Voltage formula in terms of field strength. electrostatic dipole. electrostatic field. Tension. Electric field strength units
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 an idea of the lines of tension and graphic representation 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 start on positive charge(or at infinity) and the end on the negative (or at infinity), that is, 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- electric charge(Cl);
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 geometric sum tension vectors, which is the principle of 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.
Along with Coulomb's law, another description of the interaction of electric charges is also possible.
Long range and close range. Coulomb's law, like the law gravity, interprets the interaction of charges as "action at a distance", or "long-range action". Indeed, the Coulomb force depends only on the magnitude of the charges and on the distance between them. Coulomb was convinced that the intermediate medium, that is, the "emptiness" between the charges, does not take any part in the interaction.
This point of view was undoubtedly inspired by the impressive success of Newton's theory of gravitation, which was brilliantly confirmed astronomical observations. However, Newton himself wrote: “It is not clear how inanimate inert matter, without the mediation of something else that is immaterial, could act on another body without mutual contact.” Nevertheless, the concept of long-range action, based on the idea of the instantaneous action of one body on another at a distance without the participation of any intermediate medium, dominated the scientific worldview for a long time.
The idea of a field as a material medium through which any interaction of spatially distant bodies is carried out was introduced into physics in the 30s of the 19th century by the great English naturalist M. Faraday, who believed that “matter is present everywhere, and there is no intermediate space not occupied
by her." Faraday developed a consistent concept electromagnetic field, based on the idea of a finite interaction propagation velocity. A complete theory of the electromagnetic field, clothed in a rigorous mathematical form, was subsequently developed by another great English physicist, J. Maxwell.
By modern ideas electric charges endow the space around them with special physical properties- create an electric field. The main property of the field is that a certain force acts on a charged particle in this field, i.e., the interaction of electric charges is carried out through the fields they create. The field created by stationary charges does not change with time and is called electrostatic. To study the field, you need to find it physical characteristics. Consider two such characteristics - power and energy.
Electric field strength. For experimental study of the electric field, it is necessary to place a test charge in it. In practice, this will be some kind of charged body, which, firstly, must be small enough to be able to judge the properties of the field at a certain point in space, and, secondly, its electric charge must be small enough to be able to neglect the influence of this charge on the distribution of charges that create the field under study.
A test charge placed in an electric field is subjected to a force that depends both on the field and on the test charge itself. This force is greater, the larger the test charge. By measuring the forces acting on different test charges placed at the same point, one can be convinced that the ratio of the force to the test charge no longer depends on the magnitude of the charge. Hence, this relation characterizes the field itself. The power characteristic of the electric field is the intensity E - a vector quantity equal at each point to the ratio of the force acting on the test charge placed at this point to the charge
In other words, the field strength E is measured by the force acting on a single positive test charge. In general, the field strength is different in different points. A field in which the intensity at all points is the same both in absolute value and in direction is called homogeneous.
Knowing the strength of the electric field, you can find the force acting on any charge placed in given point. In accordance with (1), the expression for this force has the form
How to find the field strength at any point?
The strength of the electric field created by a point charge can be calculated using Coulomb's law. We will consider a point charge as a source of an electric field. This charge acts on a test charge located at a distance from it with a force whose modulus is equal to
Therefore, in accordance with (1), dividing this expression by we obtain the module E of the field strength at the point where the test charge is located, i.e., at a distance from the charge
Thus, the field strength of a point charge decreases with distance in inverse proportion to the square of the distance, or, as they say, according to the inverse square law. Such a field is called a Coulomb field. When approaching a point charge creating a field, the field strength of a point charge increases indefinitely: from (4) it follows that when
The coefficient k in formula (4) depends on the choice of the system of units. In CGSE k = 1, and in SI . Accordingly, formula (4) is written in one of two forms:
The unit of tension in the CGSE does not have a special name, but in SI it is called "volt per meter"
Due to the isotropy of space, i.e., the equivalence of all directions, the electric field of a solitary point charge is spherically symmetrical. This circumstance is manifested in formula (4) in that the modulus of the field strength depends only on the distance to the charge that creates the field. The intensity vector E has a radial direction: it is directed from the charge that creates the field if it is a positive charge (Fig. 6a, a), and to the charge that creates the field if this charge is negative (Fig. 6b).
The expression for the field strength of a point charge can be written in vector form. It is convenient to place the origin of coordinates at the point where the charge that creates the field is located. Then the field strength at any point characterized by the radius vector is given by the expression
This can be verified by comparing the definition (1) of the field strength vector with the formula (2) § 1, or starting from
directly from formula (4) and taking into account the above considerations about the direction of the vector E.
The principle of superposition. How to find the strength of the electric field created by an arbitrary distribution of charges?
