All formulas on electrostatics 1 course. Basic formulas and guidelines for solving problems in electrostatics
Electric charge Is a physical quantity that characterizes the ability of particles or bodies to enter into electromagnetic interactions. Electric charge is usually indicated by letters q or Q... In the SI system, electric charge is measured in Coulomb (C). A free charge of 1 C is a gigantic amount of charge that practically does not occur in nature. As a rule, you will have to deal with microcoulomb (1 µC = 10 –6 C), nanocoulomb (1 nC = 10 –9 C) and picoculones (1 pC = 10 –12 C). An electric charge has the following properties:
1. Electric charge is a kind of matter.
2. The electric charge does not depend on the movement of the particle and on its speed.
3. Charges can be transferred (for example, by direct contact) from one body to another. Unlike body weight, electric charge is not an integral characteristic of a given body. The same body in different conditions may have a different charge.
4. There are two kinds of electric charges, conventionally named positive and negative.
5. All charges interact with each other. In this case, like charges repel, unlike charges attract. The forces of interaction of charges are central, that is, lie on a straight line connecting the centers of charges.
6. There is a minimum possible (modulo) electric charge, called elementary charge... Its meaning:
e= 1.602177 · 10 –19 C ≈ 1.6 · 10 –19 C.
The electric charge of any body is always a multiple of the elementary charge:
where: N- an integer. Please note that the existence of a charge equal to 0.5 is impossible. e; 1,7e; 22,7e etc. Physical quantities that can only take a discrete (not continuous) series of values are called quantized... The elementary charge e is a quantum (the smallest portion) of the electric charge.
In an isolated system, the algebraic sum of the charges of all bodies remains constant:
The law of conservation of electric charge states that in a closed system of bodies, the processes of creation or disappearance of charges of only one sign cannot be observed. It also follows from the law of conservation of charge if two bodies of the same size and shape, possessing charges q 1 and q 2 (it does not matter what sign of the charges), bring into contact, and then dissolve back, then the charge of each of the bodies will become equal:
From the modern point of view, elementary particles are charge carriers. All ordinary bodies are composed of atoms, which include positively charged protons negatively charged electrons and neutral particles - neutrons... Protons and neutrons are part of atomic nuclei, electrons form the electron shell of atoms. The electric charges of a proton and an electron in modulus are exactly the same and equal to the elementary (that is, the minimum possible) charge e.
In a neutral atom, the number of protons in the nucleus is equal to the number of electrons in the shell. This number is called atomic number... An atom of a given substance can lose one or more electrons, or acquire an extra electron. In these cases, a neutral atom turns into a positively or negatively charged ion. Please note that positive protons are part of the nucleus of an atom, so their number can only change during nuclear reactions. Obviously, when electrizing bodies nuclear reactions not happening. Therefore, in any electrical phenomena, the number of protons does not change, only the number of electrons changes. So, the message to the body of a negative charge means the transfer of extra electrons to it. And the message of a positive charge, in spite of a common mistake, means not adding protons, but taking away electrons. Charge can be transferred from one body to another only in portions containing an integer number of electrons.
Sometimes in problems the electric charge is distributed over a certain body. To describe this distribution, the following quantities are introduced:
1. Linear charge density. Used to describe the distribution of charge along the filament:
where: L- the length of the thread. Measured in C / m.
2. Surface charge density. Used to describe the distribution of charge over the surface of the body:
where: S- body surface area. Measured in C / m2.
3. Bulk charge density. Used to describe the distribution of charge over the volume of the body:
where: V- body volume. Measured in C / m 3.
Please note that electron mass is equal to:
m e= 9.11 ∙ 10 -31 kg.
