In order to give a deeper definition of what was already familiar to us in the eighth grade physical quantity, let us remember the definition of the potential of a field point and how to calculate the work electric field.
Potential, as we remember, is the ratio of the potential energy of a charge placed at a certain point in the field to the magnitude of this charge, or this is the work that the field will do if a single positive charge is placed at this point.
Here is the potential energy of the charge; - amount of charge. As we remember from mechanics, to calculate the work done by a field on a charge: .
Let us now describe the potential energy using the definition of potential: . And let's do some algebraic transformations:
Thus, we obtain that .
For convenience, we introduce a special value denoting the difference under brackets: 
.
Definition: voltage (potential difference) - the ratio of the work performed by the field when transferring a charge from the initial point to the final point to the value of this charge.
Unit of measurement - V - volt:
.
Particular attention should be paid to the fact that, in contrast to the standard concept in physics of difference (the algebraic difference of a certain value at the final moment and the same value at the initial moment), to find the potential difference (voltage), the final potential should be subtracted from the initial potential.
To obtain the formula for this connection, as in the previous lesson, for simplicity, we will use the case of a uniform field created by two oppositely charged plates (see Fig. 1).
Fig.1. Example of a uniform field
The intensity vectors in this case of all field points between the plates have one direction and one magnitude. Now, if a positive charge is placed near the positive plate, then under the influence of the Coulomb force it will naturally move towards the negative plate. Thus, the field will do some work on this charge. Let's write down the definition of mechanical work: . Here is the force module; - movement module; - the angle between the force and displacement vectors.
In our case, the force and displacement vectors are co-directional (a positive charge is repelled by a positive charge and attracted to a negative one), so the angle is zero and the cosine is one: .
Let's write the force through tension, and denote the displacement module as d - the distance between two points - the beginning and end of the movement: .
At the same time. Equating the right-hand sides of the equalities, we obtain the desired relationship:
It follows that tension can also be measured in .
Moving away from our homogeneous field model, special attention should be given to the inhomogeneous field that is created by a charged metal ball. From experiments it is clear that the potential of any point inside or on the surface of a ball (hollow or solid) does not change its value, namely:
.
Here is the electrostatic coefficient; - full charge of the ball; - radius of the ball.
The same formula is also valid for calculating the field potential of a point charge at a distance from this charge.
Energy of interaction of two charges
How to determine the energy of interaction between two charged bodies located at some distance from each other (see Fig. 2).
Rice. 2. Interaction of two bodies located at some distance r
To do this, let’s imagine the whole situation: as if body 2 is in the external field of body 1. Accordingly, now the interaction energy can be called the potential energy of charge 2 in the external field, the formula for which we know: .
Now, knowing the nature of the external field (the field of a point charge), we know the formula for calculating the potential at a point at a certain distance from the field source:
.
Substitute the second expression into the first and get the final result:
.
If we had initially imagined that charge 1 was in the external field of charge 2, then, of course, the result would not have changed.
In electrostatics, it is interesting to identify all points in space that have the same potential. Such points form certain surfaces, which are called equipotential.
Definition: equipotential surfaces are surfaces in which each point has the same potential. If you draw such surfaces and draw the field lines of intensity of the same electric field, you will notice that equipotential surfaces are always perpendicular to the field lines, and, in addition, the field lines are always directed in the direction of decreasing potential (see Fig. 3).
Rice. 3. Examples of equipotential surfaces
Another important fact about equipotential surfaces: based on the definition, the potential difference between any points on such a surface is zero (the potentials are equal), which means that the work done by the field to move a charge from one point on the equipotential surface to another is also zero.
In the next lesson we will take a closer look at the field of two charged plates, namely the device of a capacitor and its properties.
1) Tikhomirova S.A., Yavorsky B.M. Physics (basic level) M.: Mnemosyne. 2012
2) Gendenshtein L.E., Dick Yu.I. Physics 10th grade. M.: Ilexa. 2005
3) Kasyanov V.A. Physics 10th grade. M.: Bustard. 2010
1) Website "Physikon" ()
Homework
1) Page 95: No. 732 - 736. Physics. Problem book. 10-11 grades. Rymkevich A.P. M.: Bustard 2013 ()
2) At a point with a potential of 300 V, a charged body has a potential energy of -0.6 μJ. What is the charge of the body?
3) What kinetic energy did the electron receive after passing through an accelerating potential difference of 2 kV?
