1. Vector lines. To graphically depict electrostatic fields, vector lines are used - they are drawn so that at each point of the line the vector is directed tangentially to it (Fig. 3.6). The lines do not intersect anywhere; they begin on positive charges, end on negative charges, or go to infinity. Examples of graphical representations of the fields of point charges are shown in Fig. 3.6, b, c, d. It is clear that
for a single point charge, the lines are straight lines leaving or entering the charge. In the case of a uniform electric field (Fig. 3.6, e), at each point of which the vector is the same both in magnitude and direction, the lines are straight lines, parallel to each other and spaced at the same distance from each other.
Graphic representation fields using lines allows you to clearly see the direction Coulomb force, acting on point charge, placed in this point fields, which is convenient for qualitative analysis charge behavior.
Usually lines are drawn so that their density (the number of lines piercing perpendicular to them) flat surface fixed area) at each point of the field determined numeric value vector Therefore, by the degree of proximity of the lines to each other, one can judge the change in the modulus and, accordingly, the change in the modulus of the Coulomb force acting on a charged particle in an electric field.
2. Equipotential surfaces. An equipotential surface is a surface of equal potential; at each point on the surface the potential φ remains constant. Therefore, the elementary work of moving a charge q on such a surface will be equal to zero: . It follows from this that the vector at each point of the surface will be perpendicular to it, i.e. will be directed along the normal vector (Fig. 3.6, d). Indeed, if this were not so, then there would be a component of the vector () directed tangentially to the surface, and, therefore, the potential at different points of the surface would be different ( 
¹const), which contradicts the definition of an equipotential surface.
Figure 3.6 shows a graphical representation of electric fields using equipotential surfaces (dashed lines) for a point charge (Figure 3.6, b, c, these are spheres in the center of which there is a point charge), for a field created simultaneously by negative and positive charges ( Fig. 3.6, d), for a uniform electric field (Fig. 3.6, d, these are planes perpendicular to the lines).
We agreed to carry out equipotential surfaces so that the potential difference between adjacent surfaces is the same. This allows you to clearly see the change potential energy charge as it moves in an electric field.
The fact that the vector is perpendicular to the equipotential surface at each point makes it quite simple to move from a graphical representation of the electric field using lines to equipotential surfaces and vice versa. Thus, by drawing dotted lines in Fig. 3.6, b, c, d, e, perpendicular to the lines, you can obtain a graphical representation of the field using equipotential surfaces in the plane of the drawing.
A graphical representation of a field using tension vectors at various points of the field is very inconvenient. The tension vectors overlap each other, and the result is a very confusing picture. The method of depicting electric fields using lines of force, proposed by Faraday, is more visual.
Lines of tension (field lines) are lines drawn in the field so that the tangents to them at each point coincide in direction with the field strength vector at a given point (Fig. 8).
The lines of tension do not intersect, because at each point in the field the intensity vector has only one direction. Figure 9 shows the electrostatic fields of point charges and a dipole and an infinitely large plane.
Let charge q move along a uniformly charged infinite plane from point 1 to point 2. Power lines electrostatic field and the intensity vector of this field are directed perpendicular to the plane (Fig. 9). Let's calculate the work electrical forces when moving a charge.
, because
But the same work could be determined using the equation. And since it is equal to zero, the field potentials at points 1 and 2 are equal. Consequently, surfaces of equal potential, i.e. equipotential and surfaces located along the plane and normal to the lines of tension. This is also true for the field of a point charge, the field of a ball charged either over the surface or over the volume, and other fields.
Thus, tension lines are always normal to equipotential surfaces, i.e. surfaces of equal potential.
Figure 9 shows that the fields of point charges have central symmetry. Tension lines are straight lines, they leave the charge if it is positive and enter the charge if it is negative. Therefore, a positive charge can be considered the beginning of the tension lines, and a negative charge can be considered the place where they end. The tangents to the lines of force coincide with the lines themselves and are directed at each point of the field in the same direction as the tension.
In the case of a dipole, these lines are curved. It is worth noting that in all these cases the electrostatic fields are non-uniform - at each point of the field the intensity differs both in magnitude and direction. It is obvious that the lines uniform field are straight lines parallel to the tension vector.