Coulomb's law
Point charge is called a charged body, the dimensions of which can be neglected under the conditions of this problem. Based on numerous experiments, Coulomb established the following law:
The forces of interaction of stationary point charges are directly proportional to the product of charge modules and inversely proportional to the square of the distance between them:
where: ε - dielectric constant of a medium - a dimensionless physical quantity that shows how many times the force of electrostatic interaction in a given medium will be less than in a vacuum (that is, how many times the medium weakens the interaction). Here k- coefficient in Coulomb's law, a value that determines the numerical value of the force of interaction of charges. In the SI system, its value is taken equal to:
k= 9 ∙ 10 9 m / F.
The forces of interaction of point stationary charges obey Newton's third law, and are forces of repulsion from each other with the same signs of charges and forces of attraction to each other at different signs... The interaction of stationary electric charges is called electrostatic or Coulomb interaction. The branch of electrodynamics that studies the Coulomb interaction is called electrostatics.
Coulomb's law is valid for point charged bodies, uniformly charged spheres and balls. In this case, the distance r take the distance between the centers of the spheres or balls. In practice, Coulomb's law is well satisfied if the dimensions of the charged bodies are much less than the distance between them. Coefficient k in the SI system is sometimes written in the form:
where: ε 0 = 8.85 ∙ 10 –12 F / m - electrical constant.
Experience shows that the forces of the Coulomb interaction obey the principle of superposition: if a charged body interacts simultaneously with several charged bodies, then the resulting force acting on a given body is equal to the vector sum of the forces acting on this body from all other charged bodies.
Remember also two important definitions:
Conductors- substances containing free carriers of electric charge. Inside the conductor, the free movement of electrons - charge carriers is possible ( electricity). Conductors include metals, solutions and molten electrolytes, ionized gases, and plasma.
Dielectrics (insulators)- substances in which there are no free charge carriers. Free movement of electrons inside dielectrics is impossible (electric current cannot flow through them). It is dielectrics that have a certain dielectric constant that is not equal to unity ε .
For the dielectric constant of a substance, the following is true (about what an electric field is a little lower):
Electric field and its strength
By modern ideas, electric charges do not act directly on each other. Each charged body creates in the surrounding space electric field... This field exerts a forceful effect on other charged bodies. Main property electric field- action on electric charges with some force. Thus, the interaction of charged bodies is carried out not by their direct impact on each other, but through the electric fields surrounding the charged bodies.
Electric field surrounding a charged body can be investigated using the so-called test charge - a small point charge, which does not introduce a noticeable redistribution of the charges under study. To quantify the electric field, a force characteristic is introduced - electric field strength E.
The electric field strength is called physical quantity, equal to the ratio of the force with which the field acts on the test charge placed at a given point of the field to the value of this charge:
Electric field strength is a vector physical quantity. The direction of the tension vector coincides at each point in space with the direction of the force acting on the positive test charge. The electric field of charges that are stationary and do not change over time is called electrostatic.
For a visual representation of the electric field, use lines of force... These lines are drawn so that the direction of the tension vector at each point coincides with the direction of the tangent to the line of force. Lines of force have the following properties.
- The lines of force of the electrostatic field never intersect.
- The lines of force of the electrostatic field are always directed from positive charges to negative charges.
- When displaying an electric field using lines of force, their density should be proportional to the modulus of the field strength vector.
- Lines of force start at positive charge or infinity and end at negative or infinity. The density of the lines is the greater, the greater the tension.
- At a given point in space, only one line of force can pass, since the electric field strength at a given point in space is set uniquely.
An electric field is called uniform if the intensity vector is the same at all points of the field. For example, a uniform field creates a flat capacitor - two plates charged with equal in magnitude and opposite in sign charge, separated by a dielectric layer, and the distance between the plates is much smaller sizes plates.
At all points uniform field per charge q introduced into a uniform field with a strength E, a force of the same magnitude and direction acts, equal to F = Eq... Moreover, if the charge q positive, then the direction of the force coincides with the direction of the tension vector, and if the charge is negative, then the vectors of force and tension are oppositely directed.
