The purpose of the lesson: give the concept of electric field strength and its definition at any point in the field.
Lesson objectives:
- formation of the concept of electric field strength; give the concept of tension lines and a graphical representation of the electric field;
- teach students to apply the formula E \u003d kq / r 2 in solving simple problems for calculating tension.
The electric field is special form matter, the existence of which can only be judged by its action. It has been experimentally proved that there are two types of charges around which there are electric fields characterized by lines of force.
Graphically depicting the field, it should be remembered that the electric field strength lines:
- do not intersect with each other anywhere;
- have a beginning on a positive charge (or at infinity) and an end on a negative charge (or at infinity), i.e., they are open lines;
- between charges are not interrupted anywhere.
Fig.1
Positive charge lines of force:
Fig.2
Negative charge lines of force:
Fig.3
Force lines of like interacting charges:
Fig.4
Force lines of opposite interacting charges:
Fig.5
The power characteristic of the electric field is the intensity, which is denoted by the letter E and has units of measurement or. The tension is a vector quantity, as it is determined by the ratio of the Coulomb force to the value of a unit positive charge
As a result of the transformation of the Coulomb law formula and the strength formula, we have the dependence of the field strength on the distance at which it is determined relative to a given charge
where: k– coefficient of proportionality, the value of which depends on the choice of units of electric charge.
In the SI system 
N m 2 / Cl 2,
where ε 0 is an electrical constant equal to 8.85 10 -12 C 2 /N m 2;
q is the electric charge (C);
r is the distance from the charge to the point where the intensity is determined.
The direction of the tension vector coincides with the direction of the Coulomb force.
An electric field whose strength is the same at all points in space is called homogeneous. In a limited region of space, an electric field can be considered approximately uniform if the field strength within this region changes insignificantly.
The total field strength of several interacting charges will be equal to the geometric sum of the strength vectors, which is the principle of the superposition of fields:
Consider several cases of determining tension.
1. Let two opposite charges interact. We place a point positive charge between them, then at this point two intensity vectors will act, directed in the same direction:
According to the principle of superposition of fields, the total field strength at a given point is equal to the geometric sum of the strength vectors E 31 and E 32 .
The tension at a given point is determined by the formula:
E \u003d kq 1 / x 2 + kq 2 / (r - x) 2
where: r is the distance between the first and second charge;
x is the distance between the first and the point charge.
Fig.6
2. Consider the case when it is necessary to find the intensity at a point remote at a distance a from the second charge. If we take into account that the field of the first charge is greater than the field of the second charge, then the intensity at a given point of the field is equal to the geometric difference between the intensity E 31 and E 32 .
The formula for tension at a given point is:
E \u003d kq1 / (r + a) 2 - kq 2 / a 2
Where: r is the distance between interacting charges;
a is the distance between the second and the point charge.
Fig.7
3. Consider an example when it is necessary to determine the field strength at some distance from both the first and the second charge, in this case at a distance r from the first and at a distance b from the second charge. Since charges of the same name repel and unlike charges attract, we have two tension vectors emanating from one point, then for their addition you can apply the method to the opposite corner of the parallelogram will be the total tension vector. We find the algebraic sum of vectors from the Pythagorean theorem:
E \u003d (E 31 2 + E 32 2) 1/2
Consequently:
E \u003d ((kq 1 / r 2) 2 + (kq 2 / b 2) 2) 1/2
Fig.8
Based on this work, it follows that the intensity at any point of the field can be determined by knowing the magnitude of the interacting charges, the distance from each charge to a given point and the electrical constant.
4. Fixing the topic.
Verification work.
Option number 1.
1. Continue the phrase: “electrostatics is ...
2. Continue the phrase: the electric field is ....
3. How are the lines of force of this charge directed?
4. Determine the signs of the charges:
Home tasks:
1. Two charges q 1 = +3 10 -7 C and q 2 = −2 10 -7 C are in vacuum at a distance of 0.2 m from each other. Determine the field strength at point C, located on the line connecting the charges, at a distance of 0.05 m to the right of the charge q 2 .
