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)?
Decision. 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.
A charged body constantly transfers part of the energy, transforming it into another state, one of the parts of which is an electric field. Tension is the main component that characterizes the electrical part of electromagnetic radiation. Its value depends on the current strength and acts as a power characteristic. It is for this reason that high-voltage wires are placed at a greater height than wiring for less current.
Definition of the concept and calculation formula
The intensity vector (E) is the force acting on an infinitesimal current at the point under consideration. The formula for determining the parameter is as follows:
- F is the force that acts on the charge;
- q is the amount of charge.
The charge taking part in the study is called a test charge. It should be small so as not to distort the results. Under ideal conditions, the role of q is played by the positron.
It should be noted that the value is relative, its quantitative characteristics and direction depend on the coordinates and will change with a shift.
Based on the Coulomb law, the force acting on a body is equal to the product of the potentials divided by the square of the distance between the bodies.
F=q 1* q 2 /r 2
It follows from this that the intensity at a given point in space is directly proportional to the potential of the source and inversely proportional to the square of the distance between them. In the general, symbolic case, the equation is written as follows:
Based on the equation, the unit of electric field is Volts per meter. The same designation is adopted by the SI system. Having the value of the parameter, you can calculate the force that will act on the body at the point under study, and knowing the force, you can find the electric field strength.
The formula shows that the result is absolutely independent of the test charge. This is unusual since this parameter is present in the original equation. However, this is logical, because the source is the main emitter, not the test emitter. In real conditions, this parameter has an impact on the measured characteristics and produces a distortion, which leads to the use of a positron for ideal conditions.
Since tension is a vector quantity, in addition to the value, it has a direction. The vector is directed from the main source to the investigated one, or from the trial charge to the main one. It depends on the polarity. If the signs are the same, then repulsion occurs, the vector is directed towards the point under study. If the points are charged in opposite polarities, then the sources are attracted. In this case, it is customary to assume that the force vector is directed from a positive source to a negative one.
unit of measurement
Depending on the context and application in the fields of electrostatics, electric field strength [E] is measured in two units. It can be volt/meter or newton/coulomb. The reason for this confusion seems to be obtaining it from different conditions, deriving the unit of measurement from the formulas used. In some cases, one of the dimensions is used intentionally to prevent the use of formulas that work only for special cases. The concept is present in the fundamental electrodynamic laws, so the value is basic for thermodynamics.
The source can take many forms. The formulas described above help to find the electric field strength of a point charge, but the source can be in other forms:
- several independent material points;
- distributed straight line or curve (magnet stator, wire, etc.).
For a point charge, finding the tension is as follows: E=k*q/r 2 , where k=9*10 9
When several sources act on the body, the tension at the point will be equal to the vector sum of the potentials. Under the action of a distributed source, it is calculated by the effective integral over the entire distribution area.
The characteristic may change over time due to changes in charges. The value remains constant only for the electrostatic field. It is one of the main power characteristics, therefore, for a homogeneous field, the direction of the vector and the value of q will be the same in any coordinates.
From the point of view of thermodynamics
Tension is one of the main and key characteristics in classical electrodynamics. Its value, as well as the data of electric charge and magnetic induction, are the main characteristics, knowing which it is possible to determine the parameters of the flow of almost all electrodynamic processes. It is present and plays an important role in such fundamental concepts as the Lorentz force formula and Maxwell's equations.
F-Lorenz force;
- q is the charge;
- B is the magnetic induction vector;
- C is the speed of light in vacuum;
- j is the magnetic current density;
- μ 0 - magnetic constant \u003d 1.25663706 * 10 -6;
- ε 0 - electrical constant equal to 8.85418781762039 * 10 -12
Along with the value of magnetic induction, this parameter is the main characteristic of the electromagnetic field emitted by the charge. Based on this, from the point of view of thermodynamics, the intensity is much more important than the current strength or other indicators.
These laws are fundamental; all thermodynamics is based on them. It should be noted that Ampère's law and other earlier formulas are approximate or describe particular cases. Maxwell's and Lorentz's laws are universal.
Practical value
The concept of tension has found wide application in electrical engineering. It is used to calculate the norms of signals, calculate the stability of the system, determine the effect of electrical radiation on the elements surrounding the source.
The main area where the concept has found wide application is cellular and satellite communications, television towers and other electromagnetic emitters. Knowing the radiation intensity for these devices allows you to calculate parameters such as:
- range of the radio tower;
- safe distance from source to person .
