Let us consider the field of a point charge. The strength of this field at any point is equal to According to Coulomb's law Therefore, the field strength of a point charge
Potential.
Potential difference. In addition to tension, an important characteristic electric field is the potential j. Potential j is the energy characteristic of the electric field, while intensity E is its force characteristic, because potential is equal to the potential energy possessed unit charge at a given point of the field, and the intensity is equal to the force with which the field acts on this unit charge.
Magnetic fields can be generated by moving charges and electric currents. In this section, we will focus on studying the field created by a specific load, be it. M is called magnetic permeability and depends on the medium in which the charge is located.
The modulus can be calculated using the following expression. Its value can be easily determined using correct rule. Place thumb right hand in the direction of the velocity vector. Only a moving charge creates a magnetic field. the remaining fingers will point to the meaning magnetic field. Magnetic permeability. Magnetic permeability is a constant of each medium and on which the intensity of the magnetic field depends, as we saw in the previous equation.
j=W sweat /q, Here W sweat is the potential energy of charge q at a given point in the field. The potential of the field created by a point charge - a source q or a charged ball with a charge q, is determined by the formula j=q/4pe 0 er. Here r is the distance from a field point with potential j to a point charge or to the center of a ball. If r=R, where R is the radius of the ball, then using this formula you can determine the potential of the ball on its surface. The work of moving charge A in an electric field is determined by the expression A=q(j 1 -j 2) or A=qU. Here j 1 -j 2 potential difference(or potential drop D j, or voltage U) between points with potentials, j 1 and j 2. Obviously, if a charge is moved between points with the same potential, then the work of moving the charge is zero. In the same way, the work of moving a charge along a closed path is also zero, i.e. when he returns to the starting point with the same potential. Indeed, in this case A=q(j 1 -j 2)=0. in a uniform electrostatic field, the work of moving charge q can be determined by the formula A=Eqd, (d=Scosa), where E is the strength of this field, and d is the projection of moving charge q onto the line of force of this field, the angle between the direction of movement S and the vector E. If a charge moves along a power line, then d is the displacement modulus. If the charge moves perpendicular to the lines of force, then a = 90 0, cosa = 0 and A = 0. At each point of a uniform electric field, the intensity is the same in magnitude and direction, but the potential is not, since it decreases when moving from points that are closer to positive charges - sources, to points that are closer to negative charge sources. In this case, the connection between the potential difference j 1 -j 2 or U and the tension E expresses a simple correspondence E=(j 1 -j 2)/d or E=U/d. It should be noted that in an electric field you can find points whose potentials are the same. These points are located on surfaces perpendicular to the lines of vector E. Such surfaces are called equipotential. The work of moving charge q along the equipotential surface is zero, since A = q(j 1 -j 2)=0. The surface of a conductor with stationary charges is also equipotential, therefore, when a charge moves along such a conductor, no work is done. The formula E=(j 1 -j 2)/d can be applied to the field of an infinite charged plane and to the field flat capacitor, the plates of which are charged differently (in this case, if j 1 - j 2 is the potential difference between the plates, then d is the distance between them).
In contrast, the magnetic field is not central and its field lines are closed. The field lines coincide with the dotted blue lines of the figure. Similar electric field is central and the field lines are radial to the load. Although in the previous section we focus on the field generated by moving points. both fields are inversely proportional to the distance at which they are measured and depend on the environment in which they are located. that, like a charge, an electric field or mass of a gravitational field arises.
A current element is a current that flows through a section touching a conductor of infinitesimal length and which has the meaning of electric current. In particular, we will focus on: the field generated by any electrical current. A field generated by a straight-line electric current. A field generated by an electrical current that circulates through a loop. The principle of superposition of magnetic fields.
Dipole
It is a collection of equal and opposite charges, located at a small distance from each other. When an external electric field is applied, the dipoles are oriented in such a way that the field created by the polarized charge is directed in the direction opposite to the external electric field. The electric field strength in the dielectric is equal to the difference between the voltages of the external field E 0 and the field created by the polarized charge Ep: E = Eo – Ep. In nonpolar dielectrics, in the absence of an external field, the molecules are not dipoles, since the centers of positive and negative charges coincide. When an external electric field is applied, the molecules are stretched and become dipoles, with the field of the polarized charge directed against the external field. Regardless of the nature of the dielectric, the external field strength in it is always weakened by e times: e = Eо/E. Relative dielectric constant e shows how many times the electric field strength in dielectrics is less than in vacuum.
The magnetic field created by any electric current was founded by Jean Baptiste Biot and Felix Savart shortly after Oersted made his experiment public. This is where we come to the study of the field created by electric current. Measured in meters. This means that the direction of the magnetic field touches them at every point, and its direction is given by the right-hand rule. Where is the field heading?