2. At some point of the field, a force of 3 10 -4 N acts on a charge of 5 10 -9 C. Find the field strength at this point and determine the magnitude of the charge that creates the field if the point is 0.1 m away from it.
electric field
Electric field (static) - fieldmotionless , electrically charged tel,whose charges do not change in time.
Electric field is detected how force interaction of charged bodies.
At the same time, they distinguish positive and negative charges. (types of charges )
Charges of the same sign repel each other, charges of the opposite sign attract. (charge interaction)
The description of the properties of the electric field is based on the Coulomb's law, established empirically.
Coulomb's law . Between point charges at rest there is a force proportional to the product of the charges, inversely proportional to the square of the distance between them and directed in a straight line from one charge to another(Fig. 1.1):
(1.1)
where F, is the force acting on the charge q
r 2 - squared distance between charges q 1 and q 2
F 2 is the force acting on the charge q 2
r 0 21 - unit vector directed from the second charge to the first one;
e 0 \u003d 8.854 10-12 F / m - electrical constant.
point charges we can consider charged bodies, the dimensions of which are small compared to the distance between them.
Main units :
strength in international system of units (SI) - newton(H);
charge - pendant(C): 1 C = 1 A s;
length - meter(m).
The main quantities characterizing the electric field , are
tension,
electric potentialand
potential difference, or voltage
tension electric field called the measure of the intensity of its forces, equal to the ratio of the forceF , valid for trialsolid point chargeq, introduced into the considered point of the field, to the value of the charge
(1.2)
As well as the force F, the electric field strength ε - vector quantity, i.e. characterized by meaning and direction of action.
Main SI unit of electric field strength - volt per meter(W / m).
From formula (1.1) it follows that electric field strength of a point charge q on distance r from it is equal to
(1-3)
and is directed from the point where the charge is located to the point where the tension is determined, if the charge is positive (Fig. 1.2, a),
Rice. 1.2, a
and in the opposite direction, if the charge is negative (Fig. 1.2, b).
1.2 b
If there are several charges that create an electric field, then the intensity at any point of the field is equal to the geometric sum of the intensities from each of them separately. ( tension electrostatic field multiple charges )
Example 1.1. Determine the value and direction of the electric field strength at a point BUT, located at a distance r 1 = 1m and r 2 = 2 m from point charges
q 1 = 1,11 10 -10 Cl and q 2 = -4,44- 10 -10 Cl (Fig. 1.3).
Solution. According to formula (1.3), we determine the electric field strength at the point BUT from the action of "point charges q 1 = and q 2
Tension vector directions 
coincide with the directions of action of forces on a test positive point charge, if it is placed at a point BUT
.
The strength of the resulting electric field at a point BUT is directed along the hypotenuse of a right-angled triangle, the legs of which are the stress vectors 
and it matters
You can talk about vector field and display this field vector lines -lines of force .
If the electric field strength is the same at all points, then field homogeneous , for example, the field of a uniformly charged flat plate of infinite dimensions (Fig. 1.4),
and if different, then the field is not uniform , for example, the field of two point charges (Fig. 1.5).
When moving along an arbitrary section with a length of charge q in an electric field under the influence of field forces F work is done
Wherein Work on charge transfer along an arbitrary closed contour zero .
Indeed, since all the properties of the field are determined by the relative arrangement of charges, the charge transfer along a closed circuit and return to the starting point means the initial charge distribution and energy reserve. This also means that, taking into account (1.4), the circulation of the intensity vector is equal to zero
Condition (1.5) allows us to characterize the electric field at each point by the function of its coordinates - electrical potential .
Electrical potential in this electric field point taking into account (1.4) is numerically equal to the work that the electric field forces can do when transferring a unit positive charge from a given point to a point whose potential is assumed to be zero.
Potential differencetwo points 1 and 2, or voltage between points 1 and 2, electric field
(1.7)
is numerically equal to the work that the electric field forces can do when transferring a unit positive charge from a point1 exactly2 .
SI unit of electrical potential - volt(AT).
Coulomb's law
point charge
0 those.