Positive and negative point charges are shown in the figure:
Superposition principle
If an electric field created by several charged bodies is investigated with the help of a test charge, then the resulting force turns out to be equal to the geometric sum of forces acting on the test charge from the side of each charged body separately. Consequently, the intensity of the electric field created by the system of charges at a given point in space is equal to the vector sum of the intensities of the electric fields created at the same point by the charges separately:
This property of the electric field means that the field obeys superposition principle... In accordance with Coulomb's law, the strength of the electrostatic field created by a point charge Q on distance r from it, is equal in modulus:
This field is called Coulomb. In a Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: if Q> 0, then the tension vector is directed from the charge, if Q < 0, то вектор напряженности направлен к заряду. Величина напряжённости зависит от величины заряда, среды, в которой находится заряд, и уменьшается с увеличением расстояния.
The strength of the electric field, which creates a charged plane near its surface:
So, if in the task it is required to determine the field strength of the system of charges, then one must act as follows algorithm:
- Draw a drawing.
- Display the field strength of each charge separately at the desired point. Remember that the tension is directed towards the negative charge and away from the positive charge.
- Calculate each of the tensions using the appropriate formula.
- Add the tension vectors geometrically (i.e. vectorially).
Potential energy of interaction of charges
Electric charges interact with each other and with the electric field. Any interaction describes potential energy. Potential energy of interaction of two point electric charges calculated by the formula:
Pay attention to the lack of modules in the charges. For opposite charges, the interaction energy has a negative value. The same formula is valid for the interaction energy of uniformly charged spheres and balls. As usual, in this case, the distance r is measured between the centers of the balls or spheres. If there are more than two charges, then the energy of their interaction should be considered as follows: break the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.
The problems on this topic are solved, as are the problems on the law of conservation of mechanical energy: first, the initial interaction energy is found, then the final one. If the problem is asked to find work on the movement of charges, then it will be equal to the difference between the initial and final total energy of interaction of charges. The energy of interaction can also pass into kinetic energy or other types of energy. If the bodies are very great distance, then the energy of their interaction is set equal to 0.
Please note: if in a task you need to find the minimum or maximum distance between bodies (particles) when moving, then this condition will be fulfilled at that moment in time when the particles move in one direction with the same speed. Therefore, the solution must begin with writing down the law of conservation of momentum, from which this identical velocity is found. And then the law of conservation of energy should be written taking into account the kinetic energy of particles in the second case.
Potential. Potential difference. Voltage
The electrostatic field has an important property: the work of the forces of the electrostatic field when the charge moves from one point of the field to another does not depend on the shape of the trajectory, but is determined only by the position of the starting and ending points and the magnitude of the charge.
A consequence of the independence of the work from the shape of the trajectory is the following statement: the work of the forces of the electrostatic field when the charge moves along any closed trajectory is equal to zero.
The property of potentiality (independence of work from the shape of the trajectory) of an electrostatic field allows us to introduce the concept of potential energy of a charge in an electric field. And a physical quantity equal to the ratio of the potential energy of an electric charge in an electrostatic field to the value of this charge is called potential φ electric field:
Potential φ is the energy characteristic of the electrostatic field. In the International System of Units (SI), the unit of potential (and hence the potential difference, ie voltage) is volt [V]. Potential is a scalar quantity.
In many electrostatic problems, when calculating the potentials for the reference point, where the values of the potential energy and potential vanish, it is convenient to take an infinitely distant point. In this case, the concept of potential can be defined as follows: the potential of the field at a given point in space is equal to the work that electric forces perform when a single positive charge is removed from a given point to infinity.
Remembering the formula for the potential energy of interaction of two point charges and dividing it by the value of one of the charges in accordance with the definition of the potential, we obtain that potential φ point charge fields Q on distance r from it relative to the point at infinity is calculated as follows:
The potential calculated by this formula can be positive or negative, depending on the sign of the charge that created it. The same formula expresses the potential of the field of a uniformly charged ball (or sphere) at r ≥ R(outside of a ball or sphere), where R Is the radius of the ball, and the distance r measured from the center of the ball.
