Coulomb's law:
where F is the strength of the electrostatic interaction between two charged bodies;
q 1 , q 2 - electric charges of bodies;
ε is the relative, dielectric permittivity of the medium;
ε 0 \u003d 8.85 10 -12 F / m - electrical constant;
r is the distance between two charged bodies.
Linear charge density:
where d q- elementary charge per section of length d l.
Surface charge density:
where d q- elementary charge per surface d s.
Bulk charge density:
where d q- elementary charge, in volume d v.
Electric field strength:
where F – force acting on a charge q.
Gauss theorem:
where E is the strength of the electrostatic field;
d S – vector , the modulus of which is equal to the area of the penetrating surface, and the direction coincides with the direction of the normal to the site;
q is the algebraic sum of enclosed inside the surface d S charges.
Tension vector circulation theorem:
Electrostatic field potential:
where W p is the potential energy of a point charge q.
Point charge potential:
Field strength of a point charge:
.
The intensity of the field created by an infinite straight line of a uniformly charged line or an infinitely long cylinder:
where τ is the linear charge density;
r is the distance from the filament or the axis of the cylinder to the point where the field strength is determined.
The intensity of the field created by an infinite uniform charged plane:
where σ is the surface charge density.
Relationship of potential with tension in the general case:
E=- gradφ = 
.
Relationship between potential and strength in the case of a uniform field:
E= ,
where d– distance between points with potentials φ 1 and φ 2 .
Relationship between potential and strength in the case of a field with central or axial symmetry:
The work of the field forces to move the charge q from a point of the field with a potential φ 1 to the point of potential φ2:
A=q(φ 1 - φ 2).
Conductor capacitance:
where q is the charge of the conductor;
φ is the potential of the conductor, provided that at infinity the potential of the conductor is assumed to be zero.
Capacitor capacitance:
where q is the charge of the capacitor;
U is the potential difference between the capacitor plates.
Electric capacitance of a flat capacitor:
where ε is the permittivity of the dielectric located between the plates;
d is the distance between the plates;
S is the total area of the plates.
Capacitor battery capacity:
b) with parallel connection:
Energy of a charged capacitor:
,
where q is the charge of the capacitor;
U is the potential difference between the plates;
C is the capacitance of the capacitor.
DC power:
where d q- the charge flowing through the cross section of the conductor during the time d t.
current density:
where I- current strength in the conductor;
S is the area of the conductor.
Ohm's law for a circuit section that does not contain EMF:
where I- current strength in the area;
U
R- section resistance.
Ohm's law for a circuit section containing EMF:
where I- current strength in the area;
U- voltage at the ends of the section;
R- the total resistance of the section;
ε – source emf.
Ohm's law for a closed (complete) circuit:
where I- current strength in the circuit;
R- external resistance of the circuit;
r is the internal resistance of the source;
ε – source emf.
Kirchhoff's laws:
2. 
,
where is the algebraic sum of the strengths of the currents converging in the node;
- algebraic sum of voltage drops in the circuit;
is the algebraic sum of the EMF in the circuit.
Conductor resistance:
where R– conductor resistance;
ρ is the resistivity of the conductor;
l- conductor length;
S
Conductor conductivity:
where G is the conductivity of the conductor;
γ is the specific conductivity of the conductor;
l- conductor length;
S is the cross-sectional area of the conductor.
Conductor system resistance:
a) in series connection:
a) in parallel connection:
Current work:
,
where A– current work;
U- voltage;
I– current strength;
R- resistance;
t- time.
Current power:
.
Joule–Lenz law
where Q is the amount of heat released.
Ohm's law in differential form:
j=γ E ,
where j is the current density;
γ – specific conductivity;
E is the electric field strength.
Relationship of magnetic induction with magnetic field strength:
B=μμ 0 H ,
where B is the magnetic induction vector;
μ is the magnetic permeability;
H is the strength of the magnetic field.
Biot-Savart-Laplace law:
,
where d B is the induction of the magnetic field created by the conductor at some point;
μ is the magnetic permeability;
μ 0 \u003d 4π 10 -7 H / m - magnetic constant;
I- current strength in the conductor;
d l – conductor element;
r is the radius vector drawn from the element d l conductor to the point where the magnetic field induction is determined.
The total current law for a magnetic field (theorem of the circulation of the vector B):
,
where n- the number of conductors with currents covered by the circuit L arbitrary shape.
Magnetic induction at the center of the circular current:
where R is the radius of the circle.
Magnetic induction on the axis of circular current:
,
where h is the distance from the center of the coil to the point at which the magnetic induction is determined.
