Graphical representation of the electric field. Field lines of electric field strength

lines of force tension electric field- lines, the tangents to which at each point coincide with the vector E. By their direction, one can judge where the positive (+) and negative (-) charges are located, creating an electric field. The density of lines (the number of lines penetrating a unit surface area perpendicular to them) is numerically equal to the modulus of the vector E.

The lines of force of the electric field strength The lines of force of the electric field are not closed, they have a beginning and an end. We can say that the electric field has "sources" and "sinks" of lines of force. Field lines start on positive (+) charges (Fig. a), end on negative (-) charges (Fig. b). The lines of force do not intersect.

Electric field strength vector flux Arbitrary area dS. The flow of the electric field strength vector through the area dS: is a pseudovector, the modulus of which is equal to dS, and the direction coincides with the direction of the vector n towards the area dS. E = constdФ E = N - the number of lines of the electric field strength vector E, penetrating the area dS.

Flux of the electric field strength vector If the surface is not flat and the field is inhomogeneous, then a small element dS is selected, which is considered flat, and the field is homogeneous. Flux of the electric field strength vector: The sign of the flux coincides with the sign of the charge.

Law (theorem) of Gauss in integral form. A solid angle is a part of space bounded by conical surface. The measure of the solid angle is the ratio of the area S of the sphere cut out on the surface of the sphere by a conical surface to the square of the radius R of the sphere. 1 steradian - a solid angle with a vertex in the center of the sphere, cutting out an area on the surface of the sphere, equal to the area a square with a side equal in length to the radius of this sphere.

Gauss' theorem in integral form Electric field is created by a point charge +q in a vacuum. The flow d Ф Е, created by this charge, through an infinitely small area dS, the radius vector of which is r. dS n is the projection of the area dS onto the plane perpendicular to the vector r. n is the unit vector of the positive normal to the area dS.

If an arbitrary surface surrounds k- charges, then according to the principle of superposition: Gauss theorem: for an electric field in vacuum, the flow of the electric field strength vector through an arbitrary closed surface is equal to algebraic sum charges enclosed inside this surface, divided by ε 0.

The method of applying the Gauss theorem to calculate electric fields - the second way to determine the electric field strength E The Gauss theorem is used to find the fields created by bodies with geometric symmetry. Then the vector equation is reduced to a scalar one.

The method of applying the Gauss theorem for calculating electric fields is the second way to determine the strength of the electric field E 1) The flux Ф E of the vector E is found by determining the flux. 2) The flow Ф Е is found by the Gauss theorem. 3) From the condition of equality of flows, the vector E is found.

Examples of application of the Gauss theorem 1. Field of an infinite uniformly charged filament (cylinder) with a linear density τ (τ = dq/dl, C/m). The field is symmetrical, directed perpendicular to the thread, and for reasons of symmetry at the same distance from the axis of symmetry of the cylinder (thread) has the same value.

2. The field of a uniformly charged sphere of radius R. The field is symmetrical, the lines of strength E of the electric field are directed in the radial direction, and at the same distance from the point O the field has the same value. The unit normal vector n to the sphere of radius r coincides with the intensity vector E. Let us cover the charged (+q) sphere with an auxiliary spherical surface of radius r.

2. The field of a uniformly charged sphere When the field of the sphere is found as the field of a point charge. For r

(σ = dq/dS, C/m 2). The field is symmetric, the vector Е is perpendicular to the plane with the surface charge density +σ and has the same value at the same distance from the plane. 3. The field of a uniformly charged infinite plane with a surface charge density + σ As a closed surface, we take a cylinder whose bases are parallel to the plane and which is divided by a charged plane into two equal halves.

Earnshaw's theorem A system of fixed electric charges cannot be in stable equilibrium. The charge + q will be in equilibrium if, when it moves over a distance dr, a force F acts from the side of all other charges of the system located outside the surface S, returning it to its original position. There is a system of charges q 1, q 2, … q n. One of the charges q of the system is covered by a closed surface S. n is the unit vector of the normal to the surface S.

Earnshaw's Theorem The force F is due to the field E created by all the other charges. The field of all external charges E must be directed opposite to the direction of the displacement vector dr, that is, from the surface S to the center. According to the Gauss theorem, if the charges are not covered by a closed surface, then Ф E = 0. A contradiction proves Earnshaw's theorem.