For a visual representation of the electric field, along with the lines of force, use equipotential surfaces... The surface, at all points of which the potential of the electric field has the same values, is called the equipotential surface or the surface of equal potential. The lines of force of the electric field are always perpendicular to the equipotential surfaces. Equipotential surfaces of the Coulomb field of a point charge are concentric spheres.
Electrical voltage it's just a potential difference, i.e. the definition of electric voltage can be given by the formula:
In a uniform electric field, there is a relationship between field strength and voltage:
Electric field work can be calculated as the difference between the initial and final potential energy of the system of charges:
The work of the electric field in the general case can also be calculated using one of the formulas:
In a uniform field, when a charge moves along its lines of force, the work of the field can also be calculated using the following formula:
In these formulas:
- φ Is the potential of the electric field.
- ∆φ - potential difference.
- W- potential energy of a charge in an external electric field.
- A- the work of the electric field to move the charge (charges).
- q- a charge that is moved in an external electric field.
- U- voltage.
- E- electric field strength.
- d or ∆ l- the distance that the charge is moved along the lines of force.
In all the previous formulas, it was just about the work of the electrostatic field, but if the problem says that "the work must be done", or in question about the "work of external forces", then this work should be considered in the same way as the work of the field, but with the opposite sign.
Potential Superposition Principle
The principle of superposition for potentials follows from the principle of superposition of field strengths created by electric charges (in this case, the sign of the field potential depends on the sign of the charge that created the field):
Notice how much easier it is to apply the principle of superposition of potential than of tension. Potential is a scalar quantity with no direction. Adding potentials is simply adding numbers.
Electric capacity. Flat capacitor
When a conductor is imparted a charge, there is always a certain limit beyond which it will not be possible to charge the body. To characterize the body's ability to accumulate an electric charge, the concept is introduced electrical capacity... The capacity of a solitary conductor is called the ratio of its charge to potential:
In SI, capacitance is measured in Farads [F]. 1 Farad - extremely large capacity... For comparison, the capacity of only the globe significantly less than one farad. The capacity of a conductor does not depend on its charge or on the potential of the body. Likewise, density does not depend on either the mass or the volume of the body. Capacity depends only on the shape of the body, its size and the properties of its environment.
Electric capacity a system of two conductors is called a physical quantity, defined as the ratio of the charge q one of the conductors to the potential difference Δ φ between them:
The amount of electrical capacity of conductors depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are such configurations of conductors in which the electric field is concentrated (localized) only in a certain region of space. Such systems are called capacitors, and the conductors that make up the capacitor are called covers.
The simplest capacitor is a system of two flat conducting plates located parallel to each other at a small distance compared to the dimensions of the plates and separated by a dielectric layer. Such a capacitor is called flat... The electric field of a flat capacitor is mainly localized between the plates.
Each of the charged plates of a flat capacitor creates an electric field near its surface, the modulus of strength of which is expressed by the ratio already given above. Then the modulus of the final field strength inside the capacitor, created by two plates, is equal to:
Outside the capacitor, the electric fields of the two plates are directed towards different sides, and therefore the resulting electrostatic field E= 0. can be calculated by the formula:
Thus, the electrical capacity of a flat capacitor is directly proportional to the area of the plates (plates) and inversely proportional to the distance between them. If the space between the plates is filled with a dielectric, the capacitance of the capacitor increases by ε once. note that S in this formula there is the area of only one plate of the capacitor. When one speaks of the "area of the plates" in the problem, they mean exactly this value. It is never necessary to multiply or divide it by 2.