Experience shows that electric fields satisfy the principle of superposition. The strength of the field created by several charges is equal to vector sum field strengths created by each charge separately:
The principle of superposition actually means that the presence of other electric charges has no effect on the field created by this charge. This property, when separate sources act independently and their actions simply add up, is inherent in the so-called linear systems, and this property itself physical systems called linearity. The origin of this name is due to the fact that such systems are described linear equations(first degree equations).
We emphasize that the validity of the superposition principle for an electric field is not a logical necessity or something taken for granted. This principle is a generalization of experimental facts.
The principle of superposition makes it possible to calculate the strength of the field created by any distribution of immobile electric charges. In the case of several point charges, the recipe for calculating the resulting intensity is obvious. Any non-point charge can be mentally divided into such small parts that each of them can be considered as a point charge. The electric field strength at an arbitrary point is found as
the vector sum of the tensions created by these "point" charges. The corresponding calculations are greatly simplified in cases where there is a certain symmetry in the distribution of the charges creating the field.
Tension lines. A visual graphical representation of electric fields is given by lines of tension or lines of force.
Rice. 7. Field strength lines of positive and negative point charges
These electric field lines are drawn in such a way that at each point the tangent to the line coincides in direction with the intensity vector at that point. In other words, at any place the tension vector is directed tangentially to the line of force passing through this point. The lines of force are assigned a direction: they come from positive charges or come from infinity. They either end in negative charges or go to infinity. In the figures, this direction is indicated by arrows on the field line.
A line of force can be drawn through any point in the electric field.
The lines are drawn thicker in those places where the field strength is greater, and less often where it is less. Thus, the density of field lines gives an idea of the modulus of tension.
Rice. 8. Lines of field strength of opposite identical charges
On fig. 7 shows the field lines of a solitary positive and negative point charge. It is obvious from the symmetry that these are radial lines distributed with the same density in all directions.
More complex view has a pattern of field lines created by two charges of opposite signs. Such a field is obviously
has axial symmetry: the whole picture remains unchanged when rotated through any angle around an axis passing through the charges. When the modules of the charges are the same, the pattern of lines is also symmetrical with respect to a plane passing perpendicular to the segment connecting them through its middle (Fig. 8). In this case, the lines of force come out of the positive charge and they all terminate in the negative, although in Fig. 8 it is impossible to show how the lines going far from the charges are closed.
Coulomb's law:
where F is the force of interaction of two point charges q 1 and q 2; r is the distance between charges; is the dielectric constant of the medium; 0 - electrical constant
.
The law of conservation of charge:
,
where 
is the algebraic sum of the charges included in the isolated system; n is the number of charges.
Strength and potential of the electrostatic field:
;
, or 
,
where 
is the force acting on a point positive charge q 0 placed at a given point of the field; P is the potential energy of the charge; And ∞ is the work spent on moving the charge q 0 from a given point of the field to infinity.
Tension Vector Flow 
electric field:
a) through an arbitrary surface S placed in an inhomogeneous field:
, or 
,
where is the angle between the intensity vector 
and normal 
to the surface element; dS is the area of the surface element; E n is the projection of the stress vector onto the normal;
b) through a flat surface placed in a uniform electric field:
.
Tension Vector Flow 
through a closed surface
(integration is carried out over the entire surface).
Ostrogradsky-Gauss theorem. The flow of the intensity vector through any closed surface covering the charges q1, q2, ..., qn, -
,
where 
is the algebraic sum of charges enclosed inside a closed surface; n is the number of charges.
tension electrostatic field, created by a point charge q at a distance r from the charge, –
.
The strength of the electric field created by a sphere having a radius R and carrying a charge q, at a distance r from the center of the sphere is as follows:
inside the sphere (r R) E=0;
on the surface of the sphere (r=R) 
;
outside the sphere (r R) 
.
The principle of superposition (superposition) of electrostatic fields, according to which the intensity 
of the resulting field created by two (or more) point charges is equal to the vector (geometric) sum of the strengths of the added fields, is expressed by the formula
In the case of two electric fields with strengths 
and 
the absolute value of the intensity vector is
where is the angle between the vectors 
and 
.
The intensity of the field created by an infinitely long and uniformly charged thread (or cylinder) at a distance r from its axis is
,
where is the linear charge density.
The linear charge density is a value equal to its ratio to the length of the thread (cylinder):
.
The intensity of the field created by an infinite uniformly charged plane is
,
where - surface density charge.
The surface charge density is a value equal to the ratio of the charge distributed over the surface to its area:
.
The strength of the field created by two infinite and parallel planes, charged uniformly and differently, with the same absolute value of the surface density of the charge (the field of a flat capacitor) -
.
The above formula is valid when calculating the field strength between the plates of a flat capacitor (in its middle part) only if the distance between the plates is much less than the linear dimensions of the capacitor plates.
electrical displacement 
associated with tension 
electric field ratio
,
which is valid only for isotropic dielectrics.