The number of power lines conducted in space is not limited in any way. Tension lines, while characterizing the direction of tension, do not characterize the magnitude of tension. However, you can introduce a condition that connects the magnitude of the tension with the number of conducted lines of force. Where there is more tension, the lines are drawn thicker, and where there is less tension, the lines are drawn less densely. It is accepted that the number of lines passing through a unit surface, which is located perpendicular to the lines of force, is equal to numerical value tension.
Total number lines of tension penetrating a certain surface will be called the flow of tension through this surface.
We obtain an equation for calculating the tension flow – N E . First, we determine the tension flow through an elementary area located at a certain angle to the tension vector (Fig. 10).
Studying the electrostatic field using electrically conductive paper
Electric field and its characteristics. Graphic representation of the electric field. Field lines and equipotential surfaces.
Electric field – special kind matter that exists around bodies or particles with an electric charge, as well as when the magnetic field changes (for example, in electromagnetic waves). The electric field is not directly visible, but can be detected due to its forceful effect on charged bodies.
The main property of the electrostatic field is its effect on stationary electric charges.
For quantification electric field, a force characteristic is introduced - the electric field strength.
Electric field strength– vector physical quantity, numerically equal to strength, acting on a unit positive point charge placed at a given point in the field:
The direction of the vector E coincides at each point in space with the direction of the force acting on a unit positive charge.
Electric potential - This energy characteristic electric field, which expresses its intensity. It determines the “potential”, the supply of energy, the work that can be done.
The potential is numerically equal to the potential energy of a single point positive charge, placed at a given point in the field:
Each point of the electric field has a potential, and between two different points a potential difference is formed and voltage. It characterizes the supply of energy that can be released when a charge moves between these two points within the electric field under consideration.
Voltage is determined by the ratio of the work done by the electric field A to the amount of charge q, which moves in it:
For visual purposes graphical representation field, it is convenient to use force lines - directed lines, the tangents to which at each point coincide with the direction of the electric field strength vector (Fig. 153).
The field lines of force created by a point charge are a set of straight lines leaving (for positive) or entering (for negative) the point where the charge is located (Fig. 154).
Properties of electric field lines:
1. Lines of force do not intersect.
2. The power lines have no kinks.
3.Electrostatic field lines begin and end at charges or go to infinity.
Equipotential surface– a surface at all points of which the electric field potential has same value:
φ ( X; y; z) = const.
Equipotential surfaces are closed and do not intersect. Between any two points on an equipotential surface, the potential difference is zero. This means that the force vector at any point in the trajectory of the charge along the equipotential surface is perpendicular to the velocity vector. Consequently, the electrostatic field strength lines are perpendicular to the equipotential surface.
The work of electric field forces for any movement of a charge along an equipotential surface is dA = 0, since dφ = 0.
The equipotential surfaces of the field of a point electric charge are spheres in the center of which the charge is located (Fig. 136).
Equipotential surfaces of a uniform electric field are planes perpendicular to the lines of tension (Fig. 137).
It is convenient to represent the electrostatic field graphically using lines of force and equipotential surfaces.
power line– this is a line, at each point of which the tangent coincides with the direction of the tension vector (see figure). Lines of force are given direction by an arrow. Properties of power lines:
1 ) The lines of force are continuous. They have a beginning and an end - they begin on positive charges and end on negative charges.
2 ) Field lines cannot intersect each other, because tension is force, and there cannot be two forces at a given point from one charge.
3 ) Lines of force are drawn so that their number through a unit perpendicular area is proportional to the magnitude of the tension.
4 ) Lines of force “exit” and “enter” are always perpendicular to the surface of the body.
5 ) A line of force should not be confused with the trajectory of a moving charge. The tangent to the trajectory coincides with the direction of the velocity, and the tangent to the line of force coincides with the force and, therefore, the acceleration.
Equipotential surface called a surface at each point of which the potential has the same value j = const.
