Concepts electric field Similar to the concepts of the gravitational field. If you put a trial charge at a distance from the field source - charge then acting on from side the force will be equal

. (2.12)

Sharing

on the , we get the value characterizing :

. (2.13)

This new value equal to force acting on a single charge is called electric field tensioncreated by the source . Denote through . Then the tension of the electrical field of a uniform spherical charge at a distance from him

(2.14)

Of course, the magnitude characterizing the electric field is a vector. Direction conditionally chosen along the direction of force, which acts in the field on a positive test charge. Therefore, the field of field strength created by a positive charge is directed from the field source, and the field created by a negative charge is to a source (Fig. 2.10).

Dimension equal

[] \u003d units. SGSE charge / cm 2 \u003d

Units. GSE potential / cm \u003d

Dean / Unit. SGSE charge.

If you enter a test charge in this electric field then the strength will act on him

. (2.15)

Electric field strength vector obeys superposition principle: full vector

where

- The vectors of the fields of individual charges at a given point, calculated independently for each of the charges. Gravitational and electric fields are independent of each other. These fields can coexist at this point of space, and one of the fields in no way affects the other. Total force acting on a test particle, possessing and mass and charge, there is a vector sum of two forces and but does not make sense to summarize vectors and Since they have different dimensions, i.e. incommensurable. Are measurable and therefore have physical meaning Only forces.

Potential energy charge attendance from another charge (let's call it the source of the field) equal

. (2.17)

We divide this expression on and let's call a new amount electric potential

:

. (2.18)

there is a potential energy of charge unit and has dimension

[

] \u003d units. SGSE charge / cm \u003d

Units. SSSE Potential

Erg / units. SGSE charge.

Electric potential satisfies the principle of superposition: full potential

From the classical theory, it is known that work on the movement of charge from one point to another in the electrostatic field is equal to the difference in potential energies

in these two points. The corresponding work required to move between these dots of a single charge is equal to the change in the potential

.

, (2.20)

where there is a difference in potentials or voltagebetween two points.

Units of measurement of various electrical values \u200b\u200bare presented in Table 2.1.

Units of measuring electrical values \u200b\u200b- Table 2.1

Value	Unit in the SGSE system	Unit in SI system
		newton (H)
Communication between voltage and electric field In the form of the equation, the general relationship between the voltage and the electric field is equal. The electric field is called a gradient of electrical potential. Scalar: physical quantity with magnitude, but without direction. Vector: Physical size with a value and direction. Their unity, thus coincides with the unit σ, but the density of the electric flow is a vector value other than density surface charge. Between them are so-called equipotential surfaces. These are closed surfaces with constant potential. Electrical lines Flows are at a right angle to these equipotential surfaces.	units. SGSE charge	pendant (CL)
	units. GSE potential / cm	volt / meter (in / m)
	units. SSSE Potential
1 H \u003d 10 5 DIN; 1 unit. GSE potential \u003d 300V; 1 CL \u003d 3 · 10 9 units. SGSE charge

The expressions obtained above for tensions, forces and potentials of both gravitational and electric fields are valid in cases where the masses or charges of sources of these fields are distributed in the sphere or are point, i.e. They have infinitely small sizes.

However, real physical bodies do not have the right spherical form And are not point. Therefore, the ratio obtained above is not suitable for them. However, thanks to the principle of the superposition of the fields, any extended body can be viewed as a set of a large number of "point" bodies and calculate the fields by summing off contributions from all of them.

The trial mass in the gravitational field is always experiencing the force of attraction to the source of this field, because the power lines of the field of gravity are always directed to the source. The electric test charge may either repel, or attract the charge to the source of the field depending on the signs of both charges. There was also agreed to choose a direction silest lines The electric field so that it coincides with the direction of force acting with any source charge sign on a positive test charge. The power lines in the case of a positively charged source of the field are directed along radius from it, and in the case of a negatively charged source - by radius to it. This agreement coincides with what is accepted for the electric field strength vector.

Where do the power lines stretch? If we had an isolated charge, the power lines in the form of straight lines would be in infinity. But the existence of an isolated charge is physically impossible. The whole universe as a whole consists of the same number of positive and negative charges and therefore electrically neutral. Separate bodies can be charged, but this is achieved by the spatial separation of positive and negative charges in the original neutral bodies.