Magnetic induction of direct current field:
where r 0 is the distance from the wire axis to the point where the magnetic induction is determined.
Solenoid field magnetic induction:
B=μμ 0 ni,
where n is the ratio of the number of turns of the solenoid to its length.
Amp power:
d F =I,
where d F – Ampere power;
I- current strength in the conductor;
d l - conductor length;
B– magnetic field induction.
Lorentz force:
F=q E +q[v B ],
where F is the Lorentz force;
q is the particle charge;
E is the electric field strength;
v is the speed of the particle;
B– magnetic field induction.
Magnetic Flux:
a) in the case of a uniform magnetic field and a flat surface:
Φ=B n S,
where Φ – magnetic flux;
B n is the projection of the magnetic induction vector onto the normal vector;
S is the contour area;
b) in the case of an inhomogeneous magnetic field and an arbitrary projection:
Flux linkage (full flow) for toroid and solenoid:
where Ψ – full flow;
N is the number of turns;
Φ - magnetic flux penetrating one turn.
Loop inductance:
Solenoid inductance:
L=μμ 0 n 2 V,
where L is the inductance of the solenoid;
μ is the magnetic permeability;
μ 0 is the magnetic constant;
n is the ratio of the number of turns to its length;
V is the volume of the solenoid.
Faraday's law of electromagnetic induction:
where ε i– EMF of induction;
– change in total flow per unit time.
The work of moving a closed loop in a magnetic field:
A=IΔ Φ,
where A- work on moving the contour;
I- current strength in the circuit;
Δ Φ – change in the magnetic flux penetrating the circuit.
EMF of self-induction:
Magnetic field energy:
Volumetric energy density of the magnetic field:
,
where ω is the volumetric energy density of the magnetic field;
B– magnetic field induction;
H– magnetic field strength;
μ is the magnetic permeability;
μ 0 is the magnetic constant.
3.2. Concepts and definitions
? List the properties of an electric charge.
1. There are two types of charges - positive and negative.
2. Charges of the same name repel, unlike charges attract.
3. Charges have the property of discreteness - all are multiples of the smallest elementary.
4. The charge is invariant, its value does not depend on the frame of reference.
5. The charge is additive - the charge of the system of bodies is equal to the sum of the charges of all the bodies of the system.
6. The total electric charge of a closed system is a constant value
7. A stationary charge is a source of an electric field, a moving charge is a source of a magnetic field.
? Formulate Coulomb's law.
The force of interaction between two fixed point charges is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force is directed along the line connecting the charges.
? What is an electric field? Electric field strength? Formulate the principle of superposition of electric field strength.
An electric field is a type of matter associated with electric charges and transmitting the action of one charge to another. Tension is the power characteristic of the field, equal to the force acting on a unit positive charge placed at a given point in the field. The principle of superposition - the field strength created by a system of point charges is equal to the vector sum of the field strengths of each charge.
? What is called the lines of force of the electrostatic field? List the properties of lines of force.
The line, the tangent at each point of which coincides with the direction of the field strength vector, is called the force line. Properties of lines of force - start on positive, end on negative charges, do not interrupt, do not intersect with each other.
? Define an electric dipole. dipole field.
A system of two equal in absolute value, opposite in sign, point electric charges, the distance between which is small compared to the distance to the points where the action of these charges is observed. The intensity vector has a direction opposite to the dipole electric moment vector (which, in turn, is directed from negative charge to positive).
? What is the potential of an electrostatic field? Formulate the principle of potential superposition.
A scalar quantity numerically equal to the ratio of the potential energy of an electric charge placed at a given point in the field to the magnitude of this charge. The principle of superposition - the potential of a system of point charges at a certain point in space is equal to the algebraic sum of the potentials that these charges would create separately at the same point in space.
? What is the relationship between tension and potential?
E=- (E - field strength at a given point of the field, j - potential at this point.)
? Define the concept of "flux of the electric field strength vector". Formulate the electrostatic theorem of Gauss.
For an arbitrary closed surface, the intensity vector flux E electric field F E= . Gauss theorem:
= (here Q i are charges covered by a closed surface). Valid for a closed surface of any shape.
? What substances are called conductors? How are charges and electrostatic field distributed in a conductor? What is electrostatic induction?
Conductors are substances in which, under the action of an electric field, free charges can move in an orderly manner. Under the action of an external field, the charges are redistributed, creating their own field, equal in absolute value to the external one and directed oppositely. Therefore, the resulting tension inside the conductor is 0.
Electrostatic induction is a type of electrization in which, under the action of an external electric field, the redistribution of charges between parts of a given body occurs.