4) Along what trajectory should a charge be moved in an electric field so that its work is minimal?
5) *Draw equipotential surfaces of the field created by two unlike charges.
The potential difference between points 1 and 2 is the work done by field forces when moving a unit positive charge along an arbitrary path from point 1 to point 2. For potential fields, this work does not depend on the shape of the path, but is determined only by the positions of the starting and ending points
the potential is determined up to an additive constant. The work done by the electrostatic field forces when moving a charge q along an arbitrary path from the starting point 1 to the ending point 2 is determined by the expression
The practical unit of potential is the volt. A volt is the potential difference between such points that when moving one coulomb of electricity from one point to another electric field does one joule of work.
1 and 2 are infinitely close points located on the x axis, so X2 - x1 = dx.
The work done when moving a unit of charge from point 1 to point 2 will be Ex dx. The same work is equal. Equating both expressions, we get 
- scalar gradient
Gradient function 
there is a vector directed towards the maximum increase of this function, and its length is equal to the derivative of the function in the same direction. The geometric meaning of a gradient is equipotential surfaces (surfaces of equal potential) - a surface on which the potential remains constant.
13 Charge potential
Field potential of a point charge q in a homogeneous dielectric.
- electrical displacement of a point charge in a homogeneous dielectric D – vector of electrical induction or electrical displacement
We should take zero as the integration constant so that when the potential becomes zero, then
System field potential point charges in a homogeneous dielectric.
Using the superposition principle we get:
Potential of continuously distributed electrical charges.
-
elements of volume and charged surfaces with centers at a point
If the dielectric is inhomogeneous, then the integration must be extended to polarization charges. Inclusion of such
charges automatically takes into account the influence of the environment, and the value does not need to be entered
14 Electric field in matter
Electric field in matter. A substance introduced into an electric field can significantly change it. This is due to the fact that matter consists of charged particles. In the absence of an external field, particles are distributed inside a substance in such a way that the electric field they create, on average over volumes that include a large number of atoms or molecules, is zero. In the presence of an external field, a redistribution of charged particles occurs, and its own electric field arises in the substance. The total electric field is composed in accordance with the principle of superposition from the external field and the internal field created by charged particles of matter. The substance is diverse in its electrical properties. The broadest classes of substances are conductors and dielectrics. A conductor is a body or material in which electrical charges begin to move under the influence of an arbitrarily small force. Therefore, these charges are called free. In metals, free charges are electrons, in solutions and melts of salts (acids and alkalis) - ions. A dielectric is a body or material in which, under the influence of arbitrarily large forces, charges are displaced only by a small distance, not exceeding the size of an atom, relative to their equilibrium position. Such charges are called bound. Free and bound charges. FREE CHARGES 1) excess electric. charges imparted to a conducting or non-conducting body and causing a violation of its electrical neutrality. 2) Electric. current carrier charges. 3) put. electric charges of atomic residues in metals. ASSOCIATED CHARGES Electric. charges of particles that make up the atoms and molecules of the dielectric, as well as charges of ions in the crystalline. dielectrics with an ionic lattice.
Potential difference
It is known that one body can be heated more, and another less. The degree to which a body heats up is called its temperature. Likewise, one body can be electrified more than another. The degree of electrification of a body is characterized by a quantity called electrical potential or simply the potential of the body.
What does it mean to electrify the body? This means telling him electric charge , that is, add a certain number of electrons to it if we charge the body negatively, or subtract them from it if we charge the body positively. In both cases, the body will have a certain degree of electrification, i.e., one or another potential, and a body charged positively has a positive potential, and a body charged negatively has a negative potential.
Difference in electric charge levels two bodies are usually called electrical potential difference or just potential difference.
It should be borne in mind that if two identical bodies are charged with the same charges, but one is larger than the other, then there will also be a potential difference between them.
In addition, a potential difference exists between two such bodies, one of which is charged and the other has no charge. So, for example, if a body isolated from the earth has a certain potential, then the potential difference between it and the earth (the potential of which is considered to be zero) is numerically equal to the potential of this body.
So, if two bodies are charged in such a way that their potentials are unequal, a potential difference inevitably exists between them.
Everyone knows electrification phenomenon rubbing a comb against hair is nothing more than creating a potential difference between the comb and human hair.