Draw a radius vector r r from charge q to q r r. He is equal r r /r.
Force ratio F q tension and denoted by E r. Then:
1 N/C = 1/1 C, those. 1 N/Cl-
The field strength of a point charge.
Let's find the tension E electrostatic field generated by a point charge q, located in a homogeneous isotropic dielectric, at a point separated from it, at a distance r. Let's mentally place a test charge at this point q 0 . Then 
.
Hence we get that
radius vector drawn from the charge q to the point at which the field strength is determined. From the last formula it follows that the modulus of the field strength:
Thus, the modulus of tension at any point of the electrostatic field created by a point charge in vacuum is proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge to the point at which the tension is determined.
Superposition of fields
If the electric field is created by a system of point charges, then its intensity is equal to the vector sum of the field strengths created by each charge separately, i.e. . This ratio is called the principle of superposition (overlay) of fields. It also follows from the principle of superposition of fields that the potential ϕ created by a system of point charges at a certain point is equal to the algebraic sum of the potentials created at the same point by each charge separately, i.e. The sign of the potential is the same as the sign of the charge qi individual charges of the system.
Tension lines
For a visual representation of the electric field, use tension lines or lines of force , i.e. lines, at each point of which the electric field strength vector is directed tangentially to them. The easiest way to understand this is with an example uniform electrostatic field, those. field, at each point of which the intensity is the same in magnitude and direction. In this case, the tension lines are drawn so that the number of lines F E passing through a unit area of a flat area S located perpendicular to these
lines, would be equal to the modulus E the strength of this field, i.e.
If the field is inhomogeneous, then it is necessary to choose an elementary area dS, perpendicular to the lines of tension, within which the field strength can be considered constant.
where dФ E is the number of tension lines penetrating this area, i.e. the electric field strength modulus is equal to the number of lines of tension per unit area of the area perpendicular to it.
Gauss theorem
Theorem: the flow of the electrostatic field strength through any closed surface is equal to the algebraic sum of the charges enclosed inside it, divided by the electrical constant and the permittivity of the medium.
If integration is performed over the entire volume V, along which the charge is distributed. Then, with a continuous distribution of charge on some surface S 0 the Gauss theorem is written as:
In case of volumetric distribution:
Gauss's theorem relates the magnitude of the charge and the strength of the field that it creates. This determines the significance of this theorem in electrostatics, since it allows you to calculate the intensity, knowing the location of the charges in space.
Electric field circulation.
From expression
it also follows that when the charge is transferred along a closed path, i.e., when the charge returns to its original position, r 1 = r 2 and A 12 =
0. Then we write
Force acting on a charge q 0 is equal to . Therefore, we rewrite the last formula in the form
News electrostatic field per direction Dividing both sides of this equality by q 0 , we find:
The first equality is electric field strength circulation .
Capacitors
Capacitors are two conductors very close to each other and separated by a dielectric layer. Capacitor capacitance - the ability of a capacitor to accumulate charges on itself. those. the capacitance of a capacitor is a physical quantity, equal to the ratio of the charge of the capacitor to the potential difference between its plates. The capacitance of a capacitor, like the capacitance of a conductor, is measured in farads (F): 1 F is the capacitance of such a capacitor, when a charge of 1 C is imparted to it, the potential difference between its plates changes by 1 V.
Electric energy fields
The energy of charged conductors is stored in the form of an electric field. Therefore, it is advisable to express it through the tension that characterizes this field. This is easiest to do for a flat capacitor. In this case where d- the distance between the plates, and
. Here ε0 is the electrical constant, ε is the permittivity of the dielectric filling the capacitor, S- the area of each lining. Substituting these expressions, we get 
Here V=Sd- volume occupied by the field, equal to the volume capacitor.
Work and current power.
The work of electric current The work done by the forces of an electric field created in an electric circuit is called when a charge moves along this circuit.
Let a constant potential difference (voltage) be applied to the ends of the conductor U=ϕ1− ϕ2.
A=q(ϕ1−ϕ2) = qU.