? What is the electric capacitance of a solitary conductor, a capacitor. How to determine the capacitance of a flat capacitor, a bank of capacitors connected in series, in parallel? Unit of measure for electrical capacity.
Solitary conductor: where With-capacity, q- charge, j - potential. The unit of measure is farad [F]. (1 F is the capacitance of the conductor, in which the potential increases by 1 V when a charge of 1 C is imparted to the conductor).
Capacitance of a flat capacitor. Serial connection: 
. Parallel connection: C total = C 1 +C 2 +…+С n
? What substances are called dielectrics? What types of dielectrics do you know? What is dielectric polarization?
Dielectrics are substances in which, under normal conditions, there are no free electric charges. There are dielectrics polar, non-polar, ferroelectric. Polarization is the process of orientation of dipoles under the influence of an external electric field.
? What is an electrical displacement vector? Formulate Maxwell's postulate.
Electrical displacement vector D characterizes the electrostatic field created by free charges (i.e. in vacuum), but with such a distribution in space, which is available in the presence of a dielectric. Maxwell's postulate: . Physical meaning - expresses the law of creating electric fields by the action of charges in arbitrary media.
? Formulate and explain the boundary conditions for the electrostatic field.
When the electric field passes through the interface between two dielectric media, the intensity and displacement vectors change abruptly in magnitude and direction. The relations characterizing these changes are called boundary conditions. There are 4 of them:
(3), (4)
? How is the energy of an electrostatic field determined? Energy density?
Energy W= ( E- field strength, e-dielectric constant, e 0 - electrical constant, V- field volume), energy density
? Define the concept of "electric current". Types of currents. Characteristics of electric current. What condition is necessary for its occurrence and existence?
Current is the ordered movement of charged particles. Types - conduction current, ordered movement of free charges in a conductor, convection - occurs when a charged macroscopic body moves in space. For the emergence and existence of a current, it is necessary to have charged particles capable of moving in an orderly manner, and the presence of an electric field, the energy of which, being replenished, would be spent on this ordered movement.
? Give and explain the continuity equation. Formulate the condition of current stationarity in integral and differential forms.
Continuity equation. Expresses in differential form the law of conservation of charge. The condition of stationarity (constancy) of the current in integral form: and differential -.
? Write down Ohm's law in integral and differential forms.
Integral form - ( I-current, U- voltage, R-resistance). Differential form - ( j - current density, g - electrical conductivity, E - field strength in the conductor).
? What are third party forces? EMF?
External forces separate charges into positive and negative. EMF - the ratio of work to move the charge along the entire closed circuit to its value
? How is work and power determined?
When moving charge q through an electrical circuit at the ends of which voltage is applied U, electric field does work , current power (t-time)
? Formulate Kirchhoff's rules for branched chains. What conservation laws are incorporated in Kirchhoff's rules? How many independent equations should be made on the basis of the first and second laws of Kirchhoff?
1. The algebraic sum of the currents converging in the node is 0.
2. In any arbitrarily chosen closed circuit, the algebraic sum of the voltage drops is equal to the algebraic sum of the EMF occurring in this circuit. Kirchhoff's first rule follows from the law of conservation of electric charge. The number of equations in the sum should be equal to the number of sought values (all resistances and EMF should be included in the system of equations).
? Electric current in gas. Processes of ionization and recombination. The concept of plasma.
Electric current in gases is the directed movement of free electrons and ions. Under normal conditions, gases are dielectrics, they become conductors after ionization. Ionization is the process of forming ions by separating electrons from gas molecules. Occurs due to the influence of an external ionizer - strong heating, X-ray or ultraviolet radiation, electron bombardment. Recombination is a process that is the reverse of ionization. Plasma is a fully or partially ionized gas in which the concentrations of positive and negative charges are equal.
? Electric current in vacuum. Thermionic emission.
Current carriers in vacuum are electrons emitted due to emission from the surface of the electrodes. Thermionic emission is the emission of electrons by heated metals.
? What do you know about the phenomenon of superconductivity?
The phenomenon in which the resistance of some pure metals (tin, lead, aluminum) drops to zero at temperatures close to absolute zero.
? What do you know about the electrical resistance of conductors? What is resistivity, its dependence on temperature, electrical conductivity? What do you know about series and parallel connection of conductors. What is a shunt, additional resistance?
Resistance - a value directly proportional to the length of the conductor l and inversely proportional to the area S cross-section of the conductor: (r-specific resistance). Conductivity is the reciprocal of resistance. Resistivity (resistance of a conductor 1 m long with a cross section of 1 m 2). Resistivity is temperature dependent, where a is the temperature coefficient, R and R 0 , r and r 0 are resistances and specific resistances at t and 0 0 С. Parallel - 
, sequential R=R 1 +R 2 +…+R n. A shunt is a resistor connected in parallel with an electrical measuring instrument to divert part of the electric current in order to expand the measurement limits.
