The purpose of the lesson: give an idea of tension electric field and its definitions at any point of 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.
An electric field is a special form of 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 area of space electric field can be considered approximately homogeneous if the field strength within this region varies insignificantly.
The total field strength of several interacting charges will be equal to geometric sum tension vectors, which is the principle of superposition of fields:
Consider several cases of determining tension.
1. Let two opposite charges interact. Let's put a point positive charge between them, then at this point there will be two tension vectors 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
Hence:
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.
Forces acting at a distance are sometimes called field forces. If you charge an object, it will create an electric field - an area with changed characteristics surrounding it. An arbitrary charge that has fallen into the zone of an electric field will be subjected to the action of its forces. These forces are affected by the degree of charge of the object and the distance to it.
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EP strength measurement
Forces and charges
Suppose there is some initial electric charge Q that creates an electric field. The strength of this field is measured by the electric charge in the immediate vicinity. This electric charge is called a test charge, since it serves as a test charge in determining the intensity and is too small to influence the generated electric field.
The control electric charge will be called q and have some quantitative value. When placed in an electric field, it is subjected to attractive or repulsive forces F.
As a formula for the electric field strength, indicated by the Latin letterE, serves as a mathematical notation:
Force is measured in newtons (N), charge is measured in coulombs (C). Accordingly, a unit is used for tension - N / C.
Another frequently used unit for homogeneous EP in practice is V/m. This is a consequence of the formula:
That is, E depends on the voltage of the electric field (the potential difference between its two points) and the distance.
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EP tension
Does the intensity depend on the quantitative value of the electric charge? It can be seen from the formula that an increase in q entails a decrease in E. But according to Coulomb's law, more charge also means more electrical force. For example, a twofold increase in electric charge will cause a twofold increase in F. Therefore, there will be no change in tension.
Important! The intensity of the electric field is not affected by the quantitative indicator of the test charge.
How is the electric field vector directed
For a vector quantity, two characteristics are necessarily applied: quantitative value and direction. The initial charge is affected by a force directed towards it or in the opposite direction. The choice of a reliable direction is determined by the charging sign. To resolve the question in which direction the lines of tension are directed, the direction of the force F acting on the positive electric charge was taken.
Important! The lines of the field strength created by the electric charge are directed from the charge with the "plus" sign to the charge with the "minus" sign. If you imagine an arbitrary positive initial charge, then the lines will come out of it in all directions. For a negative charge, on the contrary, the occurrence of lines of force from all surrounding sides is observed.
A visual display of the vector quantities of the electric field is made by means of lines of force. The simulated EP sample can consist of an infinite number of lines, which are arranged according to certain rules, giving as much information as possible about the nature of the EP.
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Lines and intensity vectors of EP
Rules for drawing lines of force:
- Larger electric charges have the strongest electric field. In a schematic drawing, this can be shown by increasing the frequency of the lines;
- In the areas of connection with the surface of the object, the lines are always perpendicular to it. On the surface of objects regular and irregular shape there is never an electric force parallel to it. If such a force existed, any excess charge on the surface would begin to move, and there would be electricity inside an object, which is never the case in static electricity;
- When leaving the surface of an object, the force can change direction due to the influence of the EP of other charges;
- Electrical lines must not cross. If they intersect at some point in space, then at this point there should be two EPs with their own individual direction. This is an impossible condition, since each place of the EP has its own intensity and direction associated with it.
The lines of force for the capacitor will run perpendicular to the plates, but become convex at the edges. This indicates a violation of the homogeneity of the EP.
Taking into account the condition of a positive electric charge, it is possible to determine the direction of the electric field strength vector. This vector is directed towards the force acting on the electric charge with the plus sign. In situations where the electric field is created by several electric charges, the vector is found as a result of the geometric summation of all the forces that the test charge is exposed to.
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Construction of the resulting stress vector
At the same time, the electric field strength lines are understood as a set of lines in the EF coverage area, to which the vectors E will be tangent at any arbitrary point.