Taking this into account, we get
Applying Ohm's law for a homogeneous section of the circuit
U=IR, where R- the resistance of the conductor, we write:
A=I 2 Rt.
Work A completed in time t, will be equal to the sum of elementary works, i.e.
By definition, the power of an electric current is equal to P = A/t. Then:
In the SI system of units, work and power of an electric current are measured in joules and watts, respectively.
Joule-Lenz law.
Electrons moving in a metal under the action of an electric field, as already noted, continuously collide with ions crystal lattice, transferring to them its kinetic energy of ordered motion. This leads to an increase in the internal energy of the metal, i.e. to heat it up. According to the law of conservation of energy, all the work of the current A goes to the release of heat Q, i.e. Q=A. We find This ratio is called Joule law − Lenz .
Full current law.
Induction circulation magnetic field along an arbitrary closed circuit is equal to the product of the magnetic constant, magnetic permeability and the algebraic sum of the strengths of the currents covered by this circuit.
The current strength can be found using the current density j:
where S- cross-sectional area of the conductor. Then the total current law is written as: 
magnetic flux.
Magnetic flux through some surface call the number of lines of magnetic induction penetrating it.
Let there be a surface with area S. To find the magnetic flux through it, we mentally divide the surface into elementary sections with an area dS, which can be considered flat, and the field within them is homogeneous. Then the elementary magnetic flux dФ B through this surface is equal to:
Magnetic flux across the entire surface is equal to the sum these streams: , i.e.:
. In SI units, magnetic flux is measured in webers (Wb).
Inductance.
Let a constant current flow through a closed circuit with a force I. This current creates a magnetic field around itself, which permeates the area covered by the conductor, creating a magnetic flux. It is known that the magnetic flux F B is proportional to the modulus of the magnetic field B, and the modulus of induction of the magnetic field arising around the current-carrying conductor is proportional to the current strength I. Therefore F B ~B~I, i.e. F B = LI.
The coefficient of proportionality L between the strength of the current and the magnetic flux created by this current through the area bounded by the conductor, called conductor inductance .
In the SI system, inductance is measured in henries (H).
solenoid inductance.
Consider the inductance of a solenoid with a length l, with cross section S and with total number turns N, filled with a substance with a magnetic permeability μ. In this case, we take a solenoid of such a length that it can be considered as infinitely long. When a current flows through it with a force I a uniform magnetic field is created inside it, directed perpendicular to the planes of the coils. The magnetic induction modulus of this field is found by the formula
B=μ0μ ni,
magnetic flux F B through any turn of the solenoid is F B= BS(see (29.2)), and the total Ψ flux through all turns of the solenoid will be equal to the sum of the magnetic fluxes through each turn, i.e. Ψ = NF B= NBS.
N = nl, we get: Ψ = μ0μ = n 2 lSI =μ0μ n 2 VI
We conclude that the inductance of the solenoid is equal to:
L =μμ0 n 2 V
The energy of the magnetic field.
Let a direct current flow in an electric circuit with a force I. If you turn off the current source and close the circuit (switch P move into position 2
), then a decreasing current will flow in it for some time, due to the emf. self-induction 
.
The elementary work done by the emf. self-induction by transfer along the chain elementary charge dq = I dt, equal to Current strength varies from I to 0. Therefore, integrating this expression within the indicated limits, we obtain the work done by the emf. self-induction for the time during which the disappearance of the magnetic field occurs: 
. This work is spent on increasing the internal energy of the conductors, i.e. to heat them up. The performance of this work is also accompanied by the disappearance of the magnetic field, which originally existed around the conductor.
The energy of the magnetic field that exists around current-carrying conductors is
W B = LI 2 / 2.
we get that
The magnetic field inside the solenoid is uniform. Therefore, the volumetric energy density w B magnetic field, i.e. the energy of a unit volume of the field inside the solenoid is equal to .
Vortex electr. field.
From Faraday's law for electromagnetic induction it follows that with any change in the magnetic flux penetrating the area covered by the conductor, an emf arises in it. induction, under the action of which an induction current appears in the conductor if the conductor is closed.
