in one second when the conductor goes d.C. by one ampere.

Saving charge.

The law of saving charge: in an isolated system, the algebraic amount of charges remains a constant value with all changes inside the system.

Isolatedit is called a system of bodies that does not exchange charges with an external environment.

Invariance Charge to different inertial reference systems.

Experience shows that the amount of charge does not depend on the speed of the charged body. The same charge, measured in different inertial reporting systems, the same.

Z. ankon, which is subject to the strength of interaction of point charges, was established experimentally in 1785. Pendant.

The law of Kulon. The strength of the interaction of two fixed point charges is directly proportional to the charges, inversely proportional to the square of the distance between them, is directed along the direct connecting charges, and depends on the medium in which they are located (Fig. 1).

where

- values \u200b\u200bof charges; - distance between charges;

8.85 · 10 -12 CL 2 / (N · m 2) - electrical constant;

 - dielectric permeability of the medium.

the dielectric permeability of the substance shows how many times the strength of the interaction of charges in this dielectric is less than in vacuum, vacuum \u003d 1, is a dimensionless value.

Electrostatic field strength. The principle of superposition of electric fields. Gaussian theorem.

Any charge changes the properties of the surrounding space - creates an electric field in it.

The electric field is one of the forms of the existence of matter surrounding electric charges. This field manifests itself in the fact that an electric charge placed in some kind of it is under force.

The idea of \u200b\u200bthe electric field was introduced into science in the 1930s of the XIX century English scientists by Michael Faraday.

According to Faraday, each electrical charge is surrounded by the electric field created by it, so such a charge is sometimes called a source charge. The charge with which the source charge field is examined, called trial charge.

In order for the force acting on the trial charge characterized the field at this point, the test charge must be point.

Point charge Called the charged body, the sizes of which can be neglected under the conditions of this task, i.e. The dimensions of which are small compared to distances to the other bodies with which it interacts. In this case, the proper electric field of the test charge should be so little so that it does not change the charge field - source. The smaller the size of the charged body and the weaker its own field compared to the source field, the more accurately this charged body satisfies the condition of the test charge.

The electric field spreads in vacuo at a speed C \u003d 3 · 10 8.

Field of fixed electrical charges - electrostatic.

Explore with the help of a test charge field created by a fixed charge - source

.

The force acting on the trial charge at this point point depends on the value of the test charge. If you take various test charges, then the force acting on them at this point of the field will be different.

However, the ratio of force to the magnitude of the test charge remains constant and characterizes the field itself. This ratio is called the electric field strength at this point.

Electric field tension- This is a vector value that is numerically equal to force with which the field acts on a single positive test charge at a given point of the field and coated with this force (Fig. 2).

Tensions is the main characteristic of the field and fully characterizes the field at each it point in size and direction.

Tension field of point charge.

According to the law of Coulomb

therefore \u003d.

.

Consequently, the electric field charge electric field on distance from this charge is expressed by the formula:

The electrical field is conveniently graphically displayed using a painting of so-called power lines or tension lines.

Line of tension A line is called the tangent to which at each point coincides in the direction of tension at this point.

The field strength of the field created by fixed charges (Fig. 3 A, b) is always starting and ended on charges (or in infinity) and are never closed. A stronger field is depicted by more densely located lines of tension. The lines thickness is chosen so that the number of lines penetrating the unit surface of the platform perpendicular to the lines was equal to the numerical value of the vector. Tension lines never intersect, because Their intersection would mean two different directions of the field strength vector in the same point that it does not make sense.

ABOUT dnorodno called the field in which the tension at all points has the same magnitude and the same direction,

. In such a field power lines Parallel and the density of them everywhere is the same, i.e. They are located at the same distance from each other (Fig. 3B).

The principle of superposition.

If the electrical field at a given point is created by several charges, the tension of the resulting field is equal to the vector sum of field tensions created by each charge separately (Fig. 4).

The principle of superposition is an experienced fact, just up to very strong fields. Therefore, the law is not only static, but also rapidly changing electromagnetic fields

Flow

vector tension through an elementary surface ds. called a scalar piece

where

- vector, the module of which is equal

and the direction coincides with the direction of normal to the site;

- The angle between vectors and;

E P is a projection on the direction of normal.

Having arise the streams through all the elementary platforms to which the surface S is broken, we obtain the flow of the vector through the surface S.

Thread vector Through the surface S is called integral

For a closed surface.

Theorem Gaussa For electrostatic field in vacuum:

The stream of the intensity of the electrostatic field in vacuum through any closed surface is directly proportional to the algebraic amount of charges covered by this surface.