0 flows out more than flows in. Ф 0 flows out more than flows in. F 33 Gauss's law differential form Vector divergence is the number of field lines per unit volume, or the flux density of field lines. Example: water flows out and flows in from a volume. Ф > 0 more flows out than flows in. Ф 0 flows out more than flows in. Ф 0 flows out more than flows in. Ф 0 flows out more than flows in. Ф 0 flows out more than flows in. Ф title="(!LANG: Gauss's law in differential form Vector divergence is the number of field lines per unit volume, or the flux density of field lines. Example: water flows out and flows out of a volume. Ф > 0, more flows out than flows in. Ф

1. Electric charge. Coulomb's law.

2. Electric field. Tension, potential, potential difference. Graphic image electric fields.

3. Conductors and dielectrics, relative permittivity.

4. Current, current strength, current density. Thermal action current.

5. Magnetic field, magnetic induction. Power lines. The action of a magnetic field on conductors and charges. The action of a magnetic field on a circuit with current. 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 the 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 is made up of the charges of its electrons and protons, whose own charges are the same in magnitude and opposite in sign (e = 1.67x10 -19 C).

Bodies in which the number of electrons and protons is the same are called uncharged.

If for some reason the equality between the number of electrons and protons is violated, the body is called charged and its electric charge is given by

Coulomb's law

Interaction motionless point charges obey Coulomb's law and called Coulomb or electrostatic.

The power of interaction two fixed point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them:

10.2. Electric field. Tension, potential, potential difference. Graphical representation of electric fields

Electric field is a form of matter through which the interaction between electric charges is carried out.

The electric field is created by charged bodies. The power characteristic of the electric field is a vector quantity called the field strength.

Electric field strength(E) at some point in space is equal to the force acting on a unit point charge placed at this point:

Potential, potential difference

When moving a charge from one point of the field to another, the field forces do work that does not depend on the shape of the path. To calculate this work, a special physical quantity is used, called potential.

Graphical representation of electric fields

For a graphical representation of the electric field, use lines of force or equipotential surfaces(usually one or the other). force line- a line, the tangents to which coincide with the direction of the tension vector at the corresponding points.

The density of field lines is proportional to the field strength. Equipotential surface- a surface, all points of which 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

The lines of force are perpendicular to the equipotential surfaces.

Figure 10.1 shows lines of force and equipotential surfaces for the fields of charged spheres.

Figure 10.2, a shows lines of force and equipotential surfaces for a field created by two plates, the charges of which are the same in magnitude and opposite in sign. Figure 10.2, b shows the lines of force 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 permittivity

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 solute. In plasma, free charges are electrons that are separated from atoms when high temperatures, and positive ions.

Substances that do not have 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. The water molecule and its image as a dipole

If there is a conductor in the electrostatic field (charged or uncharged - it does not matter), then the free charges are redistributed in such a way that the electric field created by them compensates outer field. Therefore, the electric field strength inside the conductor equals zero.

If a dielectric is in an electrostatic field, then its polar molecules "tend" to settle down along the field. This leads to a decrease in the field inside the dielectric.

The dielectric constant (ε) - a dimensionless scalar value showing how many times the electric field strength in the 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 matter. The direction of current is taken as the direction of movement positive charges.

An electric current occurs in a conductor, between the ends of which an electric voltage (U) is maintained.

Quantitatively, the electric current is characterized using a special quantity - current strength.

current strength in a conductor is called a scalar quantity showing what charge passes through the cross section of the conductor in 1 s.

In order to show the distribution of current in conductors of complex shape, use the current density (j).

current density in the conductor is equal to the ratio of the current strength to the cross-sectional area of ​​the conductor:

Here R is a characteristic of the conductor, 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 the cross-sectional area (S):

The coefficient of proportionality ρ is called specific electrical resistance conductor material; its dimension is ohm.

The flow of current through the conductor is accompanied by the release of heat Q. The amount of heat released in the conductor during time t is calculated by the formulas

The thermal effect of the current at a certain point of the conductor is characterized by specific thermal power q.

Specific thermal power - the amount of heat released per unit volume of the conductor per unit time.

To find this value, you need to calculate or measure the amount of heat dQ released in a small neighborhood of the point, and then divide it by the time and volume of the neighborhood:

where ρ is the resistivity of the conductor.

10.5. Magnetic field, magnetic induction. Power lines. Magnetic permeability

A magnetic field is a form of matter through which the interaction of moving electric charges is carried out.

In the microcosm, magnetic fields are created individual moving charged particles. At chaotic movement of charged particles in matter, their magnetic fields compensate each other and the magnetic field in the macrocosm does not occur. If the movement of particles in a substance is somehow arrange, 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 force characteristic of the magnetic field is the vector magnetic inductionb. Unit of magnetic induction - tesla(Tl).

lines of force

The magnetic field is graphically represented using lines of magnetic induction(magnetic lines of force). Tangents to lines of force show the direction of the vector AT at the respective points. The density of lines is proportional to the modulus of the vector AT. Unlike lines of force electrostatic field, the lines of magnetic induction are closed (Fig. 10.4).