? A magnetic field. What sources can create a magnetic field?
A magnetic field is a special kind of matter through which moving electric charges interact. The reason for the existence of a constant magnetic field is a fixed conductor with a constant electric current, or permanent magnets.
? Formulate Ampère's law. How do conductors interact in which current flows in one (opposite) direction?
Ampere's force is acting on a current-carrying conductor.
B - magnetic induction, I- conductor current, D l is the length of the conductor section, a is the angle between the magnetic induction and the conductor section. In one direction they attract, in the opposite direction they repel.
? Define the ampere force. How to determine its direction?
This is the force acting on a current-carrying conductor placed in a magnetic field. We define the direction as follows: we position the palm of the left hand so that it includes the lines of magnetic induction, and four outstretched fingers are directed along the current in the conductor. The bent thumb will show the direction of Ampere's force.
? Explain the movement of charged particles in a magnetic field. What is the Lorentz force? What is its direction?
A moving charged particle creates its own magnetic field. If it is placed in an external magnetic field, then the interaction of the fields will manifest itself in the emergence of a force acting on the particle from the external field - the Lorentz force. Direction - according to the rule of the left hand. For positive charge - vector B enters the palm of the left hand, four fingers are directed along the movement of the positive charge (velocity vector), the bent thumb shows the direction of the Lorentz force. On a negative charge, the same force acts in the opposite direction.
(q-charge, v-speed, B- induction, a - angle between the direction of velocity and magnetic induction).
? Frame with current in a uniform magnetic field. How is magnetic moment determined?
The magnetic field has an orienting effect on the frame with current, turning it in a certain way. The torque is given by: M =p m x B , where p m- the vector of the magnetic moment of the loop with current, equal to IS n (current per contour surface area, per unit normal to the contour), B - vector of magnetic induction, quantitative characteristic of the magnetic field.
? What is the magnetic induction vector? How to determine its direction? How is a magnetic field shown graphically?
The magnetic induction vector is the power characteristic of the magnetic field. The magnetic field is visualized using lines of force. At each point of the field, the tangent to the field line coincides with the direction of the magnetic induction vector.
? Formulate and explain the Biot-Savart-Laplace law.
The Biot-Savart-Laplace law allows you to calculate for a current-carrying conductor I magnetic induction of the field d B
, created at an arbitrary point of the field d l
conductor: (here m 0 is the magnetic constant, m is the magnetic permeability of the medium). The direction of the induction vector is determined by the rule of the right screw, if the translational movement of the screw corresponds to the direction of the current in the element.
? Formulate the principle of superposition for a magnetic field.
Superposition principle - the magnetic induction of the resulting field created by several currents or moving charges is equal to the vector sum of the magnetic inductions of the added fields created by each current or moving charge separately:
? Explain the main characteristics of a magnetic field: magnetic flux, magnetic field circulation, magnetic induction.
magnetic flux F through any surface S call the value equal to the product of the modulus of the magnetic induction vector and the area S and the cosine of the angle a between the vectors B and n (outer normal to the surface). Vector circulation B along a given closed contour is called an integral of the form , where d l - vector of elementary contour length. Vector circulation theorem B : vector circulation B along an arbitrary closed circuit is equal to the product of the magnetic constant and the algebraic sum of the currents covered by this circuit. The magnetic induction vector is the power characteristic of the magnetic field. The magnetic field is visualized using lines of force. At each point of the field, the tangent to the field line coincides with the direction of the magnetic induction vector.
? Write down and comment on the condition of solenoidality of the magnetic field in integral and differential forms.
Vector fields in which there are no sources and sinks are called solenoidal. The condition of solenoidality of the magnetic field in integral form: and differential form:
? Magnetics. Types of magnets. Feromagnets and their properties. What is hysteresis?
A substance is magnetic if it is capable of acquiring a magnetic moment (be magnetized) under the action of a magnetic field. Substances that are magnetized in an external magnetic field against the direction of the field are called diamagnets. Those that are magnetized in an external magnetic field in the direction of the field are called paramagnets. These two classes are called weakly magnetic substances. Strongly magnetic substances that are magnetized even in the absence of an external magnetic field are called ferromagnets. . Magnetic hysteresis - the difference in the values of the magnetization of a ferromagnet at the same intensity H of the magnetizing field, depending on the value of the preliminary magnetization. Such a graphical dependence is called a hysteresis loop.
? Formulate and explain the law of total current in integral and differential forms (basic equations of magnetostatics in matter).