If an EP is created from two or more charges, lines appear surrounding their configuration. Such constructions are cumbersome and are performed using computer graphics. When solving practical problems, the resulting electric field strength vector for given points is used.
Coulomb's Law
Coulomb's law defines the electrical force:
F = (K x q x Q)/r², where:
- F is the electric force directed along the line between two electric charges;
- K - constant of proportionality;
- q and Q are the quantitative values of the charges (C);
- r is the distance between them.
Constant proportionality is found from the ratio:
K = 1/(4π x ε).
The value of the constant depends on the medium in which the charges are located (permittivity).
Then F \u003d 1 / (4π x ε) x (q x Q) / r².
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Coulomb's Law
The law operates in the natural environment. For the theoretical calculation, it is initially assumed that the electric charges are in free space (vacuum). Then the value ε = 8.85 x 10 (to the -12th power), and K = 1/(4π x ε) = 9 x 10 (to the 9th power).
Important! Formulas describing situations where there is spherical symmetry (most cases) have 4π in their composition. If there is cylindrical symmetry, 2π appears.
To calculate the tension modulus, you need to substitute the mathematical expression for Coulomb's law into the formula for E:
E \u003d F / q \u003d 1 / (4π x ε) x (q x Q) / (r² x q) \u003d 1 / (4π x ε) x Q / r²,
where Q is the initial charge that creates the EF.
To find the electric field intensity at a particular point, it is necessary to place a test charge at this point, determine the distance to it and calculate E using the formula.
Inverse square law
In the formulaic representation of Coulomb's law, the distance between electric charges appears in the equation as 1/r². Hence, the application of the inverse square law will be fair. Another well-known such law is Newton's law of gravity.
If another charge is introduced into the space surrounding an electric charge, then the Coulomb force will act on it; This means that in the space surrounding electric charges, there exists force field. According to the concepts of modern physics, the field really exists and, along with matter, is one of the forms of existence of matter, through which certain interactions are carried out between macroscopic bodies or particles that make up the substance. In this case, we speak of an electric field - a field through which interact electric charges. We consider electric fields that are created by stationary electric charges and are called electrostatic.
For discovery and experimental research electrostatic field used test point positive charge - such a charge that does not distort the field under study (does not cause a redistribution of charges that create the field). If in the field created by the charge Q, place test charge Q 0 , then a force acts on it F, different at different points of the field, which, according to Coulomb's law, is proportional to the test charge Q 0 . Therefore, the ratio F/ Q 0 does not depend on Q 0 and characterizes the electrostatic field at the point where the test charge is located. This value is called tension and is power characteristic of the electrostatic field.
Electrostatic field strength at this point is physical quantity, determined by the force acting on a test unit positive charge placed at this point of the field:
Field strength of a point charge in vacuum
The direction of the vector E coincides with the direction of the force acting on the positive charge. If the field is created by a positive charge, then the vector E is directed along the radius vector from the charge to the outer space (repulsion of a test positive charge); if the field is created by a negative charge, then the vector E is directed towards the charge (Fig.).
The unit of electrostatic field strength is newton per pendant (N/C): 1 N/C is the intensity of such a field that acts on a point charge of 1 C with a force of 1 N; 1 N/Cl= 1 V/m, where V (volt) is the unit of the potential of the electrostatic field. Graphically, the electrostatic field is depicted using tension lines - lines, the tangents to which at each point coincide with the direction of the vector E (Fig.).
Since at each given point in space the tension vector has only one direction, the lines of tension never intersect. For uniform field(when the tension vector at any point is constant in magnitude and direction) tension lines are parallel to the tension vector. If the field is created by a point charge, then the lines of tension are radial straight lines coming out of the charge if it is positive (Fig. a), and included in it if the charge is negative (Fig. b). Due to the great visibility graphic way representation of the electrostatic field is widely used in electrical engineering.
In order to be able to characterize not only the direction, but also the value of the electrostatic field strength with the help of tension lines, we agreed to draw them with a certain density: the number of tension lines penetrating a unit surface area perpendicular to the tension lines should be equal to the modulus of the vector E. Then the number of tension lines , penetrating the elementary area d S, normal n which forms an angle a with the vector E, equals E d Scos a = E n d S, where E p-vector projection E to normal n to site d S(rice.).