Field lines are always perpendicular to equipotential surfaces. Let's prove it. Let a point charge move along the equipotential surface q. Elementary work, performed in this case is equal to dA=qE×cosa×dl = q×dj = 0, because dj = 0. Because q ,E And ×dl¹ 0, therefore
cosa = 0 And a= 90 o.
 | The figure shows the electrostatic field of two identical point charges. Lines with arrows are lines of force, closed curves are equipotential surfaces. At the center of the axial line connecting the charges, the voltage is 0. At very long distance from the charges, the equipotential surfaces become spherical. . |
 | This figure shows a homogeneous field - this is a field at each point of which the intensity vector remains constant in magnitude and direction. Equipotential surfaces are planes perpendicular to the lines of force. The tension vector is always directed towards decreasing potential. |
Topic 1. Question 6.
Superposition principle.
Based on experimental data, it was obtained superposition principle ( overlays ) fields: “If an electric field is created by several charges, then the intensity and potential of the resulting field add up independently, i.e. without affecting each other." At discrete distribution charges, the resulting field strength is equal to the vector sum, and the potential is the algebraic (taking into account the sign) sum of the fields created by each charge separately. At continuous distribution charge in the body, vector sums are replaced by integrals, where dE And dj– intensity and potential of the field of an elementary (point) charge allocated in the body. Mathematically, the principle of superposition can be written as follows.
Topic 2. Question 1.
Gauss's theorem.
First we introduce the concept “ vector flow" - This scalar quantity
 (N×m 2 /Cl = V×m) | elementary flux of the tension vector E, n – normal to the site, dS– an elementary site is a small site within which E= const; E n– vector projection E to the normal direction n | ||
 | tension vector flow through the terminal site S | ||
 | -²- -²- -²-through a closed surface S | ||
1) Sphere, charged with surface density charge s(C/m 2)
Let's consider the areas: 1) outside the sphere () and inside it (). Let's select surfaces: 1) S 1 and 2) S 2– both surfaces are spheres, concentric with the charged sphere. First, let's find the vector flows E through the selected surfaces, and then use the theorem.
 (¨)
| Vector threads E through S 1() And S 2. () E^n, a = 0, cosa = 1. |  | |
 (¨¨)
| by Gauss's theorem; F 2= 0, because S 2 does not cover any charges. Equating the flows from (¨) and (¨¨), we find E(r). | ||
 | |||
 q = s×2pR 2– full charge of the sphere | Outside the sphere the field is the same as the field of a point charge. At the boundary of the sphere there is a jump in tension. |
Topic 2. Question 2.
Gauss's theorem.
2) Thin long thread, charged with linear charge density t(C/m)
In this case, the “Gaussian” surface is a cylinder of length coaxial with the thread l.
First, let's find the flow, then use Gauss's theorem.
Topic 2. Question 3.
Gauss's theorem.
3) Thin-walled long cylinder, charged:
1) with linear charge density t or
2) with surface charge density s.
This example is similar to the previous one. We select a Gaussian surface in the form of a coaxial cylinder, and divide the surface into a lateral one and two torsional ones. In the first case, for a given linear density t we get the same formula as for a long thread. In the second case, the covered charge is equal to ( s×2p×R×l) and formula for E somewhat different, although depending on r- the same.
  |  |  |
Topic 2. Question 4.
1. Electric charge. Coulomb's law.
2. Electric field. Tension, potential, potential difference. Graphic representation of electric fields.
3. Conductors and dielectrics, relative dielectric constant.
4. Current, current strength, current density. Thermal effect current
5. Magnetic field, magnetic induction. Power lines. The effect of a magnetic field on conductors and charges. The effect of a magnetic field on a current-carrying circuit. Magnetic permeability.
6. Electromagnetic induction. Toki Fuko. Self-induction.
7. Capacitor and inductor. Energy of electric and magnetic fields.
8. Basic concepts and formulas.
9. Tasks.
The characteristics of electric and magnetic fields that are created by biological systems or act on them are a source of information about the state of the organism.
10.1. Electric charge. Coulomb's law
The charge of a body consists of the charges of its electrons and protons, whose own charges are equal in magnitude and opposite in sign (e = 1.67x10 -19 C).