Indeed, when a comb rubs against hair, some of the electrons transfer to the comb, charging it negatively, while the hair, having lost some electrons, becomes charged to the same extent as the comb, but positively. The potential difference created in this way can be reduced to zero by touching the hair with a comb. This reverse transition of electrons is easily detected by ear if an electrified comb is brought close to the ear. A characteristic crackling sound will indicate a discharge is occurring.
Speaking above about the potential difference, we meant two charged bodies, however A potential difference can also be obtained between different parts (points) of the same body.
So, for example, let's consider what will happen if, under the influence of some external force, we manage to move the free electrons located in the wire to one end of it. Obviously, at the other end of the wire there will be a shortage of electrons, and then a potential difference will arise between the ends of the wire.
As soon as we stop the action of the external force, the electrons will immediately, due to the attraction of opposite charges, rush to the positively charged end of the wire, i.e., to the place where they are missing, and electrical equilibrium will again occur in the wire.
Electromotive force and voltage
D to maintain electric current the conductor needs some kind of external source energy, which would always maintain a potential difference at the ends of this conductor.
These energy sources are the so-called electric current sources, having a certain electromotive force, which creates and long time maintains the potential difference at the ends of the conductor.
Electromotive force (abbreviated EMF) is denoted by the letter E. The unit of measurement for EMF is the volt. In our country, the volt is abbreviated as "B", and in the international designation - by the letter "V".
So, to obtain a continuous flow, you need an electromotive force, that is, you need a source of electric current.
The first such source of current was the so-called “voltaic column,” which consisted of a series of copper and zinc circles lined with leather soaked in acidified water. Thus, one of the ways to obtain electromotive force is the chemical interaction of certain substances, as a result of which chemical energy is converted into electrical energy. Current sources in which electromotive force is created in this way are called chemical current sources.
Currently, chemical current sources are galvanic cells and batteries - widely used in electrical engineering and power engineering.
Another main source of current, widely used in all areas of electrical engineering and power engineering, are generators.
Generators are installed on power stations and serve as the only source of current for supplying electricity to industrial enterprises, electric lighting of cities, electrical railways, tram, metro, trolleybuses, etc.
Both with chemical sources of electric current (cells and batteries) and with generators, the action of electromotive force is exactly the same. It lies in the fact that the EMF creates a potential difference at the terminals of the current source and maintains it for a long time.
These terminals are called current source poles. One pole of the current source always experiences a lack of electrons and, therefore, has a positive charge, the other pole experiences an excess of electrons and, therefore, has a negative charge.
Accordingly, one pole of the current source is called positive (+), the other - negative (-).
Current sources are used to supply electric current to various devices -. Current consumers are connected using conductors to the poles of the current source, forming a closed circuit. electrical circuit. The potential difference that is established between the poles of a current source in a closed electrical circuit is called voltage and is designated by the letter U.
The unit of measurement for voltage, like EMF, is the volt.
If, for example, it is necessary to write down that the voltage of the current source is 12 volts, then they write: U - 12 V.
A device called a voltmeter is used to measure voltage.
To measure the EMF or voltage of a current source, you need to connect a voltmeter directly to its poles. In this case, if it is open, the voltmeter will show EMF source current If you close the circuit, the voltmeter will no longer show the EMF, but the voltage at the terminals of the current source.
The EMF developed by a current source is always greater than the voltage at its terminals.
To study electrostatic field from an energy point of view, a positively charged point body - a test charge - is introduced into it, as in the case of considering tension. Let us assume that a uniform electric field, moving from point 1 to point 2 a body introduced into it with a charge q and along the path l, does work A = qEl(Fig. 62, a). If the amount of charge introduced is 2q, 3q, ..., nq, then the field will do the work accordingly: 2A, 3A, ..., nA. These works are different in magnitude, and therefore cannot serve as a characteristic of the electric field. If we take, respectively, the ratios of the values of these works to the values of the charge of the body, it turns out that these ratios for two points (1 and 2) are constant quantities:
If we study the electric field between any two of its points in a similar way, we will come to the conclusion that for any two points of the field the ratio of the amount of work to the amount of charge of the body moved by the field between the points is a constant value, but it is different depending on the distance between the points. The quantity measured by this Ratio is called the potential difference between two points of the electric field (denoted by φ 2 - φ 1) or the voltage U between the points of the field. Scalar quantity, which is an energy characteristic of an electric field and measured by the work performed by it when moving a point body, the charge of which is +1, from one point of the field to another, is called the potential difference between two points of the field, or the voltage between these points. From the definition of potential difference 
voltage U = φ 2 - φ 1 = Δφ.