With the help of the Gauss Theorem, formulas defining tensions in a number of special cases were obtained.

Field of an infinite uniformly charged plane(Fig. 5):

Surface density charge - The physical value equal to the charge occurring per unit area of \u200b\u200ba uniformly charged surface.

If the surface is charged unevenly,

P ole 2 endless variepealed charged planes(Fig. 6):

The resulting result is fair for the planes of the final dimensions, the distance between which is not enough compared to their dimensions (condenser).

In addition to the main vector in the theory of electricity it turns out to be introduced yet. electric induction (or bias vector).

,

=1.

The potential and difference of the potentials of the electrostatic field.

Any point in the field created by a fixed charge q. 0 , Charge q. Power acts

If the charge q moves from point 1 to point 2, then this force makes a job. It can be proved that this work is determined by the formula:

From the resulting expression it follows that the work in the electrostatic field does not depend on the trajectory of the charge movement, but is determined only by the initial positions 1 and the final 2 points. Consequently, the electrostatic field is potential, and the electrostatic forces are conservative.

The body located in a potential field of forces (and the electrostatic field is potential), has potential energy, due to which work is performed by the field. As is known from the mechanics, the work of conservative forces is performed due to the decrease of potential energy. Therefore, work can be represented as a difference in potential energies that have a charge q. In the initial (1) and ultimate (2) points:

from where it follows that the potential energy charge in the charge field equal

which, as in the mechanics, is not defined unequivocally, but up to an arbitrary constant FROM. If we assume that when the charge is removed infinity (

) Potential energy refers to zero (

), T. FROM=0.

Potential energy charge in the charge field on distance from him: .

Different charges will possess in the same point of the field different energy

, however, attitude At this point, all charges will be equally and called the potential of the electrostatic field at this point:

The field potential at this point is a scalar physical value, numerically equal to the potential energy, which has a single positive charge at this point point.

1 B is the potential of the field point in which the positive charge of 1 CL has a potential energy of 1 J.

The potential is the energy characteristics of the field. Conditionally it is considered to be a higher potential of the point of the field, which is closer to positive charge - Field source.

The potential at some point of the field created by the charge system is equal to the algebraic amount of field potentials created by each charge separately: if.

Potential Pattern Point Charge on distance from him (or spheres by radius with charge ):

Because .

At a point with potential  1 charge q. possesses potential energy

at the point with the potential  2:

The operation of moving the charge from the point with the potential  1 to the point C  2 can be represented as a change in potential energy:

A \u003d w p1 - w p2 \u003d q 1 - q 2 \u003d q ( 1 - 2),

- Potential differences (voltage) between points.

The potential difference between the two points of the electrostatic field is a physical value, numerically equal to work on moving the trial charge between these points.

The relationship between the tensions and the potential of the electrostatic field.

For a homogeneous field (Fig. 7):

In the electric field, you can spend the surface, all points of which have the same potential.

Imaginary surface, all points of which have the same potential called equipotential.

The operation of the charging of the charge on the equipotential surface is zero, because  1 \u003d  2.

Equipotential surfaces are always perpendicular to the lines (Fig. 8). Their agreed in such a way that  for two neighboring surfaces was the same.

Electrical capacity of conductors and capacitors. Energy electrostatic field.

Under the word "conductor" in physics means a conductive body of any size and a form containing free charges (electrons or ions). For certainty, we will consider metals in the future.

Electrical capacity of a secluded conductor. Electrical capacity of the ball.

If the charge on the conductor is increased several times, the potential at each point of the field surrounding the conductor will increase:

The electrical capacity of the conductor is numerically equal to the charge, which must be reported to the conductor to change its potential per unit.

1 F - Conductor capacity to whom you need to inform 1 CL charge to change the potential by 1 V.

The conductor capacity does not depend on the metal from which it is manufactured.

Capacity depends on the size and shape of the conductor, the environment and the presence of near other conductors. In the dielectric, the container increases by  times.

Calculate the container of the ball with a radius R.:

Capacitors and their electrical capacity. Sequential and parallel connection of capacitors.

The capacity of the secluded conductors is small, but it increases sharply if there is a number of other conductors.

Capacitor - a system of two conductors separated by a dielectric located at a short distance from each other.

The field is concentrated in space between the plates.

Capacitors are divided:

form: flat, cylindrical, spherical;

by the kind of dielectric between the plates:

air, paper, saliva, ceramic;

according to the type of capacity: constant and variable tank.

Designations on Radioshechum

The capacitance of the capacitor is numerically equal to the charge that one of the plates must be reported to change the potential difference between them by one.