Bodies in which the number of electrons and protons are equal are called uncharged.
If for some reason the equality between the number of electrons and protons is violated, the body is called charged and him electric charge is determined by the formula
Coulomb's law
Interaction stationary point charges obey Coulomb's law and is called Coulomb or electrostatic.
The power of interaction two point stationary charges is directly proportional to the product of their values and inversely proportional to the square of the distance between them:
10.2. Electric field. Tension, potential, potential difference. Graphic representation of electric fields
Electric field is a form of matter through which interaction between electric charges occurs.
An electric field is created by charged bodies. The strength characteristic of the electric field is vector quantity, called field strength.
Electric field strength(E) at a certain point in space is equal to the force acting on a unit point charge placed at this point:
Potential, potential difference
When a charge moves from one point in the field to another, the field forces do work that does not depend on the shape of the path. To calculate this work, use a special physical quantity, called potential.
Graphic representation of electric fields
To graphically represent the electric field, use power lines or equipotential surfaces(usually one thing). power line- a line whose tangents coincide with the direction of the tension vector at the corresponding points.
The density of the field lines is proportional to the field strength. Equipotential surface- a surface in which all points have the same potential.
These surfaces are carried out so that the potential difference between adjacent surfaces is constant.
Rice. 10.1. Field lines and equipotential surfaces of charged spheres
Field lines are perpendicular to equipotential surfaces.
Figure 10.1 shows the field lines and equipotential surfaces for the fields of charged spheres.
Figure 10.2, a shows the field lines and equipotential surfaces for the field created by two plates, the charges of which are equal in magnitude and opposite in sign. Figure 10.2, b shows the field lines and equipotential surfaces for the Earth's electric field near standing man.
Rice. 10.2. Electric field of two plates (a); electric field of the Earth near a standing person (b).
10.3. Conductors and dielectrics, relative dielectric constant
Substances that have free charges are called conductors.
The main types of conductors are metals, electrolyte solutions and plasma. In metals, free charges are the electrons of the outer shell separated from the atom. In electrolytes, the free charges are the ions of the dissolved substance. In plasma, free charges are electrons, which are separated from atoms when high temperatures, and positive ions.
Substances that do not contain free charges, are called dielectrics.
All gases are dielectrics low temperatures, resins, rubber, plastics and many other non-metals. Dielectric molecules are neutral, but the centers of positive and negative charges do not match. Such molecules are called polar and are depicted as dipoles. Figure 10.3 shows the structure of a water molecule (H 2 O) and its corresponding dipole.
Rice. 10.3. Water molecule and its image in the form of a dipole
If there is a conductor in an electrostatic field (charged or uncharged - it makes no difference), then the free charges are redistributed in such a way that the electric field created by them compensates external field. Therefore, the electric field strength inside the conductor equal to zero.
If there is a dielectric in an electrostatic field, then its polar molecules “tend” to position themselves along the field. This leads to a decrease in the field inside the dielectric.
Permittivity (ε) - dimensionless scalar quantity showing how many times the electric field strength in a dielectric decreases compared to the field in vacuum:
10.4. Current, current strength, current density. Thermal effect of current
Electric shock called the ordered movement of free charges in a substance. The direction of current is taken to be the direction of movement positive charges.
Electric current arises in a conductor between the ends of which is supported electrical voltage(U).
Quantitatively electric current characterized using a special quantity - current strength.
Current strength in a conductor is a scalar quantity that shows how much charge passes through the cross section of the conductor in 1 s.
In order to show the distribution of current in conductors complex shape, use current density (j).
Current Density in a conductor is equal to the ratio of the current to the cross-sectional area of the conductor:
Here R is a conductor characteristic called resistance. Unit of measurement - Ohm.
The resistance value of a conductor depends on its material, shape and size. For a cylindrical conductor, the resistance is directly proportional to its length (l) and inversely proportional to area cross section(S):
The proportionality coefficient ρ is called specific electrical resistance conductor material; its dimension is Omm.