Consider the case of two bodies with equal and variepete charges. As usual, you can construct a picture of power lines, measuring or calculating the amount and direction of force acting on a positively charged trial body. The power lines come from the body with a positive charge and for smooth curves included in a negatively charged body. Therefore, electric power lines begin on positive and ends on negative charges. This is one of the most important results of theoretical electrostatics. It differs from the case of a gravitational field where there are no specific points, where the power lines would begin, and at the same time they extend to infinity.

The form of power lines for various geometric configurations can be determined by various methods that give illustrative fields of fields. For example, in oil weigh pollen of plants, and then this suspension is poured around the studied charge system. The electric field causes the separation of charges on the particles. One end of the particle becomes negative, and the other is positively charged, but in general the particle remains electrically neutral. This phenomenon is called electric polarization.Polarized pollen particles are oriented along the power lines, thereby making their visible shape. A number of configurations obtained in this method are shown in Fig.2.11.

When learning Fig. 2.11, but it can be seen that the picture of the power lines of the field of two equally charged bodies is the same as for the case of two identical masses (Fig. 2.7). Fig.2.11 illustrates two general results.

1. The electric field inside the solid or hollow conductor, according to which the current does not flow, is zero(Fig.2.11, B and D). Consider at first a solid conductor.

If there is some charge inside such a conductor and if the charges can move freely, then as a result of mutual repulsion, they scatter to the surface. If this surface charge creates the field inside the conductor, it will force the conduction electrons and then the electric current will appear. But it is in contradiction with the assumption that there is no current in the conductor.

Now consider the hollow conductor in the form of a ball. If the ball is charged, the charge will evenly distribute over its surface. On a trial charge, placed in the center of the ball, the force will not act. At this point, the resulting force and field strength are zero. However, what can be said about the field at other points of the cavity? Consider the case submitted fig. 2.12.

We define the resulting force acting on the trial charge at the point .

Fig.2.11. The power lines of the electric field of various charged bodies: a - two charges of the same name; b - two different charges; in - charged ring; r - charged conductor arbitrary shape; d - charged plate; E - a pair of differently charged plates

We construct two concentric cones aimed at opposite sides with a total vertex at the point . These cones cut the platforms on the opposite sides of the sphere. Since the charge is distributed uniformly, the force acting on the test charge from each of the cut segments of the ball is proportional to the segment area. Both forces are directed opposite to each other. But the segment is from the larger area further than a segment with a smaller area. An increase in force acting on a trial charge with an increase in the segment area (proportionally ) is compensated by a decrease in its due to the greater remoteness of the segment (proportionally

). As a result, both strengths turn out to be equal in such a way that the resulting force is zero. This argument can be extended to the rest of the surface of the sphere. As a result, it turns out that the force will not act on the trial charge. Exactly the same conclusion is also obtained for all other points inside the ball. Consequently, the electric field inside the spherical shell is missing. Applying a more complex mathematical method, you can get the same result not only for the sphere, but also for any other closed surface.

Infinite plane charged with surface density Charge: To calculate the tension of the electric field created by an infinite plane, we highlight the cylinder in the space, the axis of which is perpendicular to the charged plane, and the base is parallel to it and one of the bases passes through the field the field you are interested in. According to the Gauss Theorem, the stream of the electric field strength vector through the closed surface is:

F \u003d, on the other hand, it is: f \u003d e

We equate the right parts of the equations:

Express \u003d - through the surface density of charge and find the electric field strength:

We will find the tension of the electric field between the variemlessly charged plates with the same surface density:

(3)

Find the field outside the plates:

; ; (4)

The tension of the field of the charged sphere

(1)

F \u003d (2) t. Gauss

for R.< R

; because (There are no charges inside the sphere)

For R \u003d R

( ; ; )

For R\u003e R

The intensity of the field created by a ball charged uniformly throughout

Volumetric charge density

Distributed by the ball:

For R.< R

(; F \u003d)

For R \u003d R

For R\u003e R

The work of the electrostatic field to move charge

Electrostatic field - Al. Field of still charge.
Fel, acting on the charge, moves it, making a slave.
In a homogeneous electric field FEL \u003d QE - constant value

Work field (email) does not dependfrom the form of the trajectory and on a closed trajectory \u003d zero.