The first parameter is extremely important for those who install satellite television broadcasting, as well as mobile communications. The second makes it possible to determine the permissible standards for radiation, thereby protecting users from the harmful effects of electrical appliances. The application of these properties of electromagnetic radiation is not limited to communications. Power generation, household appliances, partly the production of mechanical products (for example, dyeing with electromagnetic pulses) are built on these basic principles. Thus, understanding the magnitude is also important for the production process.
Interesting experiments that allow you to see the pattern of electric field lines: video
The electric field that surrounds the charge is a reality independent of our desire to change something and somehow influence it. From this we can conclude that the electric field is one of the forms of existence of matter, as well as matter.
The electric field of charges at rest is called electrostatic. To detect the electrostatic field of a certain charge, you need to introduce another charge into its field, on which a certain force will act in. However, without the presence of a second charge, the electrostatic field of the first charge exists, but does not manifest itself in any way.
Tension E characterize the electrostatic field. The intensity at a certain point of the electric field is a physical quantity that is equal to the force acting on a unit positive charge at rest placed at a certain point in the field, and directed in the direction of the force.
If a “trial” positive point charge q pr is introduced into the electric field created by the charge q, then, according to the Coulomb law, a force will act on it:
If different test charges q / pr, q // pr and so on are placed at one point of the field, then different forces proportional to the magnitude of the charge will act on each of them. The ratio F / q pr for all charges introduced into the field will be identical, and will also depend only on q and r, which determine the electric field at a given point. This value can be expressed by the formula:
If we assume that q pr \u003d 1, then E \u003d F. From this we conclude that the strength of the electric field is its power characteristic. From formula (2), taking into account the expression of the Coulomb force (1), it follows:
It can be seen from formula (2) that the unit of tension is taken to be the intensity at a certain point in the field, where a unit of force will act on a unit of charge. Therefore, in the CGS system, the unit of tension is dyn / CGS q, and in the SI system it will be N / Cl. The ratio between the given units is called the absolute electrostatic unit of tension (CGS E):
The intensity vector is directed from the charge along the radius with a positive charge forming the field q +, and with a negative charge q - towards the charge along the radius.
If the electric field is formed by several charges, then the forces that will act on the test charge are added according to the vector addition rule. Therefore, the strength of a system consisting of several charges at a given point in the field will be equal to the vector sum of the strengths of each charge separately:
This phenomenon is called the principle of superposition (superposition) of electric fields.
The intensity at any point of the electric field of two point charges - q 2 and + q 1 can be found using the principle of superposition:
According to the parallelogram rule, the vectors E 1 and E 2 will be added. The direction of the resulting vector E is determined by the construction, and its absolute value can be calculated using the formula below:
Where α is the angle between the vectors E 1 and E 2.
Let's consider the electric field that the dipole creates. Electric dipole - this is a system of equal in magnitude (q \u003d q 1 \u003d q 2), but opposite in sign, charges, the distance between which is very small when compared with the distance to the considered points of the electric field.
The electric dipole moment p, which is the main characteristic of the dipole and is defined as a vector directed from a negative charge to a positive one, and equal to the product of the dipole arm l and the charge q:
Also, the vector is the arm of the dipole l, directed from the negative charge to the positive, and determines the distance between the charges. The line that passes through both charges is called - dipole axis.
Let's determine the electric field strength at a point that lies on the dipole axis in the middle (Figure below a)):
At point B, the intensity E will be equal to the vector sum of the intensities E / and E // , which are created by positive and negative charges but separately. Between the charges –q and +q, the intensity vectors E / and E // are directed in the same direction, therefore, in absolute value, the resulting intensity E will be equal to their sum.
If we need to find E at point A, which lies on the continuation of the dipole axis, then the vectors E / and E // will be directed in different directions, respectively, in absolute value, the resulting intensity will be equal to their difference:
Where r is the distance between the point that lies on the axis of the dipole and where the intensity is determined, and the midpoint of the dipole.
In the case of r>>l, the value (l/2) in the denominator can be neglected, then we get the following relation:
Where p is the electric dipole moment.
This formula in the CGS system will take the form:
Now you need to calculate the electric field strength at point C (figure above b)) lying on the perpendicular restored from the midpoint of the dipole.