? What is electromagnetic induction? Formulate and explain the basic law of electromagnetic induction (Faraday's law). Formulate Lenz's rule.
The phenomenon of the occurrence of an electromotive force (EMF of induction) in a conductor located in an alternating magnetic field or moving in a constant in a constant magnetic field is called electromagnetic induction. Faraday's law: whatever the reason for the change in the flux of magnetic induction, covered by a closed conducting circuit, that occurs in the EMF circuit
The minus sign is determined by the Lenz rule - the induction current in the circuit always has such a direction that the magnetic field it creates prevents a change in the magnetic flux that caused this induction current.
? What is the phenomenon of self-induction? What is inductance, units of measurement? Currents during the closing and opening of the electrical circuit.
The occurrence of induction EMF in a conducting circuit under the influence of its own magnetic field when it changes, which occurs as a result of a change in the current strength in the conductor. Inductance is a proportionality factor depending on the shape and dimensions of the conductor or circuit, [H]. In accordance with the Lenz rule, the EMF of self-induction prevents the increase in current strength when the circuit is turned on and the decrease in current strength when the circuit is turned off. Therefore, the magnitude of the current cannot change instantly (the mechanical analogue is inertia).
? The phenomenon of mutual induction. Mutual induction coefficient.
If two fixed circuits are located close to each other, then when the current strength in one circuit changes, an emf occurs in the other circuit. This phenomenon is called mutual induction. Proportionality coefficients L 21 and L 12 is called the mutual inductance of the circuits, they are equal.
? Write Maxwell's equations in integral form. Explain their physical meaning.
; 
;
; .
It follows from Maxwell's theory that the electric and magnetic fields cannot be considered as independent - a change in time of one leads to a change in the other.
? The energy of the magnetic field. Magnetic field energy density.
Energy, L-inductance, I- current strength.
Density 
, AT- magnetic induction, H is the magnetic field strength, V-volume.
? The principle of relativity in electrodynamics
The general laws of electromagnetic fields are described by Maxwell's equations. In relativistic electrodynamics, it is established that the relativistic invariance of these equations takes place only under the condition of relativity of electric and magnetic fields, i.e. when the characteristics of these fields depend on the choice of inertial frames of reference. In a moving system, the electric field is the same as in a stationary system, but in a moving system there is a magnetic field, which is not present in a stationary system.
Vibrations and waves
Electric charge is a physical quantity that characterizes the ability of particles or bodies to enter into electromagnetic interactions. Electric charge is usually denoted by the 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, practically not found in nature. As a rule, you will have to deal with microcoulombs (1 μC = 10 -6 C), nanocoulombs (1 nC = 10 -9 C) and picocoulombs (1 pC = 10 -12 C). 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 mass, electric charge is not an inherent characteristic of a given body. The same body in different conditions can have a different charge.
4. There are two types of electric charges, conventionally named positive and negative.
5. All charges interact with each other. At the same time, like charges repel each other, unlike charges attract. The forces of interaction of charges are central, that is, they lie on a straight line connecting the centers of charges.
6. There is the smallest 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 is an integer. Please note that it is impossible to have a charge equal to 0.5 e; 1,7e; 22,7e etc. Physical quantities that can take only 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 processes of the birth 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 that have charges q 1 and q 2 (it doesn’t matter what sign the charges are), bring into contact, and then back apart, then the charge of each of the bodies will become equal:
From the modern point of view, charge carriers are elementary particles. All ordinary bodies are made up 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 the proton and electron modulo 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 the atomic number. An atom of a given substance can lose one or more electrons, or acquire an extra electron. In these cases, the 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 electrifying bodies, nuclear reactions do not occur. Therefore, in any electrical phenomena, the number of protons does not change, only the number of electrons changes. So, giving a body a negative charge means transferring extra electrons to it. And the message of a positive charge, contrary to a common mistake, does not mean the addition of protons, but the subtraction of 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 some 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- thread length. Measured in C/m.
2. Surface charge density. Used to describe the distribution of charge over the surface of a body:
where: S is the surface area of the body. Measured in C / m 2.
3. Bulk charge density. Used to describe the distribution of charge over the volume of a body:
where: V- volume of the body. Measured in C / m 3.
Please note that electron mass is equal to:
me\u003d 9.11 ∙ 10 -31 kg.