The value of dФ E \u003d E n dS \u003d E dS is called tension vector flow through platform d S. Here d S=d Sn- a vector whose modulus is equal to d S, and the direction is the same as the direction of the normal n to the site. Selecting the direction of the vector n(and hence also d S) is conditional, since it can be directed in any direction. The unit of the electrostatic field strength vector flux is 1 V×m.
For an arbitrary closed surface S flow vector E through this surface
,
where the integral is taken over a closed surface S. Vector flow E is an algebraic value: depends not only on the configuration of the field E, but also on the choice of direction n. For closed surfaces, the positive direction of the normal is taken outward normal, i.e., a normal directed outward of the area covered by the surface.
The principle of independence of the action of forces is applicable to the Coulomb forces, i.e. the resulting force F acting from the side of the field on the trial charge Q 0 is equal to the vector sum of the forces Fi applied to it from the side of each of the charges Q i: . F = Q 0 E and F i = Q 0 E i , where E is the strength of the resulting field, and E i is the strength of the field generated by the charge Q i . Substituting this into the expression above, we get . This formula expresses the principle of superposition (superposition) of electrostatic fields, according to which the strength E of the resulting field created by a system of charges is equal to the geometric sum of the field strengths created at a given point by each of the charges separately.
The principle of superposition is applicable to calculate the electrostatic field of an electric dipole. An electric dipole is a system of two point charges equal in absolute value (+Q, –Q), the distance l between which is much less than the distance to the considered points of the field. According to the principle of superposition, the strength E of the dipole field at an arbitrary point 
, where E+ and E– are the strengths of the fields created respectively by positive and negative charges.
>>Physics: Electric field strength. Principle of superposition of fields
It is not enough to say that an electric field exists. It is necessary to enter a quantitative characteristic of the field. After that, the electric fields can be compared with each other and continue to study their properties.
The electric field is detected by the forces acting on the charge. It can be argued that we know everything we need about the field if we know the force acting on any charge at any point in the field.
Therefore, it is necessary to introduce such a characteristic of the field, the knowledge of which will allow us to determine this force.
If we alternately place small charged bodies at the same point of the field and measure the forces, it will be found that the force acting on the charge from the field is directly proportional to this charge. Indeed, let the field be created by a point charge q 1. According to Coulomb's law (14.2) for a charge q2 there is a force proportional to the charge q2. Therefore, the ratio of the force acting on a charge placed at a given point of the field to this charge for each point of the field does not depend on the charge and can be considered as a characteristic of the field. This characteristic is called the electric field strength. Like a force, field strength - vector quantity
; it is denoted by a letter. If the charge placed in the field is denoted by q instead of q2, then the stress will be:
Hence the force acting on the charge q from the side of the electric field, is equal to:
Field strength of a point charge. Find the strength of the electric field created by a point charge q0. According to Coulomb's law, this charge will act on a positive charge q with a force equal to
if at a given point in space various charged particles create electric fields, the strengths of which
etc., then the resulting field strength at this point is equal to the sum of the strengths of these fields:
moreover, the field strength created by a single charge is defined as if there were no other charges creating the field.
Thanks to the principle of superposition, to find the field strength of a system of charged particles at any point, it is enough to know the expression (14.9) for the field strength of a point charge. Figure 14.8 shows how the field strength at the point A, created by two point charges q 1 and q 2 , q 1 > q 2
???
1. What is called the strength of the electric field?
2. What is the field strength of a point charge?
3. How is the charge field strength q 0 directed if q0>0
? if q0<0
?
4. How is the principle of superposition of fields formulated?
G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics Grade 10
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5. Electrostatics
Coulomb's Law
1. Charged bodies interact. In nature, there are two types of charges, they are conditionally called positive and negative. Charges of the same sign (like) repel, charges of opposite signs (opposite) attract. The unit of charge in the SI system is the coulomb (denoted
2. In nature, there is a minimum possible charge. He is called
elementary and denoted by e . The numerical value of the elementary charge e ≈ 1.6 10–19 C, Electron charge q electr = –e, proton charge q proton = +e. All charges
v nature are multiples of the elementary charge.