To explain the emf. Induction, Maxwell hypothesized that an alternating magnetic field creates an electric field in the surrounding space. This field acts on the free charges of the conductor, bringing them into ordered motion, i.e. creating an inductive current. Thus, a closed conducting circuit is a kind of indicator, with the help of which this electric field is detected. Let us denote the strength of this field through E r. Then the emf induction 
it is known that the circulation of the electrostatic field strength is zero, i.e.
It follows that i.e. an electric field excited by a time-varying magnetic field is a vortex(not potential).
It should be noted that the lines of the electrostatic field strength begin and end on the charges that create the field, and the lines of the vortex electric field strength are always closed.
Bias current
Maxwell hypothesized that an alternating magnetic field creates a vortex electric field. He also made the opposite assumption: an alternating electric field should induce a magnetic field. Subsequently, these both hypotheses received experimental confirmation in the experiments of Hertz. The appearance of a magnetic field with a change in the electric field can be interpreted as if an electric current arises in space. This current was named by Maxwell bias current .
Displacement current can occur not only in a vacuum or a dielectric, but also in conductors through which an alternating current flows. However, in this case it is negligible compared to the conduction current.
Maxwell introduced the concept of total current. Strength I total current is equal to the sum of forces I at I see conduction and displacement currents, i.e. I= I pr + I see We get:
Maxwell's equation.
First equation.
It follows from this equation that the source of the electric field is a magnetic field that changes with time.
Maxwell's second equation.
Second equation. Full current law 
This equation shows that a magnetic field can be generated by both moving charges ( electric shock) and an alternating electric field.
Fluctuations.
fluctuations called processes characterized by a certain repeatability over time. The process of propagation of oscillations in space called wave . Any system capable of oscillating or in which oscillations can occur is called vibrational . The fluctuations that take place in oscillatory system, taken out of equilibrium and presented to itself, is called free vibrations .
Harmonic vibrations.
Harmonic oscillations are called oscillations in which the oscillating physical quantity changes according to the Sin or Cos law. Amplitude - this is the largest value that a fluctuating value can take. Equations of harmonic oscillations: and
same thing with sine. Period of non-damped oscillations is called the time of one complete oscillation. The number of oscillations per unit time is called oscillation frequency . The oscillation frequency is measured in hertz (Hz).
Oscillatory circuit.
An electrical circuit consisting of inductance and capacitance is called oscillatory circuit
total energy electromagnetic oscillations there is a constant value in the circuit, just like the total energy of mechanical vibrations.
When fluctuating, it always throws. energy is converted into potential energy and vice versa.
Energy W oscillatory circuit is made up of energy W E electric field capacitor and energy W B magnetic field inductance
damped vibrations.
Processes described by the equation 
can be considered oscillatory. They are called damped oscillations
. Smallest gap time T, through which the maxima (or minima) are repeated is called period of damped oscillations.
The expression is considered as the amplitude of damped oscillations. Value A 0 is the amplitude of the oscillation at time t = 0, i.e. this is the initial amplitude of damped oscillations. The value of β, on which the decrease in the amplitude depends, is called damping factor
.
Those. the damping coefficient is inversely proportional to the time during which the amplitude of the damped oscillations decreases by a factor of e.
Waves.
Wave- this is the process of propagation of oscillations (perturbations) in space.
Area of space, within which vibrations take place., is called wave field .
Surface, separating the wave field from the region, where there is no hesitation, called wave front .
lines, along which the wave propagates, are called rays .
Sound waves.
Sound is vibrations of air or other elastic medium perceived by our hearing organs. Sound vibrations perceived by the human ear have frequencies ranging from 20 to 20,000 Hz. Oscillations with frequencies less than 20 Hz are called infrasonic , and more than 20 kHz - ultrasonic .
Sound characteristics. We usually associate sound with its auditory perception, with the sensations that arise in the human mind. In this regard, we can distinguish three of its main characteristics: height, quality and loudness.
Physical quantity characterizing the pitch of the sound is sound wave frequency.