Coulomb's law
point charge 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 fixed 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 permittivity of the medium – a dimensionless physical quantity showing how many times the force of electrostatic interaction in a given medium will be less than in vacuum (that is, how many times the medium weakens the interaction). Here k- coefficient in the Coulomb law, the 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 fixed 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 with different signs. The interaction of fixed electric charges is called electrostatic or Coulomb interaction. The section 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, for distances r take the distance between the centers of spheres or balls. In practice, Coulomb's law is well fulfilled if the dimensions of the charged bodies are much smaller than the distance between them. Coefficient k in the SI system is sometimes written as:
where: ε 0 \u003d 8.85 10 -12 F / m - electrical constant.
Experience shows that the forces of the Coulomb interaction obey the superposition principle: if a charged body interacts simultaneously with several charged bodies, then the resulting force acting on this 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, free movement of electrons is possible - charge carriers (electric current can flow through the conductors). Conductors include metals, electrolyte solutions and melts, ionized gases, and plasma.
Dielectrics (insulators)- substances in which there are no free charge carriers. The free movement of electrons inside dielectrics is impossible (electric current cannot flow through them). It is dielectrics that have a certain permittivity not equal to unity ε .
For the permittivity of a substance, the following is true (about what an electric field is a little lower):
Electric field and its intensity
According to modern concepts, electric charges do not act directly on each other. Each charged body creates in the surrounding space electric field. This field has a force effect on other charged bodies. The main property of an electric field is the action on electric charges with a certain force. Thus, the interaction of charged bodies is carried out not by their direct influence on each other, but through the electric fields surrounding the charged bodies.
The electric field surrounding a charged body can be investigated using the so-called test charge - a small point charge that does not introduce a noticeable redistribution of the investigated charges. To quantify the electric field, a force characteristic is introduced - electric field strength E.
The electric field strength is called a physical quantity equal to the ratio of the force with which the field acts on a test charge placed at a given point in the field to the magnitude of this charge:
The 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 stationary and unchanging charges with 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. Force lines have the following properties.
- The lines of force of an electrostatic field never intersect.
- The lines of force of an electrostatic field are always directed from positive charges to negative ones.
- When depicting an electric field using lines of force, their density should be proportional to the modulus of the field strength vector.
- The lines of force start at a positive charge, or infinity, and end at a negative charge, 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, because the strength of the electric field at a given point in space is uniquely specified.
An electric field is called homogeneous if the intensity vector is the same at all points in the field. For example, a flat capacitor creates a uniform field - two plates charged with an equal and opposite charge, separated by a dielectric layer, and the distance between the plates is much less than the size of the plates.
At all points of a uniform field per charge q, introduced into a uniform field with intensity E, there is a force of the same magnitude and direction 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 force and tension vectors 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 using a test charge, then the resulting force turns out to be equal to the geometric sum of the forces acting on the test charge from each charged body separately. Consequently, the strength of the electric field created by the system of charges at a given point in space is equal to the vector sum of the strengths 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 modulo:
This field is called the Coulomb field. In the Coulomb field, the direction of the intensity vector depends on the sign of the charge Q: if Q> 0, then the intensity vector is directed away 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 electric field strength that a charged plane creates near its surface:
So, if in the task it is required to determine the field strength of the system of charges, then it is necessary to act according to the following algorithm:
- Draw a drawing.
- Draw the field strength of each charge separately at the desired point. Remember that tension is directed towards the negative charge and away from the positive charge.
- Calculate each of the tensions using the appropriate formula.
- Add the stress vectors geometrically (i.e. vectorially).
Potential energy of interaction of charges
Electric charges interact with each other and with an electric field. Any interaction is described by 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 also 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 balls or spheres. If there are more than two charges, then the energy of their interaction should be considered as follows: divide the system of charges into all possible pairs, calculate the interaction energy of each pair and sum up all the energies for all pairs.
Problems on this topic are solved, as well as problems on the law of conservation of mechanical energy: first, the initial interaction energy is found, then the final one. If the task asks to find the work on the movement of charges, then it will be equal to the difference between the initial and final total energy of the interaction of charges. The interaction energy can also be converted into kinetic energy or into other types of energy. If the bodies are at a very large distance, then the energy of their interaction is assumed to be 0.
Please note: if the task requires finding the minimum or maximum distance between bodies (particles) during movement, then this condition will be satisfied at the moment when the particles move in the same direction at the same speed. Therefore, the solution must begin with writing the law of conservation of momentum, from which this same speed is found. And then you should write the law of conservation of energy, taking into account the kinetic energy of the particles in the second case.
Potential. Potential difference. Voltage
An electrostatic field has an important property: the work of the forces of an electrostatic field when moving a charge 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 moving the charge 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 the 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, i.e. voltage) is the volt [V]. Potential is a scalar quantity.