3. In an electrically isolated system, the algebraic sum of the charges remains unchanged. For example, if you connect two identical metal balls with charges q 1 \u003d 5 nCl \u003d 5 10–9 C and q 2 \u003d - 1 nC, then the charges will be distributed
between the balls equally and the charge q of each of the balls becomes equal
q \u003d (q 1 + q 2) / 2 \u003d 2 nC.
4. A charge is called a point charge if its geometric dimensions are much smaller than the distances at which the effect of this charge on other charges is studied.
5. Coulomb's law determines the magnitude of the electrical interaction force of two fixed point charges q 1 and q 2 located at a distance r from each other (Fig. 1)
k|q| |q | ||||||
F=| F | |= |F | |||||
Here F 12 is the force acting on the first charge from the second, F 21 is the force,
acting on the second charge from the side of the first, k ≈ 9 10 9 N m2 /Cl2 is a constant in Coulomb's law. In the SI system, this constant is usually written as
k = 4 πε 1 0 ,
where ε 0 ≈ 8.85 10 − 12 F/m is the electrical constant.
6. The force of interaction of two point charges does not depend on the presence of other charged bodies near these charges. This statement is called the principle of superposition.
Electric field strength vector
1. Place a point charge q near a motionless charged body (or several bodies). We will assume that the magnitude of the charge q is so small that it does not cause the movement of charges in other bodies (such a charge is called a trial charge).
From the side of a charged body, a force F will act on a stationary test charge q. In accordance with Coulomb's law and the principle of superposition, the force F will be proportional to the magnitude of the charge q. This means that if the value of the test charge is increased, for example, by 2 times, then the value of the force F will also increase by 2 times, if the sign of the charge q is reversed, then the force will change direction to the opposite. This proportionality can be expressed by the formula
F = qE.
The vector E is called the electric field strength vector. This vector depends on the distribution of charges in the bodies that create the electric field, and
on the position of the point at which the vector E is defined in the indicated way. We can say that the electric field strength vector is equal to the force acting on a unit positive charge placed at a given point in space.
The definition of E G = F G /q can also be generalized to the case of variable (time-dependent) fields.
2. Calculate the electric field strength vector created by a fixed point charge Q . We choose some point A located at a distance r from the point charge Q . To determine the intensity vector at this point, we mentally place a positive test charge q in it. On the
a test charge from a point charge Q will act as an attractive or repulsive force, depending on the sign of the charge Q. The magnitude of this force is
F = k| Q| q. r2
Therefore, the modulus of the electric field strength vector created by a fixed point charge Q at a point A remote from it at a distance r is equal to
E = k r |Q 2 |.
The vector E G starts at point A and is directed from the charge Q if Q > 0 and to the charge Q ,
if Q< 0 .
3. If the electric field is created by several point charges, then the intensity vector at an arbitrary point can be found using the principle of superposition of fields.
4. Force line (vector line E ) is called a geometric line,
the tangent to which at each point coincides with the vector E at this point.
In other words, the vector E is directed tangentially to the line of force at each of its points. The line of force is assigned a direction - along the vector E. The picture of the lines of force is a clear way force field, gives an idea of the spatial structure of the field, its sources, allows you to determine the direction of the intensity vector at any point.
5. A field is called a uniform electric field, vector E which is the same (in magnitude and direction) at all points. Such a field is created, for example, by a uniformly charged plane at points located fairly close to this plane.
6. The field of a sphere uniformly charged over the surface is zero inside the sphere,
a outside the ball coincides with the field of a point charge Q located in the center of the ball:
k | Q| | for r > R | ||
E = r2 | |||
at r< R | |||
where Q is the charge of the ball, R is its radius, r is the distance from the center of the ball to the point, in
which defines the vector E .