Let us give the formula for capacitor charge... The charge of a capacitor is understood only as the charge of its positive plate:
The force of attraction of the capacitor plates. The force acting on each plate is determined not by the total field of the capacitor, but by the field created by the opposite plate (the plate itself does not act on itself). The strength of this field is equal to half the strength of the total field, and the force of interaction of the plates:
Capacitor energy. It is also called the energy of the electric field inside the capacitor. Experience shows that a charged capacitor contains a reserve of energy. The energy of a charged capacitor is equal to the work of external forces, which must be expended to charge the capacitor. There are three equivalent forms of writing the formula for the energy of a capacitor (they follow one from the other if we use the relation q = CU):
Pay particular attention to the phrase: "The capacitor is connected to the source." This means that the voltage across the capacitor does not change. And the phrase "The capacitor was charged and disconnected from the source" means that the charge on the capacitor will not change.
Electric field energy
Electrical energy should be considered as potential energy stored in a charged capacitor. According to modern ideas, Electric Energy the capacitor is localized in the space between the capacitor plates, that is, in the electric field. Therefore, it is called the energy of the electric field. The energy of charged bodies is concentrated in space, in which there is an electric field, i.e. we can talk about the energy of the electric field. For example, in a capacitor, energy is concentrated in the space between its plates. Thus, it makes sense to introduce a new physical characteristics- volumetric energy density of the electric field. Using a flat capacitor as an example, one can obtain the following formula for the volumetric energy density (or the energy of a unit volume of an electric field):
Capacitor connections
Parallel connection of capacitors- to increase the capacity. Capacitors are connected by like charged plates, as if increasing the area of equally charged plates. The voltage across all capacitors is the same, the total charge is equal to the sum of the charges of each of the capacitors, and the total capacity is also equal to the sum of the capacitances of all capacitors connected in parallel. Let's write out the formulas for parallel connection of capacitors:
At series connection of capacitors the total capacity of the capacitor bank is always less than the capacity of the smallest capacitor included in the bank. A series connection is used to increase the breakdown voltage of the capacitors. Let's write out the formulas for the series connection of capacitors. The total capacitance of series-connected capacitors is found from the ratio:
From the law of conservation of charge it follows that the charges on adjacent plates are equal:
The voltage is equal to the sum of the voltages across the individual capacitors.
For two capacitors connected in series, the formula above will give us the following expression for the total capacitance:
For N identical series-connected capacitors:
Conductive sphere
The field strength inside a charged conductor is zero. Otherwise, an electric force would act on the free charges inside the conductor, which would force these charges to move inside the conductor. This movement, in turn, would lead to warming up of the charged conductor, which in fact does not happen.
The fact that there is no electric field inside the conductor can be understood in another way: if it were, then the charged particles would move again, and they would move exactly in such a way as to bring this field to zero with their own field, because in fact, they would not want to move, because every system strives for balance. Sooner or later, all moving charges would stop exactly in that place so that the field inside the conductor would become equal to zero.
On the surface of the conductor, the electric field strength is at its maximum. The magnitude of the electric field strength of a charged ball outside of it decreases with distance from the conductor and is calculated using a formula similar to the formulas for the field strength of a point charge, in which the distances are measured from the center of the ball.
Since the field strength inside a charged conductor is zero, the potential at all points inside and on the surface of the conductor is the same (only in this case, the potential difference, and hence the strength, is zero). The potential inside a charged ball is equal to the potential on the surface. The potential outside the ball is calculated using a formula similar to the formulas for the potential of a point charge, in which distances are measured from the center of the ball.
Radius R:
If the ball is surrounded by a dielectric, then:
Properties of a conductor in an electric field
- Inside the conductor, the field strength is always zero.
- The potential inside the conductor at all points is the same and equal to the potential of the surface of the conductor. When the problem says that "the conductor is charged to the potential ... B", then they mean the potential of the surface.
- Outside the conductor near its surface, the field strength is always perpendicular to the surface.