There is an electric field around every charged body. As the distance from the body to any point in the field increases, the force with which it acts on the charge introduced into it decreases (Coulomb's law) and at some point in space practically becomes equal to zero. The place where the action of the electric field of a given charged body is not detected is called infinitely distant from him.
If the electroscope ball is placed in different points the electric field of the charged ball of the electrophore machine, then it charges the electroscope. When the electroscope ball is grounded, the electric field of the machine has no effect on the electroscope at all. The potential difference between an arbitrary point of the electric field and a point located on the Earth's surface is called the potential of this field point relative to the Earth. It is measured by work, to calculate which you need to know the starting and ending points of the path. A point on the Earth’s surface is taken as one of these points, and the work of moving the charge, and therefore the potential of the other point, is calculated relative to it.
If the electric field is formed by a positively charged body (Fig. 62, b), then it itself moves the positively charged body C brought into it to the surface of the Earth. The potentials of the points of such a field are considered positive. When the electric field is formed by a negatively charged body (Fig. 62, c), an extraneous force F post is needed to move the positively charged body C to the surface of the Earth. The potential of points of such a field is considered negative.
If the potentials of the field points φ 1 and φ 2 are known, then, based on the potential difference formula, we can calculate the work of moving a charged body from one field point to another: A = q(φ 2 - φ 1), or A = qU. Therefore, the potential difference is the energy characteristic of the electric field. Using these formulas, the work of moving a charge in homogeneous and inhomogeneous electric fields is calculated.
Let's set the unit of measurement for voltage (potential difference) in the SI system. To do this, we substitute the value into the voltage formula A = 1 J And q = 1 k:
The unit of voltage - volt - is taken to be the potential difference between two points of the electric field, when moving between which a point body with a charge of 1 to the field does 1 J of work.
An electrostatic field has energy. If there is an electric charge in an electrostatic field, then the field, acting on it with some force, will move it, doing work. Any work involves a change in some type of energy. The work of an electrostatic field to move a charge is usually expressed through a quantity called potential difference.
where q is the amount of charge being moved,
j 1 and j 2 are the potentials of the starting and ending points of the path.
For brevity, in what follows we will denote . V - potential difference.
V = A/q. THE POTENTIAL DIFFERENCE BETWEEN POINTS OF AN ELECTROSTATIC FIELD IS THE WORK THAT ELECTRIC FORCES DO WHEN THE CHARGE OF ONE COULLOMB MOVES BETWEEN THEM .
[V] = V. 1 volt is the potential difference between points, when moving a charge of 1 coulomb between them, electrostatic forces do 1 joule of work.
The potential difference between bodies is measured with an electrometer, for which one of the bodies is connected by conductors to the body of the electrometer, and the other to the arrow. In electrical circuits, the potential difference between points in the circuit is measured with a voltmeter.
With distance from the charge, the electrostatic field weakens. Consequently, the energy characteristic of the field, the potential, also tends to zero. In physics, the potential of a point at infinity is taken to be zero. In electrical engineering, it is believed that the surface of the Earth has zero potential.
If a charge moves from a given point to infinity, then
A = q(j - O) = qj => j= A/q, i.e. POTENTIAL OF A POINT IS THE WORK THAT MUST BE DONE BY ELECTRIC FORCES, MOVING A CHARGE OF ONE COULDOMS FROM A GIVEN POINT TO INFINITY .
Let a positive charge q move along the direction of the intensity vector to a distance d in a uniform electrostatic field with intensity E. The work done by the field to move a charge can be found both through the field strength and through the potential difference. Obviously, with any method of calculating the work, the same value is obtained.
A = Fd = Eqd = qV. =>
This formula connects the force and energy characteristics of the field. In addition, it gives us a unit of tension.
[E] = V/m. 1 V/m is the intensity of such a uniform electrostatic field, the potential of which changes by 1 V when moving along the direction of the intensity vector by 1 m.
OHM'S LAW FOR A CIRCUIT SECTION.
An increase in the potential difference at the ends of the conductor causes an increase in the current strength in it. Ohm experimentally proved that the current strength in a conductor is directly proportional to the potential difference across it.