The flow of current through a conductor is accompanied by the release of heat Q. The amount of heat released in the conductor during time t is calculated using the formulas
The thermal effect of current at a certain point on a conductor is characterized by specific thermal power q.
Specific thermal power - the amount of heat released per unit volume of a conductor per unit time.
To find this value, you need to calculate or measure the amount of heat dQ released in a small vicinity of the point, and then divide it by the time and volume of the vicinity:
where ρ - resistivity conductor.
10.5. Magnetic field, magnetic induction. Power lines. Magnetic permeability
Magnetic field is a form of matter through which the interaction of moving electric charges occurs.
In the microcosm, magnetic fields are created separate moving charged particles. At chaotic the movement of charged particles in matter, their magnetic fields compensate each other and the magnetic field in the macrocosm does not arise. If the movement of particles in a substance is in any way arrange, then the magnetic field also appears in the macrocosm. For example, a magnetic field arises around any current-carrying conductor. The special ordered rotation of electrons in some substances also explains the properties of permanent magnets.
The strength characteristic of the magnetic field is the vector magnetic inductionB. Unit of magnetic induction - tesla(Tl).
Power lines
The magnetic field is graphically represented using magnetic induction lines(magnetic lines of force). Tangents to field lines show the direction of the vector IN at the appropriate points. The density of the lines is proportional to the vector module IN. Unlike the electrostatic field lines, the magnetic induction lines are closed (Fig. 10.4).
Rice. 10.4. Magnetic lines of force
The effect of a magnetic field on conductors and charges
Knowing the magnitude of magnetic induction (V) in this place, you can calculate the force exerted by the magnetic field on a current-carrying conductor or a moving charge.
A) Ampere power, acting on a straight section of a current-carrying conductor is perpendicular to both direction B and the current-carrying conductor (Fig. 10.5, a):
where I is the current strength; l- length of the conductor; α is the angle between the direction of the current and vector B.
b) Lorentz force acting on a moving charge is perpendicular to both the direction B and the direction of the charge velocity (Fig. 10.5, b):
where q is the amount of charge; v- its speed; α - angle between direction v and V.
Rice. 10.5. Ampere (a) and Lorentz forces (b).
Magnetic permeability
Just as a dielectric placed in an external electric field polarizes and creates its own electric field, any substance placed in an external magnetic field, magnetized and creates its own magnetic field. Therefore, the value of magnetic induction inside a substance (B) differs from the value of magnetic induction in a vacuum (B 0). Magnetic induction in a substance is expressed through the magnetic field induction in vacuum according to the formula
where μ is the magnetic permeability of the substance. For vacuum μ = 1
Magnetic permeability of a substance(μ) is a dimensionless quantity showing how many times the magnetic field induction in a substance changes compared to the magnetic field induction in a vacuum.
Based on their ability to magnetize, substances are divided into three groups:
1) diamagnetic materials, for which μ< 1 (вода, стекло и др.);
2) paramagnets, for which μ > 1 (air, hard rubber, etc.);
3) ferromagnets, for which μ >>1 (nickel, iron, etc.).
For dia- and paramagnetic materials, the difference in magnetic permeability from unity is very insignificant (~0.0001). The magnetization of these substances when removed from a magnetic field disappears.
For ferromagnetic materials, the magnetic permeability can reach several thousand (for example, for iron μ = 5,000-10,000). When removed from a magnetic field, the magnetization of ferromagnets is partially is saved. Ferromagnets are used to make permanent magnets.
10.6. Electromagnetic induction. Toki Fuko. Self-induction
In a closed conducting loop placed in a magnetic field, under certain conditions, an electric current arises. To describe this phenomenon, a special physical quantity is used - magnetic flux. Magnetic flux through a contour of area S, the normal of which (n) forms an angle α with the direction of the field (Fig. 10.6), calculated by the formula
Rice. 10.6. Magnetic flux through the loop
Magnetic flux is a scalar quantity; unit of measurement weber[Wb].