If the driver is vertically oriented and the intensity goes up? Solution. A magnetic field created by an electric current flowing through a loop. Remember this, as we have already said. This is the metro. the right hand rule states that if we use the thumb of the said hand to indicate the meaning of the intensity of the current. This is Tesla. the remaining fingers will point to the meaning of the magnetic field. How does this happen with a magnet? The face through which they exit is named Northern face and through which they enter the South Face. the field lines will come out of one side of the loop and enter the other.
Dipole
(from di... and Greek pólos - pole) electric, a combination of two equal in absolute value opposite point charges located at some distance from each other. The main characteristic of an electric charge is its dipole moment - a vector directed from a negative charge to a positive one ( rice. 1 ) and is numerically equal to the product of the charge e to a distance l between charges: R = el. The dipole moment determines the electric field of D. at a large distance R from D. ( R"l), as well as the effect on D. of an external electric field.
Far from D. its electric field E decreases with distance as 1/ R 3, i.e. faster than the field of a point charge (~ 1/ R 2). Field strength components E along the D axis ( E p) and in a direction perpendicular to R (E┴), are proportional to the dipole moment and in the CGS (Gaussian) system of units are equal to:
where J is the angle between R and radius vector R points in space at which the D field is measured; full tension
Thus, on the D axis at J = 0 the field strength is twice as high as at J = 90°; at both these angles it has only the component E p, and at J = 0 its direction is parallel R, and at J = 90° - antiparallel ( rice. 2 ).
The effect of an external electric field on a diaphragm is also proportional to the magnitude of its dipole moment. A uniform field creates a torque M =pE sin a (a is the angle between the external electric field strength vector E and dipole moment R; rice. 3 ), tending to rotate the D. so that its dipole moment is directed along the field. In a non-uniform electric field, in addition to the torque, a force also acts on the dynamic force, tending to pull the dynamic into the region of a stronger field ( rice. 4 ).
The electric field of any generally neutral system at distances significantly greater than its size approximately coincides with the field of an equivalent dynamic - an electric dynamic with the same dipole moment as that of a system of charges (that is, the field at large distances from the system is insensitive to charge distribution details). Therefore, in many cases, electrical dynamics is a good approximation for describing such a system at distances that are large compared to its size. For example, the molecules of many substances can be approximately considered as electric molecules (in the simplest case, these are molecules of two ions with charges of opposite signs); atoms and molecules in an external electric field, which somewhat pushes their positive and negative charges apart, acquire an induced (field-induced) dipole moment and become microscopic dielectrics (see, for example, Dielectrics).
Electric D. with a time-varying dipole moment (due to changes in its length l or charges e) is a source of electromagnetic radiation (see Hertz vibrator).
D. magnetic. Study of pole interactions permanent magnets(C. Coulon, 1785) led to the idea of the existence of magnetic charges similar to electric ones. A pair of such magnetic charges, equal in magnitude and opposite in sign, was considered as a magnetic magnet (possessing a magnetic dipole moment). Later it was found that magnetic charges do not exist and that magnetic fields are created by moving electric charges, that is, electric currents (see Ampere's theorem). However, the concept of a magnetic dipole moment turned out to be expedient to retain, since at large distances from closed conductors through which currents flow, magnetic fields turn out to be the same as if they were generated by magnetic magnets (the magnetic field of a magnetic magnet at large distances from the magnet is calculated according to the same formulas as the electric field D. electric, and the electric moment dipoles must be replaced by the magnetic moment of the current). The magnetic moment of a current system is determined by the strength and distribution of currents. In the simplest case of current I, flowing along a circular contour (turn) of radius A, the magnetic moment in the SGS system is equal to p = ISn/c, Where S= p a 2 is the area of the coil, and the unit vector n, drawn from the center of the coil, is directed so that from its end the current is visible flowing counterclockwise ( rice. 5 ), With- speed of light.
The analogy between a magnetic field and a current-carrying coil can also be seen when considering the effect of a magnetic field on a current. In a uniform magnetic field, a current-carrying coil is acted upon by a moment of force that tends to orient the coil so that its magnetic moment is directed along the field; in a non-uniform magnetic field, such closed currents (“magnetic currents”) are drawn into a region with higher field strength. The interaction of an inhomogeneous magnetic field with a magnetic field is based, for example, on the separation of particles with different magnetic moments—nuclei, atoms, or molecules (the magnetic moments of which are determined by the movement of the charged elementary particles included in their composition, as well as by the magnetic moments associated with the spins of the particles). A beam of particles passing through a non-uniform magnetic field is divided, because the field more strongly changes the trajectories of particles with a large magnetic moment.
However, the analogy between a magnetic current and a coil with a current (the equivalence theorem) is not complete. So, for example, in the center of a circular coil, the magnetic field strength is not only not equal to the field strength of the “equivalent” magnetic field, but is even opposite to it in direction ( rice. 6 ). Magnetic lines of force (unlike electrical lines of force, which begin and end at charges) are closed.