To characterize the quality of sound in music, the terms timbre or tonal coloring of sound are used. Sound quality can be associated with physically measurable quantities. It is determined by the presence of overtones, their number and amplitudes.
The loudness of sound is related to a physically measurable quantity - the intensity of the wave. Measured in whites.
The laws of thermal radiation
Stefan-Boltzmann law- the law of radiation of a completely black body. Determines the dependence of the radiation power of an absolutely black body on its temperature. The wording of the law: 
Kirchhoff's radiation law
The ratio of the emissivity of any body to its absorption capacity is the same for all bodies at a given temperature for a given frequency and does not depend on their shape and chemical nature.
The wavelength at which the radiation energy of a black body is maximum is determined by Wien's displacement law: 
where T is the temperature in kelvins, and λ max is the wavelength with maximum intensity in meters.
The structure of the atom.
The experiments of Rutherford and his collaborators led to the conclusion that in the center of the atom there is a dense positively charged nucleus, the diameter of which does not exceed 10–14–10–15 m.
Studying the scattering of alpha particles when passing through gold foil, Rutherford came to the conclusion that the entire positive charge of atoms is concentrated in their center in a very massive and compact nucleus. And negatively charged particles (electrons) revolve around this nucleus. This model was fundamentally different from the Thomson model of the atom, which was widespread at that time, in which the positive charge uniformly filled the entire volume of the atom, and the electrons were embedded in it. A little later, Rutherford's model was called the planetary model of the atom (it really looks like solar system: the heavy nucleus is the Sun, and the electrons revolving around it are the planets).
Atom- the smallest chemically indivisible part of a chemical element, which is the carrier of its properties. An atom is made up of atomic nucleus and electrons. The nucleus of an atom is made up of positively charged protons and uncharged neutrons. If the number of protons in the nucleus coincides with the number of electrons, then the atom as a whole is electrically neutral. Otherwise, it has some positive or negative charge and is called an ion. Atoms are classified according to the number of protons and neutrons in the nucleus: the number of protons determines whether an atom belongs to some chemical element, and the number of neutrons is the isotope of this element.
Atoms of various kinds in different quantities, linked by interatomic bonds, form molecules.
Questions:
1. electrostatics
2. law of conservation of electric charge
3. Coulomb's law
4. electric field. electric field strength
6. superposition of fields
7. tension lines
8. flux-vector of electric field strength
9. Gauss theorem for electrostatic field
10. Gauss theorem
11. electric field circulation
12. potential. Potential difference electrostatic field
13. relationship between field voltage and potential
14.capacitors
15. energy charged capacitor
16. electric field energy
17. conductor resistance. Ohm's law for a piece of chain
18. Ohm's law for the conductor section
19. sources of electric current. Electromotive force
20. work and current power
21. joule lenz law
22. magnetic field. magnetic field induction
23. full current law
24. magnetic flux
25. Gauss theorem for magnetic field
26. work on moving a conductor with current into a magnet field
27. electromagnet induction phenomenon
28. inductance
29. solenoid inductance
30. phenomenon and law of self-induction
31. magnetic field energy
32. vortex electric field
33. bias current
34. maxwell equation
35. Maxwell's second equation
36. third and fourth Maxwell equation
37. fluctuations
40. damped vibrations
41. forced vibrations. Resonance phenomenon
43. plane monochromatic wave equation
44. sound waves
45. wave and corpuscular properties of light
46. Thermal radiation and its characteristics.
47. Laws of thermal radiation
48. The structure of the atom.
Coulomb's law
The interaction force is found for the so-called point charges.
point charge a charged body is called, the dimensions of which are negligible compared to the distance to other charged bodies with which it interacts.
The law of interaction of point charges was discovered by Coulomb and is formulated as follows: modulus F of the force of interaction between two fixed charges q and q 0 proportional to the product of these charges, inversely proportional to the square of the distance r between them, those.
where ε0 is the electrical constant, ε is the permittivity characterizing the medium. This force is directed along a straight line connecting the charges. The electrical constant is ε0 = 8.85⋅10–12 C2/(N⋅m2) or ε0 = 8.85⋅10–12 F/m, where farad (F) is the unit of electrical capacity. Coulomb's law in vector form will be written:
Draw a radius vector r r from charge q to q 0. Let us introduce a unit vector directed in the same direction as the vector r r. He is equal r r /r.