In many problems of electrostatics, when calculating potentials, it is convenient to take the point at infinity as the reference point, where the values of potential energy and potential vanish. In this case, the concept of potential can be defined as follows: the field potential at a given point in space is equal to the work that electric forces do when a unit positive charge is removed from a given point to infinity.
Recalling 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 get that potential φ point charge fields Q on distance r from it relative to a 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 field potential of a uniformly charged ball (or sphere) at r ≥ R(outside of the 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 lines of force, use equipotential surfaces. A surface at all points of which the potential of the electric field has the same values is called an equipotential surface or a surface of equal potential. The electric field lines are always perpendicular to the equipotential surfaces. The 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 electrical voltage can be given by the formula:
In a uniform electric field, there is a relationship between field strength and voltage:
The work of the electric field 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 is the potential energy of the charge in an external electric field.
- A- the work of the electric field on the movement of the charge (charges).
- q is the charge that moves in an external electric field.
- U- voltage.
- E is the electric field strength.
- d or ∆ l is the distance over which the charge is moved along the lines of force.
In all the previous formulas, it was specifically about the work of the electrostatic field, but if the problem says that “work must be done”, or it is about “the work of external forces”, then this work should be considered in the same way as the work of the field, but with opposite sign.
Potential superposition principle
From the principle of superposition of field strengths created by electric charges, the principle of superposition for potentials follows (in this case, the sign of the field potential depends on the sign of the charge that created the field):
Note how much easier it is to apply the principle of superposition of potential than of tension. Potential is a scalar quantity that has no direction. Adding potentials is simply summing up numerical values.
electrical capacitance. Flat capacitor
When a charge is communicated to a conductor, there is always a certain limit, more than which it will not be possible to charge the body. To characterize the ability of a body to accumulate an electric charge, the concept is introduced electrical capacitance. The capacitance of a solitary conductor is the ratio of its charge to potential:
In the SI system, capacitance is measured in Farads [F]. 1 Farad is an extremely large capacitance. In comparison, the capacitance of the entire globe is much less than one farad. The capacitance of a conductor does not depend on its charge or on the potential of the body. Similarly, the density does not depend on either the mass or the volume of the body. Capacity depends only on the shape of the body, its dimensions and the properties of its environment.
Electrical capacity 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 value of the electrical capacitance of the 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 facings.
The simplest capacitor is a system of two flat conductive 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 intensity modulus 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 in different directions, and therefore the resulting electrostatic field E= 0. can be calculated using the formula:
Thus, the capacitance 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 an area of only one plate of the capacitor. When in the problem they talk about the "plate area", they mean exactly this value. You should never multiply or divide by 2.
Once again, we present the formula for capacitor charge. By the charge of a capacitor is meant only the charge of its positive lining:
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 does not act on itself). The strength of this field is equal to half the strength of the full 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 store of energy. The energy of a charged capacitor is equal to the work of external forces that 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 you use the relation q = CU):
Pay special 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 of the capacitor will not change.
Electric field energy
Electrical energy should be considered as potential energy stored in a charged capacitor. According to modern concepts, the electrical energy of a capacitor is localized in the space between the capacitor plates, that is, in an 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 characteristic - the volumetric energy density of the electric field. Using the example of a flat capacitor, you can get the following formula for the volumetric energy density (or the energy per unit volume of the electric field):
Capacitor connections
Parallel connection of capacitors- to increase capacity. Capacitors are connected by similarly charged plates, as if increasing the area of equally charged plates. The voltage on all capacitors is the same, the total charge is equal to the sum of the charges of each of the capacitors, and the total capacitance is also equal to the sum of the capacitances of all capacitors connected in parallel. Let's write out the formulas for the parallel connection of capacitors:
At series connection of capacitors the total capacitance of a battery of capacitors is always less than the capacitance of the smallest capacitor included in the battery. A series connection is used to increase the breakdown voltage of 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 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 heating of the charged conductor, which actually does not occur.
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 again move, and they would move in such a way as to reduce this field to zero by their own field, because. in fact, they would not want to move, because any system tends to balance. Sooner or later, all the 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 maximum. The magnitude of the electric field strength of a charged ball outside 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 the charged conductor is zero, then 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 tension, is zero). The potential inside the charged sphere is equal to the potential on the surface. The potential outside the ball is calculated by a formula similar to the formulas for the potential of a point charge, in which the distances are measured from the center of the ball.
Radius R:
If the sphere 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 is the same at all points and is equal to the potential of the surface of the conductor. When in the problem they say that "the conductor is charged to the potential ... V", then they mean exactly the surface potential.
- Outside the conductor near its surface, the field strength is always perpendicular to the surface.