5. Polarization of dielectrics
(dielectric, what they are, how they are polarized)
According to modern ideas, electric charges do not act on each other directly. Each charged body creates in the surrounding space electric field . This field exerts a force on other charged bodies. The main property of the electric field is the effect on electric charges with some 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 studied using the so-called test charge – a small point charge that does not produce a noticeable redistribution of the charges under study.
To quantify the electric field, we introduce power characteristic electric field strength .
Electric field strength is a physical quantity equal to the ratio of the force with which the field acts on a positive test charge placed in this point space, to the magnitude of this charge:
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Electric field strength – vector physical quantity. The direction of the vector at each point in space coincides with the direction of the force acting on the positive test charge.
The electric field of stationary charges that do not change over time is called electrostatic . In many cases, for brevity, this field is denoted by a general term - electric field
If an electric field created by several charged bodies is studied 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 electric field strength created by a system of charges at a given point in space is equal to the vector sum of the electric field strengths created at the same point by charges separately:
This field is called Coulomb . In a Coulomb field, the direction of the vector depends on the sign of the charge Q: If Q> 0, then the vector is directed radially from the charge, if Q < 0, то вектор направлен к заряду.
To visually depict the electric field, use power lines . These lines are drawn so that the direction of the vector at each point coincides with the direction of the tangent to the field line (Fig. 1.2.1). When depicting an electric field using field lines, their density should be proportional to the magnitude of the field strength vector.
Power lines Coulomb fields of positive and negative point charges are shown in Fig. 1.2.2. Since the electrostatic field created by any system of charges can be represented as a superposition of Coulomb fields of point charges, shown in Fig. 1.2.2 fields can be considered as elementary structural units (“bricks”) of any electrostatic field.
Coulomb field of a point charge Q It is convenient to write in vector form. To do this, you need to draw the radius vector from the charge Q to the observation point. Then at Q> 0 the vector is parallel and when Q < 0 вектор антипараллелен Следовательно, можно записать:
Important characteristic electric dipole is the so-called dipole moment
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where is a vector directed from a negative charge to a positive one, the Dipole module can serve electric model many molecules.
For example, a neutral water molecule (H 2 O) has an electric dipole moment, since the centers of two hydrogen atoms are located not on the same straight line with the center of the oxygen atom, but at an angle of 105° (Fig. 1.2.4). Dipole moment of a water molecule p= 6.2·10 –30 C m.
3.Gauss's electrostatic theorem. Proof of Gauss's theorem for a special case (a point charge is located inside a sphere of radius R). Generalization of Gauss's theorem to N point charges. Generalization of Gauss's theorem to the case of a continuously distributed charge. Gauss's theorem in differential form.
Let's find the vector flow E through a spherical surface S, in the center of which there is a point charge q.
In this case, because directions E And n coincide at all points of the spherical surface.
Taking into account the field strength of a point charge 
and the fact that the surface area of the sphere we get
An algebraic quantity depending on the sign of the charge. For example, when q<0 линии E directed towards the charge and opposite to the direction of the outer normal n. Therefore, in this case the flux is negative<0 .
Let the closed surface around the charge q has an arbitrary shape. Obviously, the surface is intersected by the same number of lines E, same as the surface S. Therefore, the vector flux E through an arbitrary surface is also determined by the resulting formula.
If the charge is located outside the closed surface, then, obviously, how many lines enter the closed area, the same number will leave it. As a result, the vector flow E will be equal to zero.
If the electric field is created by a system of point charges 
then according to the principle of superposition,
Proof of a special case:
Gauss's theorem states:
The flow of the electrostatic field strength vector through an arbitrary closed surface is equal to the algebraic sum of the charges located inside this surface, divided by the electric constant ε 0.
Where R– radius of the sphere. The flux Φ through a spherical surface will be equal to the product E per sphere area 4π R 2. Hence,
Let us now surround the point charge with an arbitrary closed surface S and consider an auxiliary sphere of radius R 0 (Fig. 1.3.3).
Consider a cone with small solid angle ΔΩ at the top. This cone will highlight a small area Δ on the sphere S 0 , and on the surface S– pad Δ S. The elementary flows ΔΦ 0 and ΔΦ through these areas are the same. Really,
In a similar way it can be shown that if a closed surface S does not cover point charge q, then the flow Φ = 0. Such a case is shown in Fig. 1.3.2. All electric field lines of a point charge penetrate a closed surface S through. Inside the surface S there are no charges, so in this region the field lines do not break off or arise.
A generalization of Gauss's theorem to the case of an arbitrary charge distribution follows from the superposition principle. The field of any charge distribution can be represented as a vector sum of the electric fields of point charges. Flow Φ of a system of charges through an arbitrary closed surface S will consist of flows Φ i electric fields of individual charges. If the charge q i ended up inside the surface S, then it makes a contribution to the flow equal to if this charge is outside the surface, then the contribution of its electric field to the flow will be equal to zero.
Thus, Gauss's theorem is proven.