- If charge is imparted to a conductor, then it will all be distributed over a very thin layer near the surface of the conductor (it is usually said that all the charge of the conductor is distributed on its surface). This is easily explained: the fact is that by imparting a charge to a body, we transfer to it carriers of a charge of the same sign, i.e. like charges that repel. This means that they will tend to scatter from each other at the maximum distance of all possible, i.e. will accumulate at the very edges of the conductor. As a consequence, if the core is removed from the conductor, its electrostatic properties will not change in any way.
- Outside the conductor, the more curved the surface of the conductor, the greater the field strength. Maximum value tension is achieved near the edges and sharp breaks of the surface of the conductor.
Notes on solving complex problems
1. Grounding something means a connection by a conductor of this object with the Earth. In this case, the potentials of the Earth and the existing object are equalized, and the charges necessary for this run across the conductor from the Earth to the object, or vice versa. In this case, several factors must be taken into account, which follow from the fact that the Earth is incommensurably larger than any object that is not on it:
- The total charge of the Earth is conditionally equal to zero, therefore its potential is also equal to zero, and it will remain equal to zero after the object is connected to the Earth. In a word, grounding means zeroing the potential of the object.
- To zero out the potential (and hence the object's own charge, which could have been both positive and negative before), the object will have to either accept or give the Earth some (possibly even very large) charge, and the Earth can always provide such an opportunity.
2. We repeat once again: the distance between the repulsive bodies is minimal at the moment when their velocities become equal in magnitude and directed in the same direction (the relative velocity of the charges is equal to zero). At this moment, the potential energy of interaction of charges is maximum. The distance between the attracting bodies is maximum, also at the moment of equality of the velocities directed in one direction.
3. If the problem has a system consisting of a large number of charges, then it is necessary to consider and describe the forces acting on a charge that is not in the center of symmetry.
Successful, diligent and responsible implementation of these three points will allow you to show excellent results at the CT, the maximum of what you are capable of.
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Definition 1
Electrostatics is an extensive section of electrodynamics that studies and describes electrically charged bodies at rest in a certain system.
In practice, two types of electrostatic charges are distinguished: positive (glass on silk) and negative (ebonite on wool). Elementary charge is the minimum charge ($ e = 1.6 ∙ 10 ^ (-19) $ C). The charge of any physical body is a multiple of an integer number of elementary charges: $ q = Ne $.
Electrization of material bodies is the redistribution of charge between bodies. Electrifying methods: touch, friction and impact.
The law of conservation of an electric positive charge - in a closed concept, the algebraic sum of the charges of all elementary particles remains stable and unchanged. $ q_1 + q _2 + q _3 +… .. + q_n = const $. The test charge in this case is a point positive charge.
Coulomb's law
This law was established experimentally in 1785. According to this theory, the force of interaction of two stationary point charges in a medium is always directly proportional product of positive moduli and inversely the square of the total distance between them.
The electric field is unique view matter, which interacts between stable electric charges, is formed around charges, affects only charges.
Such a process of point fixed elements completely obey Newton's third law, and is considered the result of repulsion of particles from each other with the same force attraction to each other. The interconnection of stable electric charges in electrostatics is called the Coulomb interaction.
Coulomb's law is quite fair and accurate for charged material bodies, uniformly charged balls and spheres. In this case, the parameters of the centers of spaces are generally taken as distances. On practice this law performed well and quickly if the values of the charged bodies are much less than the distance between them.
Remark 1
Conductors and dielectrics also act in an electric field.
The first represent substances containing free carriers of an electromagnetic charge. Free movement of electrons can occur inside the conductor. These elements include solutions, metals and various electrolyte melts, ideal gases and plasma.
Dielectrics are substances in which there can be no free carriers of electric charge. Free movement of electrons inside the dielectrics themselves is impossible, since no electric current flows through them. It is these physical particles that have a permeability that is not equal to the dielectric unit.
Power lines and electrostatics
The lines of force of the initial strength of the electric field are continuous lines, the tangent points to which in each medium, through which they pass, completely coincide with the axis of strength.