When different consumers are connected to the same electrical circuit, the current strength in them is different. This means that different consumers hinder the passage of electric current through them in different ways. A PHYSICAL QUANTITY CHARACTERIZING THE ABILITY OF A CONDUCTOR TO PREVENT THE PASSAGE OF ELECTRIC CURRENT THROUGH IT IS CALLED ELECTRICAL RESISTANCE . The resistance of a given conductor is a constant value at constant temperature. As the temperature rises, the resistance of metals increases, and that of liquids decreases. [R] = Ohm. 1 Ohm is the resistance of a conductor through which a current of 1 A flows with a potential difference of 1 V at its ends. Metal conductors are most often used. The current carriers in them are free electrons. When moving along a conductor, they interact with positive ions crystal lattice, giving them part of their energy and losing speed. To obtain the required resistance, use a resistance magazine. A resistance store is a set of wire spirals with known resistances that can be included in a circuit in the desired combination.
Ohm experimentally established that THE CURRENT STRENGTH IN A HOMOGENEOUS SECTION OF THE CIRCUIT IS DIRECTLY PROPORTIONAL TO THE POTENTIAL DIFFERENCE AT THE ENDS OF THIS SECTION AND INVERSE PROPORTIONAL TO THE RESISTANCE OF THIS SECTION.
A homogeneous section of a circuit is a section in which there are no current sources. This is Ohm's law for a homogeneous section of a circuit - the basis of all electrical calculations.
Including conductors of different lengths, different cross-sections, made of different materials, it was established: THE RESISTANCE OF A CONDUCTOR IS DIRECTLY PROPORTIONAL TO THE LENGTH OF THE CONDUCTOR AND INVERSE PROPORTIONAL TO ITS CROSS SECTIONAL AREA. THE RESISTANCE OF A CUBE WITH AN EDGE OF 1 METER, MADE FROM SOME SUBSTANCE, IF THE CURRENT GOES PERPENDICULAR TO ITS OPPOSITE FACES, IS CALLED THE SPECIFIC RESISTANCE OF THIS SUBSTANCE . [r] = Ohm m. A non-system unit of resistivity is often used - the resistance of a conductor with a cross-sectional area of 1 mm 2 and a length of 1 m. [r] = Ohm mm 2 /m.
Resistivity substances - tabular value. The resistance of a conductor is proportional to its resistivity.
The action of slider and step rheostats is based on the dependence of the conductor resistance on its length. A slider rheostat is a ceramic cylinder with nickel wire wound around it. The rheostat is connected to the circuit using a slider, which includes a larger or smaller winding length in the circuit. The wire is covered with a layer of scale, which insulates the turns from each other.
A) SERIES AND PARALLEL CONNECTION OF CONSUMERS.
Often several current consumers are included in an electrical circuit. This is due to the fact that it is not rational for each consumer to have their own current source. There are two ways to connect consumers: serial and parallel, and their combinations in the form of a mixed connection.
a) Serial connection of consumers.
At serial connection Consumers form a continuous chain in which consumers connect one after another. With a series connection, there are no branches of connecting wires. For simplicity, consider a circuit of two series-connected consumers. An electric charge that passes through one of the consumers will also pass through the second one, because in the conductor connecting consumers there cannot be the disappearance, emergence or accumulation of charges. q=q 1 =q 2 . Dividing the resulting equation by the time the current passes through the circuit, we obtain a relationship between the current flowing throughout the entire connection and the currents flowing through its sections.
Obviously, the work to move a single positive charge throughout the compound consists of the work to move this charge across all its sections. Those. V=V 1 +V 2 (2).
The total potential difference across series-connected consumers is equal to the sum of the potential differences across consumers.
Let's divide both sides of equation (2) by the current in the circuit, we get: U/I=V 1 /I+V 2 /I. Those. The resistance of the entire series-connected section is equal to the sum of the resistances of the voltages of its components.
B) Parallel connection of consumers.
This is the most common way to enable consumers. With this connection, all consumers are connected to two points common to all consumers.
When passing parallel connection, the electric charge flowing through the circuit is divided into several parts, going to individual consumers. According to the law of conservation of charge q=q 1 +q 2. By dividing given equation for the duration of the charge, we obtain a connection between the total current flowing through the circuit and the currents flowing through individual consumers.
In accordance with the definition of potential difference V=V 1 =V 2 (2).
According to Ohm's law for a section of the circuit, we replace the current strengths in equation (1) with the ratio of the potential difference to the resistance. We get: V/R=V/R 1 +V/R 2. After reduction: 1/R=1/R 1 +1/R 2 ,
those. the reciprocal of the resistance of a parallel connection is equal to the sum of the reciprocals of the resistances of its individual branches.