Electric field. electric field strength
Force ratio F r acting on the charge to the value q 0 of this charge is constant for all introduced charges, regardless of their magnitude. Therefore, this ratio is taken as a characteristic of the electric field at a given point. They call her tension and denoted by E r. Then:
1 N/C = 1/1 C, those. 1 N/Cl- the intensity at a point in the field at which a force of 1 N acts on a charge of 1 C.
Definition
Tension vector is the power characteristic of the electric field. At some point in the field, the intensity is equal to the force with which the field acts on a unit positive charge placed at the specified point, while the direction of the force and the intensity are the same. The mathematical definition of tension is written as follows:
where is the force with which the electric field acts on a fixed, “trial”, point charge q, which is placed at the considered point of the field. At the same time, it is considered that the “trial” charge is small enough that it does not distort the field under study.
If the field is electrostatic, then its intensity does not depend on time.
If the electric field is uniform, then its strength is the same at all points in the field.
Graphically, electric fields can be represented using lines of force. Lines of force (tension lines) are lines, the tangents to which at each point coincide with the direction of the intensity vector at this point of the field.
The principle of superposition of electric field strengths
If the field is created by several electric fields, then the strength of the resulting field is equal to the vector sum of the strengths of the individual fields:
Let us assume that the field is created by a system of point charges and their distribution is continuous, then the resulting intensity is found as:
integration in expression (3) is carried out over the entire area of charge distribution.
Field strength in a dielectric
The field strength in the dielectric is equal to the vector sum of the field strengths created by free charges and bound (polarization charges):
In the event that the substance that surrounds the free charges is a homogeneous and isotropic dielectric, then the intensity is equal to:
where is the relative permittivity of the substance at the studied point of the field. Expression (5) means that for a given charge distribution, the strength of the electrostatic field in a homogeneous isotropic dielectric is less than in vacuum by a factor of.
Field strength of a point charge
The field strength of a point charge q is:
where F / m (SI system) - electrical constant.
Relationship between tension and potential
In the general case, the electric field strength is related to the potential as:
where is the scalar potential and is the vector potential.
For stationary fields, expression (7) is transformed into the formula:
Electric field strength units
The basic unit of measurement of electric field strength in the SI system is: [E]=V/m(N/C)
Examples of problem solving
Example
Exercise. What is the modulus of the electric field strength vector at a point defined by the radius vector (in meters) if the electric field creates a positive point charge (q=1C) that lies in the XOY plane and its position specifies the radius vector , (in meters)?
Solution. The voltage modulus of the electrostatic field, which creates a point charge, is determined by the formula:
r is the distance from the charge that creates the field to the point where we are looking for the field.
From formula (1.2) it follows that the modulus is equal to:
Substitute in (1.1) the initial data and the resulting distance r, we have:
Answer. 
Example
Exercise. Write down an expression for the field strength at a point, which is determined by the radius - vector, if the field is created by a charge that is distributed over the volume V with density.
Solution. Let's make a drawing.
Let us divide the volume V into small areas with the volumes of the charges of these volumes, then the field strength of the point charge at point A (Fig. 1) will be equal to:
In order to find the field that creates the whole body at point A, we use the principle of superposition:
where N is the number of elementary volumes into which the volume V is divided.
The charge distribution density can be expressed as:
From expression (2.3) we get:
We substitute the expression for the elementary charge into formula (2.2), we have:
Since the distribution of charges is given continuous, then if we tend to zero, then we can go from summation to integration, then:
physical nature electric field and its graphic representation. In the space around an electrically charged body, there is an electric field, which is one of the types of matter. Electric field has a store of electrical energy, which manifests itself in the form of electrical forces acting on charged bodies in the field.