According to Faraday's law, with any change magnetic flux piercing the circuit, an electromotive force arises in it E(induction emf), which is equal to the rate of change of the magnetic flux passing through the circuit:
E.m.f. induction occurs in a circuit that is in variable magnetic field or rotates in a constant magnetic field. In the first case, the change in flux is caused by a change in magnetic induction (B), and in the second case, by a change in angle α. The rotation of a wire frame between the poles of a magnet is used to produce electricity.
Toki Fuko
In some cases, electromagnetic induction appears even in the absence of a specially created circuit. If in variable If there is a conducting body in a magnetic field, then eddy currents arise throughout its entire volume, the flow of which is accompanied by the release of heat. Let us explain the mechanism of their occurrence using the example of a conducting disk located in a changing magnetic field. The disk can be considered as a “set” of closed contours nested within each other. In Fig. 10.7 nested contours are ring segments between
Rice. 10.7. Foucault currents in a conducting disk located in a uniform alternating magnetic field. The direction of the currents corresponds to the increase in V
circles. When the magnetic field changes, the magnetic flux also changes. Therefore, a current, shown by an arrow, is induced in each circuit. The set of all such currents is called Foucault's currents.
In technology, one has to fight with Foucault currents (loss of energy). However, in medicine these currents are used to warm tissues.
Self-induction
Phenomenon electromagnetic induction can also be observed when external there is no magnetic field. For example, if you skip along a closed contour variable current, then it will create an alternating magnetic field, which, in turn, will create an alternating magnetic flux through the circuit, and an emf will arise in it.
Self-induction called the occurrence of electromotive force in a circuit through which alternating current flows.
The electromotive force of self-induction is directly proportional to the rate of change of current in the circuit:
The “-” sign means that the self-inductive emf prevents a change in the current strength in the circuit. The proportionality factor L is a circuit characteristic called inductance. Unit of inductance - Henry (Hn).
10.7. Capacitor and inductor. Energy of electric and magnetic fields
In radio engineering to create electric and magnetic fields concentrated in small area spaces, use special devices - capacitors And inductors.
Capacitor consists of two conductors separated by a dielectric layer, on which charges of equal magnitude and opposite sign are placed. These conductors are called plates capacitor.
Charge the capacitor called the positive plate charge.
The plates have the same shape and are located at a distance very small compared to their size. In this case, the electric field of the capacitor is almost completely concentrated in the space between the plates.
Electrical capacity A capacitor is called the ratio of its charge to the potential difference between the plates:
Capacity unit - farad(F = Cl/V).
A flat capacitor consists of two parallel plates of area S, separated by a dielectric layer of thickness d with dielectric constant ε. The distance between the plates is much less than their radii. The capacity of such a capacitor is calculated by the formula:
Inductor is a wire coil with a ferromagnetic core (to enhance the magnetic field). The diameter of the coil is much less than its length. In this case, the magnetic field created by the flowing current is almost completely concentrated inside the coil. The ratio of magnetic flux (F) to current (I) is a characteristic of the coil, called its inductance(L):
Unit of inductance - Henry(Gn = Wb/A).
Energy of electric and magnetic fields
Electrical and magnetic field are material and, as a result, have energy.
Electric field energy of a charged capacitor:
where I is the current strength in the coil; L is its inductance.
10.8. Basic concepts and formulas
Continuation of the table
Continuation of the table
Continuation of the table
End of the table
10.9. Tasks
1. With what force are charges of 1 C attracted, located at a distance of 1 m from each other?
Solution
Using formula (10.1) we find: F = 9*10 9* 1*1/1 = 9x10 9 N. Answer: F = 9x10 9 N.
2. With what force does the nucleus of an iron atom ( serial number 26) attracts an electron on the inner shell with a radius r = 1x10 -12 m?
Solution
Nuclear charge q = +26e. We find the force of attraction using formula (10.1). Answer: F = 0.006 N.
3. Estimate the electric charge of the Earth (it is negative) if the electric field strength at the Earth's surface is E = 130 V/m. The radius of the Earth is 6400 km.
Solution
The field strength near the Earth is the field strength of a charged sphere:
E = k*q|/R 2, where k = 1/4πε 0 = 910 9 Nm 2 / Cl 2.
From here we find |q| = ER 2 /k = )