Main characteristics of power lines:
- do not intersect;
- not closed;
- stable;
- the end direction coincides with the direction of the vector;
- start at $ + q $ or at infinity, end at $ - q $;
- are formed near charges (where the intensity is higher);
- perpendicular to the surface of the main conductor.
Definition 2
The electrical potential difference or voltage (Ф or $ U $) is the magnitude of the potentials at the starting and ending points of the positive charge trajectory. The less the potential changes along the path segment, the lower the resulting field strength.
The electric field strength is always directed towards a decrease in the initial potential.
Figure 2. Potential energy of the system of electric charges. Author24 - online exchange of student papers
Electric capacity characterizes the ability of any conductor to accumulate the required electrical charge on its own surface.
This parameter does not depend on the electric charge, but it can be affected by the geometric dimensions of the conductors, their shape, location and properties of the medium between the elements.
A capacitor is a universal electrical device that helps to quickly accumulate an electrical charge for its release into a circuit.
Electric field and its strength
According to the modern views of scientists, electric stable charges do not directly affect each other. Each charged physical body in electrostatics creates in environment electric field. This process exerts a forceful effect on other charged substances. The main property of an electric field is to act on point charges with a certain force. Thus, the interaction of positively charged particles is carried out through the fields that surround the charged elements.
This phenomenon can be investigated by means of the so-called test charge - a small-sized electric charge, which does not introduce a significant redistribution of the studied charges. For the quantitative identification of the field, a force feature is introduced - the intensity of the electric field.
Strength is a physical indicator that is equal to the ratio of the force with which the field acts on the test charge placed at a given point of the field to the value of the charge itself.
The electric field strength is a vector physical quantity. The direction of the vector in this case coincides at each material point of the surrounding space with the direction of the force acting on the positive charge. The electric field of stationary and unchanging elements is considered electrostatic.
To understand the electric field, lines of force are used, which are drawn in such a way that the direction main axis tension in each system coincided with the direction of the tangent to the point.
Potential difference in electrostatics
The electrostatic field includes one important property: the work of the forces of all moving particles when a point charge moves from one point of the field to another does not depend on the direction of the trajectory, but is determined exclusively by the position of the initial and final lines and the charge parameter.
The result of the independence of the work from the form of movement of charges is the following statement: the functional of the forces of the electrostatic field during the transformation of the charge along any closed trajectory is always zero.
Figure 4. Potential electrostatic field. Author24 - online exchange of student papers
Property potentiality of the electrostatic field helps to introduce the concept of potential and internal energy charge. And the physical parameter, equal to the ratio of the potential energy in the field to the value of this charge, is called the constant potential of the electric field.
In many difficult tasks electrostatics, when determining the potentials for a reference material point, where the value of the potential energy and the potential itself vanish, it is convenient to use an infinitely distant point. In this case, the significance of the potential is determined as follows: the potential of the electric field at any point in space is equal to the work performed by inner strength when the positive unit charge is removed from the given system to infinity.
where F is the modulus of the force of interaction of two point charges with the value q 1 and q 2 , r- distance between charges, - dielectric permeability of the medium, 0 - dielectric constant.
Electric field strength
where
- force acting on a point charge q 0
placed at a given point in the field.
Point charge field strength (modulo)
where r- distance from charge q to the point at which the tension is determined.
The intensity of the field created by a system of point charges (principle of superposition of electric fields)
where
- the intensity at a given point of the field created by the i-th charge.
The modulus of the field strength created by an infinite uniformly charged plane:
where 
- surface charge density.
The modulus of the field strength of a flat capacitor in its middle part
.
The formula is valid if the distance between the plates is much less than the linear dimensions of the capacitor plates.