Rice. 10.4. Magnetic lines of force

The action of a magnetic field on conductors and charges

Knowing the value of magnetic induction (V) in this place, you can calculate the force acting from the magnetic field on a current-carrying conductor or a moving charge.

a) ampere power, acting on straight section current-carrying conductor, perpendicular to both direction B and current-carrying conductor (Fig. 10.5, a):

where I is the current strength; l- conductor length; α is the angle between the direction of the current and the 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 polarized and creates its own electric field, any substance placed in an external magnetic field, magnetized and creates its own magnetic field. Therefore, the magnitude of the magnetic induction inside the substance (B) differs from the magnitude of the magnetic induction in vacuum (B 0). The magnetic induction in a substance is expressed in terms of the magnetic field induction in vacuum by 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 vacuum.

According to the ability to magnetize, substances are divided into three groups:

1) diamagnets, for which μ< 1 (вода, стекло и др.);

2) paramagnets, in which μ > 1 (air, ebonite, etc.);

3) ferromagnets, for which μ >>1 (nickel, iron, etc.).

For dia- and paramagnets, the difference in magnetic permeability from unity is very insignificant (~0.0001). The magnetization of these substances when removed from a magnetic field disappears.

In ferromagnets, the magnetic permeability can reach several thousand (for example, in iron, μ \u003d 5,000-10,000). When removed from the 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 circuit placed in a magnetic field, under certain conditions, an electric current arises. To describe this phenomenon, a special physical quantity is used - magnetic flux. The magnetic flux through the contour of the area S, the normal of which (n) forms an angle α with the direction of the field (Fig. 10.6), is calculated by the formula

Rice. 10.6. Magnetic flux through the loop

Magnetic flux is a scalar quantity; unit weber[Wb].

According to Faraday's law, with any change in the magnetic flux penetrating the circuit, an electromotive force arises in it E(emf of induction), which is equal to the rate of change of the magnetic flux penetrating the circuit:

emf induction occurs in a circuit that is in variable magnetic field or revolves in a constant magnetic field. In the first case, the change in the flux is due to a change in the magnetic induction (B), and in the second case, it is due to a change in the angle α. The rotation of a wireframe between the poles of a magnet is used to generate electricity.

Toki Foucault

In some cases, electromagnetic induction manifests itself even in the absence of a specially created circuit. If in variable If there is a conducting body in the magnetic field, then eddy currents arise throughout its 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. A disk can be thought of as a "set" of nested closed loops. On fig. 10.7 nested contours are circular 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, the current shown by the arrow is induced in each circuit. The totality of all such currents is called Foucault currents.

In technology, Foucault currents have to be fought (energy losses). 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 loop variable current, then it will create an alternating magnetic field, which, in turn, will create an alternating magnetic flux through the circuit, and emf will appear in it.

Self induction is called the emergence electromotive force in a circuit through which an alternating current flows.

The electromotive force of self-induction is directly proportional to the rate of change in the current strength in the circuit:

The “-” sign means that the self-induction emf prevents a change in the current strength in the circuit. The proportionality factor L is a characteristic of the circuit, called inductance. Unit of inductance - henry (Gn).

10.7. Capacitor and inductor. Energy of electric and magnetic fields

In radio engineering, special devices are used to create electric and magnetic fields concentrated in a small area of ​​\u200b\u200bspace - capacitors and inductors.

Capacitor It consists of two conductors separated by a dielectric layer, on which charges of the same magnitude and opposite in sign are placed. These conductors are called plates capacitor.

Capacitor charge is called the charge of the positive plate.

The plates have the same shape and are located at a distance that is very small compared to their dimensions. In this case, the electric field of the capacitor is almost completely concentrated in the space between the plates.

electrical capacitance capacitor is the ratio of its charge to the potential difference between the plates:

Capacity unit - farad(F \u003d Cl / V).

A flat capacitor consists of two parallel plates of area S, separated by a dielectric layer of thickness d with permittivity ε. The distance between the plates is much less than their radii. The capacitance 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 smaller 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 strength (I) is a characteristic of the coil, called its inductance(L):

Unit of inductance - Henry(Hn = Wb/A).

Energy of electric and magnetic fields

Electrical and magnetic field material and therefore have energy.

The energy of the electric field of a charged capacitor:

where I is the current in the coil; L is its inductance.

10.8. Basic concepts and formulas

Table continuation

Table continuation

Table continuation

End of table

10.9. Tasks

1. What force attracts charges of 1 C, located at a distance of 1 m from each other?

Solution

According to the formula (10.1) we find: F \u003d 9 * 10 9 * 1 * 1/1 \u003d 9x10 9 N. Answer: F \u003d 9x10 9 N.

2. With what force does the nucleus of an iron atom (serial number 26) attract an electron on the inner shell with a radius of r = 1x10 -12 m?

Solution

Nuclear charge q = +26e. We find the force of attraction by the 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 the charged sphere:

E \u003d k * q | / R 2, where k \u003d 1/4πε 0 \u003d 910 9 Nm 2 / C 2.

From here we find |q| = ER 2 /k = )