7. In dielectrics, the field is weakened. For example, a point charge or a sphere uniformly charged over the surface, immersed in oil, creates an electric field
E = k ε |r Q 2 |,
where r is the distance from the point charge or the center of the ball to the point where the intensity vector is determined, ε is the dielectric constant of the oil. The dielectric constant depends on the properties of the substance. The permittivity of vacuum ε = 1, the permittivity of air is very close to unity (when solving problems it is usually considered equal to 1), for other gaseous, liquid and solid dielectrics ε > 1.
8. When the charges are in equilibrium (if there is no orderly movement of them), the electric field strength inside the conductors is zero.
Work in an electric field. Potential difference.
1. The field of fixed charges (electrostatic field) has an important property: the work of the forces of the electrostatic field to move the test charge from some point 1 to point 2 does not depend on the shape of the trajectory, but is determined only by the positions of the start and end points. Fields with this property are called conservative. The property of conservatism allows you to determine the so-called potential difference for any two points of the field.
Potential differenceϕ 1 − ϕ 2 at points 1 and 2 is equal to the ratio of the work A 12 of the field forces to move the test charge q from point 1 to point 2 to the value of this charge:
ϕ1 - ϕ2 =A q 12 .
Such a definition of the potential difference makes sense only because the work does not depend on the shape of the trajectory, but is determined by the positions of the initial and final points of the trajectories. In the SI system, the potential difference is measured in volts: 1V = J / C.
Capacitors
1. The capacitor consists of two conductors (they are called plates), separated from one another by a dielectric layer (Fig. 2), and the charge of one
plates Q, and the other -Q. The charge of the positive plate Q is called the charge of the capacitor.
2. It can be shown that the potential difference ϕ 1 − ϕ 2 between the plates is proportional to the charge Q, that is, if, for example, the charge Q is increased by 2 times, then the potential difference will increase by 2 times.
εS
ϕ 1ϕ 2
Fig.2 Fig.3
This proportionality can be expressed by the formula
Q \u003d C (ϕ 1 -ϕ 2),
where C is the coefficient of proportionality between the charge of the capacitor and the potential difference between its plates. This coefficient is called the capacitance or simply the capacitance of the capacitor. The capacitance depends on the geometric dimensions of the plates, their mutual arrangement and the dielectric constant of the medium. The potential difference is also called voltage, which is denoted U. Then
Q=CU.
3. A flat capacitor consists of two flat conductive plates located parallel to each other at a distance d (Fig. 3). This distance is assumed to be small compared to the linear dimensions of the plates. The area of \u200b\u200beach plate (capacitor lining) is equal to S, the charge of one plate is Q, and the other is Q.
At some distance from the edges, the field between the plates can be considered uniform. Therefore ϕ 1 -ϕ 2 = Ed, or
U = Ed.
The capacitance of a flat capacitor is determined by the formula
C = εε d 0 S ,
where ε 0 \u003d 8.85 10–12 F / m is the electrical constant, ε is the dielectric constant of the dielectric between the plates. From this formula it can be seen that to obtain a capacitor large capacity you need to increase the area of the plates and reduce the distance between them. The presence between the plates of a dielectric with a high permittivity ε also leads to an increase in capacitance. The role of the dielectric between the plates is not only to increase the dielectric constant. It is also important that good dielectrics can withstand a high electric field without allowing breakdown between the plates.
In the SI system, capacitance is measured in farads. A one farad flat capacitor would be gigantic. The area of each plate would be approximately equal to 100 km2 with a distance between them of 1 mm. Capacitors are widely used in engineering, in particular, for the accumulation of charges.
4. If the plates of a charged capacitor are closed with a metal conductor, then an electric current will appear in the conductor and the capacitor will be discharged. When a current flows in a conductor, a certain amount of heat is released, which means that a charged capacitor has energy. It can be shown that the energy of any charged capacitor (not necessarily a flat one) is given by
W = 1 2 CU2 .
Considering that Q = CU , the energy formula can also be rewritten as
W \u003d Q 2 \u003d QU.