Tension field created by an infinitely long uniformly charged thread (or cylinder) at a distance r from the thread or the axis of the cylinder modulo:
,
where 
- linear charge density.
a) through an arbitrary surface placed in an inhomogeneous field
,
where
- the angle between the vector of tension
and normal 
to a surface element, dS- area of a surface element, E n- the projection of the tension vector to the normal;
b) through a flat surface placed in a uniform electric field:
,
c) through a closed surface:
,
where the integration is over the entire surface.
Gauss's theorem. Strength vector flow through any closed surface S is equal to the algebraic sum of charges q 1 , q 2 ... q n covered by this surface divided by 0 .
.
The flux of the electric displacement vector is expressed similarly to the flux of the electric field strength vector:
a) flow through a flat surface if the field is uniform
b) in the case of an inhomogeneous field and an arbitrary surface
,
where D n- vector projection 
on the direction of the normal to an element of the surface, the area of which is equal to dS.
Gauss's theorem. Electric induction vector flux through a closed surface S covering charges q 1 , q 2 ... q n, is equal to
,
where n- the number of charges contained within a closed surface (charges with their own sign).
Potential energy of a system of two point charges Q and q provided that W = 0, is found by the formula:
W =
,
where r- distance between charges. Potential energy is positive in the interaction of like charges and negative in the interaction of unlike charges.
Potential of the electric field created by a point charge Q on distance r
=
,
Potential of the electric field created by a metal sphere of radius R carrying charge Q:
=
(r ≤ R; the field inside and on the surface of the sphere),
=
(r
>
R; field outside the sphere).
The potential of the electric field generated by the system n point charges in accordance with the principle of superposition of electric fields is equal to the algebraic sum of potentials 1 , 2 ,…, n generated by charges q 1 , q 2 , ..., q n at a given point in the field
=
.
Relationship of potentials with tension:
a) in general 
= -qrad
or 
=
;
b) in the case of a uniform field
E
= 
,
where d is the distance between equipotential surfaces with potentials 1 and 2 along the line of force;
c) in the case of a field with central or axial symmetry 
where the derivative 
taken along the line of force.
The work done by the forces of the field to move the charge q from point 1 to point 2
A = q( 1 - 2 ),
where ( 1 - 2 ) is the potential difference between the starting and ending points of the field.
The potential difference and the electric field strength are related by the relations
(
1
-
2
)
=
,
where E e- the projection of the tension vector 
on the direction of movement dl.
The electrical capacity of a solitary conductor is determined by the charge ratio q on conductor to conductor potential .
.
Capacitor capacity:
,
where ( 1 - 2 ) = U- potential difference (voltage) between capacitor plates; q- charge module on one capacitor plate.
Electrical capacity of a conducting ball (sphere) in SI
c = 4 0 R,
where R- the radius of the ball, - relative dielectric constant of the medium; 0 = 8.8510 -12 F / m.
Electric capacity of a flat capacitor in SI system:
,
where S- area of one plate; d is the distance between the plates.
Electric capacity of a spherical capacitor (two concentric spheres with radii R 1 and R 2 , the space between which is filled with a dielectric, with a dielectric constant ):
.
Electric capacity of a cylindrical capacitor (two coaxial cylinders with a length l and radii R 1 and R 2 , the space between which is filled with a dielectric with a dielectric constant )
.
Battery capacity of n capacitors connected in series is determined by the ratio
.
The last two formulas are applicable to determine the capacitance of multilayer capacitors. The arrangement of the layers parallel to the plates corresponds to the series connection of single-layer capacitors; if the boundaries of the layers are perpendicular to the plates, then it is considered that there is a parallel connection of single-layer capacitors.
Potential energy of a system of stationary point charges
.
Here i- potential of the field created at the point where the charge is q i, all charges except i th; n is the total number of charges.
The volumetric energy density of the electric field (energy per unit volume):
=
=
=
,
where D- the magnitude of the vector of electrical displacement.
Uniform field energy:
W = V.
Energy of an inhomogeneous field:
W =
.