The potential of an electric field is a quantity equal to the ratio of potential energy and a point positive charge placed at a given point in the field:
.
In other words, the electric field potential is a value equal to the ratio of the work of the field forces to move a point positive charge from a given point of the field to infinity to the value of this charge:
.
The potential of the electric field at infinity is conditionally taken equal to zero.
The potential of the electric field created by a point charge q on
distance r from the charge, –
.
The potential of the electric field created by a metal sphere having a radius R and carrying a charge q, at a distance r from the center of the sphere is as follows:
inside the sphere (r R) 
;
on the surface of a sphere (r = R) 
;
outside the sphere (r R) 
.
In all formulas given for the potential of a charged sphere, is the permittivity of a homogeneous infinite dielectric surrounding the sphere.
The potential of the electric field formed by a system of n point charges at a given point, in accordance with the principle of superposition of electric fields, is equal to the algebraic sum of the potentials 
, created by individual point charges 
:
.
Energy W of interaction of a system of point charges 
is determined by the work that this system can do when they are removed relative to each other to infinity, and is expressed by the formula
,
where 
- field potential created by all (n-1) charges (except for the i-th) at the point where the charge is located 
.
The potential is related to the electric field strength by the relation
.
In the case of an electric field with spherical symmetry, this relationship is expressed by the formula
,
or in scalar form
.
In the case of a homogeneous field, i.e. field, the intensity of which at each of its points is the same both in absolute value and in direction, -
,
where 1 and 2 are the potentials of the points of two equipotential surfaces; d is the distance between these surfaces along the electric line of force.
The work done by the electric field when moving a point charge q from one point of the field, having a potential 1, to another, having a potential 2, is equal to
, or 
,
where E 
is the vector projection 
to the direction of movement; 
- movement.
In the case of a homogeneous field, the last formula takes the form
,
where 
- displacement; - angle between vector directions 
and moving 
.
A dipole is a system of two point (equal in absolute value and opposite in sign) charges located at some distance from each other.
Electric moment 
dipole is a vector directed from a negative charge to a positive one, equal to the product of the charge 
per vector 
, drawn from a negative charge to a positive one, and called the dipole arm, i.e.
.
A dipole is called a point dipole if its arm 
much less than the distance r from the center of the dipole to the point at which we are interested in the action of the dipole ( 
r), see fig. one.
Field strength of a point dipole:
,
where p is the electric moment of the dipole; r is the absolute value of the radius vector drawn from the center of the dipole to the point where the field strength is of interest to us; - angle between the radius vector 
and shoulder 
dipole.
Field strength of a point dipole at a point lying on the axis of the dipole
(=0), is found by the formula
;
at a point perpendicular to the dipole arm reconstructed from its middle 
, - according to the formula
.
The field potential of a point dipole at a point lying on the dipole axis (=0) is
,
and at a point lying on the perpendicular to the dipole arm, reconstructed from its middle 
,
–
The strength and potential of a non-point dipole are determined in the same way as for a system of charges.
The mechanical moment acting on a dipole with an electric moment p, placed in a uniform electric field with a strength E, is
, or 
,
where is the angle between the directions of the vectors 
and 
.
The capacitance of a solitary conductor or capacitor is
,
where q is the charge imparted to the conductor; 
is the change in potential caused by this charge.
The capacitance of a solitary conducting sphere of radius R, located in an infinite medium with a permittivity , is
.
If the sphere is hollow and filled with a dielectric, then its capacitance does not change.
Electric capacitance of a flat capacitor:
,
where S is the area of each capacitor plate; d is the distance between the plates; - permittivity of the dielectric filling the space between the plates.
The capacitance of a flat capacitor filled with n layers of dielectric with thickness d i and permittivity i each (layered capacitor) is
.
The capacitance of a spherical capacitor (two concentric spheres with a radius R 1 and R 2, the space between which is filled with a dielectric with a permittivity ) is as follows:
.
The capacitance of series-connected capacitors is:
in general -
,
where n is the number of capacitors;
in the case of two capacitors -
;
.
The capacitance of capacitors connected in parallel is determined as follows:
in general -
C \u003d C 1 + C 2 + ... + C n;
in the case of two capacitors -
C \u003d C 1 + C 2;
in the case of n identical capacitors with electrical capacity C 1 each -
The energy of a charged conductor is expressed in terms of charge q, potential and electrical capacity C of the conductor as follows:
.
The energy of a charged capacitor is
,
where q is the charge of the capacitor; C is the capacitance of the capacitor; U is the potential difference on its plates.
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. Mathematical definition stress is written like this:
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, which is determined by the radius vector (in meters), if the electric field creates a positive point charge (q=1C), which lies in 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.
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.
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 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.