- If the conductor is given a charge, then it will be completely distributed over a very thin layer near the surface of the conductor (it is usually said that the entire charge of the conductor is distributed on its surface). This is easily explained: the fact is that by imparting a charge to the body, we transfer charge carriers of the same sign to it, i.e. like charges that repel each other. This means that they will strive to scatter from each other to the maximum distance possible, i.e. accumulate at the very edges of the conductor. As a consequence, if the conductor is removed from the core, then its electrostatic properties will not change in any way.
- Outside the conductor, the field strength is greater, the more curved the surface of the conductor. The maximum value of tension is reached near the tips and sharp breaks of the conductor surface.
Notes on solving complex problems
1. Grounding something means a connection by a conductor of this object with the Earth. At the same time, 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, it is necessary to take into account several factors that follow from the fact that the Earth is incommensurably larger than any object located on it:
- The total charge of the Earth is conditionally zero, so its potential is also zero, and it will remain zero after the object connects to the Earth. In a word, to ground means to nullify the potential of an object.
- To nullify the potential (and hence the object's own charge, which could have been both positive and negative before), the object will either have to accept or give the Earth some (possibly even a very large) charge, and the Earth will always be able to 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 zero). At this moment, the potential energy of the interaction of charges is maximum. The distance between the attracting bodies is maximum, also at the moment of equality of 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 an excellent result on the CT, the maximum of what you are capable of.
Found an error?
If you, as it seems to you, found an error in the training materials, then please write about it by mail. You can also write about the error on the social network (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.
where F- modulus of the force of interaction of two point charges with the value q 1 and q 2 , r- distance between charges, - dielectric permittivity of the medium, 0 - dielectric constant.
Electric field strength
where
- force acting on a point charge q 0
placed at the given point in the field.
Field strength of a point charge (modulo)
where r- distance from charge q to the point at which the tension is determined.
Field strength generated by a system of point charges (principle of superposition of electric fields)
where
- 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 
is the surface charge density.
Field strength modulus 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 axis of the cylinder modulo:
,
where 
- linear charge density.
a) through an arbitrary surface placed in an inhomogeneous field
,
where
- angle between the tension vector
and normal 
to a surface element dS- surface element area, E n- projection of the tension vector on the normal;
b) through a flat surface placed in a uniform electric field:
,
c) through a closed surface:
,
where integration is carried out over the entire surface.
Gauss theorem. The flow of the intensity vector 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 
to the direction of the normal to the surface element, the area of which is equal to dS.
Gauss theorem. Electric induction vector flux through a closed surface S covering the charges q 1 , q 2 ... q n, is equal to
,
where n- the number of charges enclosed inside 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.
The potential of the electric field created by a point charge Q on distance r
=
,
The potential of the electric field created by a metal sphere of radius R, carrying a charge Q:
=
(r ≤ R; field inside and on the surface of the sphere),
=
(r
>
R; field outside the sphere).
The potential of the electric field created 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, created 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 homogeneous field
E
= 
,
where d- distance between equipotential surfaces with potentials 1 and 2 along the power line;
c) in the case of a field with central or axial symmetry 
where is the derivative 
taken along the line of force.
The work done by the field forces to move the charge q from point 1 to point 2
A=q( 1 - 2 ),
where ( 1 - 2 ) is the potential difference between the initial and final points of the field.
The potential difference and the electric field strength are related by the relations
(
1
-
2
)
=
,
where E e- projection of tension vector 
to the direction of travel dl.
The electric capacitance of a solitary conductor is determined by the charge ratio q on conductor to conductor potential .
.
Capacitor capacitance:
,
where ( 1 - 2 ) = U- potential difference (voltage) between the capacitor plates; q- charge module on one plate of the capacitor.
Electrical capacitance of a conducting ball (sphere) in SI
c = 4 0 R,
where R- ball radius, - relative permittivity of the medium; 0 = 8.8510 -12 F/m.
Electric capacitance of a flat capacitor in the SI system:
,
where S- area of one plate; d- distance between plates.
Capacitance 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 permittivity ):
.
Capacitance of a cylindrical capacitor (two coaxial cylinders with a length l and radii R 1 and R 2 , the space between them is filled with a dielectric with a permittivity )
.
Battery capacity of n capacitors connected in series is determined by the relation
.
The last two formulas are applicable to determine the capacitance of multilayer capacitors. The arrangement of 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 fixed point charges
.
Here i- the potential of the field created at the point where the charge is located q i, by all charges except i th; n is the total number of charges.
Volumetric energy density of the electric field (energy per unit volume):
=
=
=
,
where D- magnitude of the electric displacement vector.
Uniform field energy:
W= V.
Energy of inhomogeneous field:
W=
.