Since r 1 \u003d r 2, then the equality will take place:
The dipole strength at an arbitrary point can be determined by the formula:
Where α is the angle between the dipole arm l and the radius vector r, r is the distance from the point at which the field strength is determined to the center of the dipole, p is the electric moment of the dipole.
Example
At a distance R \u003d 0.06 m from each other there are two identical point charges q 1 \u003d q 2 \u003d 10 -6 C (figure below):
It is necessary to determine the electric field strength at point A, which is located on the perpendicular restored in the center of the segment that connects the charges, at a distance h = 4 cm from this segment. It is also necessary to determine the tension at point B, located in the middle of the segment that connects the charges.
Decision
According to the principle of superposition (superposition of fields), the field strength E is determined. Thus, the vector (geometric) sum is determined by E created by each charge separately: E \u003d E 1 + E 2.
The electric field strength of the first point charge is:
Where q 1 and q 2 are the charges that form the electric field; r is the distance from the point at which the intensity is calculated to the charge; ε 0 - electrical constant; ε is the relative permittivity of the medium.
To determine the intensity at point B, you first need to build the electric field strength vectors from each charge. Since the charges are positive, the vectors E / and E // will be directed from point B in different directions. By condition q 1 = q 2:
This means that in the middle of the segment, the field strength is zero.
At point A, it is necessary to perform a geometric addition of the vectors E 1 and E 2. At point A, the tension will be equal to:
As you know, electrical voltage must have its own measure, which initially corresponds to the value that is calculated to power a particular electrical device. Exceeding or reducing the value of this supply voltage negatively affects electrical equipment, up to its complete failure. What is tension? This is the difference in electrical potential. That is, if, for ease of understanding, it is compared with water, then this will approximately correspond to pressure. According to the scientific, electrical voltage is a physical quantity that shows what work the current does in a given area when a unit charge moves through this area.
The most common formula for voltage is the one in which there are three basic electrical quantities, namely the voltage itself, current and resistance. Well, this formula is known as Ohm's law (finding the electrical voltage, potential difference).
This formula sounds as follows - the electrical voltage is equal to the product of the current strength and resistance. Let me remind you that in electrical engineering for various physical quantities there are their own units of measurement. The unit of voltage measurement is "Volt" (in honor of the scientist Alessandro Volta, who discovered this phenomenon). The unit of measurement for current is "Ampere", and resistance is "Ohm". As a result, we have - an electrical voltage of 1 volt will be equal to 1 ampere times 1 ohm.
In addition, the second most used voltage formula is the one in which this same voltage can be found knowing the electrical power and current strength.
This formula sounds as follows - the electrical voltage is equal to the ratio of power to current strength (to find the voltage, you need to divide the power by the current). The power itself is found by multiplying the current by the voltage. Well, to find the current strength, you need to divide the power by the voltage. Everything is extremely simple. The unit of electrical power is "Watt". So 1 volt is equal to 1 watt divided by 1 amp.
Well, now I will give a more scientific formula for electrical voltage, which contains "work" and "charges".
This formula shows the ratio of the work done to move the electric charge. In practice, this formula is unlikely to be needed. The most common will be the one that contains current, resistance and power (that is, the first two formulas). But, I want to warn you that it will be true only for the case of active resistances. That is, when calculations are made for an electrical circuit that has resistance in the form of conventional resistors, heaters (with a nichrome spiral), incandescent bulbs, and so on, then the above formula will work. In the case of using reactance (the presence of inductance or capacitance in the circuit), a different voltage formula will be needed, which also takes into account the voltage frequency, inductance, capacitance.
P.S. The formula of Ohm's law is fundamental, and it is from it that you can always find one unknown quantity out of two known ones (current, voltage, resistance). In practice, Ohm's law will be applied very often, so it is simply necessary for every electrician and electronics to know it by heart.
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.
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 tension and is too small to affect 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.
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 located according to certain rules, giving as much information as possible about the nature of the 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 of 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 an electric current would occur within the 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.
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 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.
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.
This expression illustrates how changing one variable can affect another. Mathematical notation of the law:
E1/E2 = r2²/r1².
The value of the field strength depends on the location of the selected point, its value decreases with distance from the charge. If we take the intensity of the EP at two different points, then the ratio of their quantitative values will be inversely proportional to the squares of the distance.
To measure the electric field strength in practical conditions, there are special devices, for example, the VX 0100 tester.
Video