Rice. 4. The simplest electric fields: a - single positive and negative charges; b - two opposite charges; c - two like charges; d - two parallel and oppositely charged plates (uniform field)
Electric field conventionally depicted in the form of electric lines of force, which show the direction of action of the electric forces created by the field. It is customary to direct the lines of force in the direction in which a positively charged particle would move in an electric field. As shown in fig. 4, electric lines of force diverge in different directions from positively charged bodies and converge at bodies with a negative charge. The field created by two flat oppositely charged parallel plates (Fig. 4, d) is called uniform.
An electric field can be made visible by placing gypsum particles suspended in liquid oil in it: they rotate along the field, located along its lines of force (Fig. 5).
Electric field strength. The electric field acts on the charge q introduced into it (Fig. 6) with a certain force F. Therefore, the intensity of the electric field can be judged by the value of the force with which a certain electric charge is attracted or repelled, taken as unity. In electrical engineering, the field intensity is characterized by the strength of the electric field E. The strength is understood as the ratio of the force F acting on a charged body at a given point in the field to the charge q of this body:
E=F/q(1)
Field with big tension E is depicted graphically by lines of force of great density; a field with low intensity - sparsely spaced lines of force. As you move away from the charged body, the lines of force of the electric field are less frequent, i.e., the field strength decreases (see Fig. 4 a, b and c). Only in a uniform electric field (see Fig. 4, d) is the intensity the same at all its points.
Electrical potential. The electric field has a certain amount of energy, i.e., the ability to do work. As you know, energy can also be stored in a spring, for which it needs to be compressed or stretched. Due to this energy, you can get a certain work. If one of the ends of the spring is released, then it will be able to move the body connected with this end for some distance. In the same way, the energy of an electric field can be realized if some charge is introduced into it. Under the action of the field forces, this charge will move in the direction of the lines of force, doing a certain amount of work.
To characterize the energy stored at each point of the electric field, a special concept is introduced - electric potential. Electrical potential? field at a given point is equal to the work that the forces of this field can do when moving a unit of positive charge from this point outside the field.
The concept of electric potential is similar to the concept of level for various points earth's surface. It is obvious that to lift the locomotive to point B (Fig. 7) it is necessary to expend more work than to raise it to point A. Therefore, the locomotive raised to the level H2 will be able to do more work during the descent than the locomotive raised to the level H2 the zero level, from which the height is measured, is usually taken as sea level.
In the same way, the zero potential is conditionally taken as the potential that the earth's surface has.
electrical voltage. Different points of the electric field have different potentials. Usually, we are of little interest in the absolute value of the potentials of individual points of the electric field, but it is very important for us to know the potential difference? 1-? 2 between two points of the field A and B (Fig. 8). The potential difference?1 and?2 of two points of the field characterizes the work expended by the forces of the field to move a unit charge from one point of the field with a large potential to another point with a lower potential. Similarly, in practice we are of little interest absolute heights H1 and H2 of points A and B above sea level (see Fig. 7), but it is important for us to know the difference in levels And between these points, since the lifting of the locomotive from point A to point B requires work depending on the value of H. The difference potentials between two points of the field is called electric voltage. Electrical voltage is denoted by the letter U (and). It is numerically equal to the ratio of the work W, which must be spent on moving a positive charge q from one point of the field to another, to this charge, i.e.
U=W/q(2)
Therefore, the voltage U acting between different points electric field, characterizes the energy stored in this field, which can be given away by moving between these points of electric charges.
Electric voltage is the most important electrical quantity that allows you to calculate the work and power developed when moving charges in an electric field. The unit of electrical voltage is the volt (V). In engineering, voltage is sometimes measured in thousandths of a volt - millivolts (mV) and millionths of a volt - microvolts (µV). For measuring high voltage use larger units - kilovolts (kV) - thousands of volts.
The electric field strength in a uniform field is the ratio of the electric voltage acting between two points of the field to the distance l between these points:
E=U/l(3)
The electric field strength is measured in volts per meter (V/m). At a field strength of 1 V/m, a force of 1 Newton (1 N) acts on a charge of 1 C. In some cases, larger units of field strength V/cm (100 V/m) and V/mm (1000 V/m) are used.