In case in the electrostatic field point charge Q From point 1 to point 2 along a trajectory (Fig. 1), another point charge Q 0 moves, then the force that is applied to the charging makes some job. The operation of the force F on the elementary movement of DL is equal as d l./ cosα \u003d dr, then Work when moving the charge Q 0 from point 1 to point 2 (1) It does not depend on the move path, but is determined only by the positions of the initial 1 and the final 2 points. It means that the electrostatic field of a point charge is potential, and the electrostatic forces - conservative from formula (1) it can be seen that the work that is performed when the electrical charge is moved in an external electrostatic field along an arbitrary closed path L, is zero, i.e. (2) If as a charge, which is moved in an electrostatic field, take a single point positive charge, the elementary operation of the field forces on the path DL is equal to EDL \u003d E L.d. l.where E. L. \u003d ECOSα is the projection of the vector E in the direction of elementary movement. Then formula (2) can be represented as (3) integral Circulated intensity vector circulation. So, circulation of tension vector electrostatic field Along any closed contour is zero. The power field that has a property (3) is called it. From the equality zero circulation of vector E it follows that the lines of the electrostatic field strength cannot be closed, they necessarily begin and end on charges (on positive or negative) or go to infinity. Formula (3) is true only for the electrostatic field. In the future, it will be shown that due to the case of moving charges, condition (3) is not true (for it, the circulation of the tension vector is different from zero).

Circulation theorem for the electrostatic field.

Since the electrostatic field is central, then the forces acting on the charge in such a field are conservative. As it is elementary workwhich field forces are produced above a single charge, the operation of conservative forces on a closed loop is equal to

Potential

The system "charge is an electrostatic field" or "charge - charge" has potential energy, just as the system "gravitational field - the body" has potential energy.

The physical scalar value characterizing the energy state of the field is called potential This point of the field. The box is placed in the field Q, it has the potential energy W. The potential is the characteristic of the electrostatic field.

Recall the potential energy in the mechanics. Potential energy is zero when the body is on Earth. And when the body raises some height, they say that the body has potential energy.

Regarding potential energy in electricity, there is no zero level of potential energy. It is chosen arbitrarily. Therefore, the potential is a relative physical value.

The potential energy of the field is the work that electrostatic force performs when the charge is moved from this point of the field to a point with zero potential.

Consider a particular case when the electrostatic field is created by an electrical charge Q. To study the potential of such a field there is no need to make a charge Q. You can calculate the potential of any point of such a field located at a distance of R from charge Q.

The dielectric permeability of the medium has a known value (tabular), characterizes the medium in which there is a field. For air it is equal to one.

Potential difference

The operation of the field to move the charge from one point to another, is called the difference in potentials

This formula can be represented in another form.

Superposition principle

The potential of the field created by several charges is equal to algebraic (taking into account the potential sign) the sum of the potentials of the fields of each field separately

This is the energy of a system of fixed point charges, the energy of a secluded charged conductor and the energy of the charged condenser.

If there is a system of two charged conductors (condenser), then the total energy of the system is equal to the amount of own potential energies Conductors and energy of their interaction:

Energy of electrostatic field Spot charge systems are equal to:

Uniformly charged plane.
The voltage of the electric field created by an infinite plane charged with the surface density of the charge can be calculated using the Gauss Theorem.

From symmetry conditions it follows that vector E. Everywhere perpendicular to the plane. In addition, in symmetrical relatively plane points vector E. It will be the same in size and opposite to the direction.
As a closed surface, we choose a cylinder whose axis is perpendicular to the plane, and the bases are arranged symmetrically relative to the plane, as shown in the figure.
Since the tension lines are parallel to the sideline surface of the cylinder, the flow through the side surface is zero. Therefore, the flow of the vector E. through the surface of the cylinder

,

where is the base area of \u200b\u200bthe cylinder. The cylinder carries out of the charge plane. If the plane is in a homogeneous isotropic medium with relative dielectric constant, then

When the field strength does not depend on the distance between the planes, this field is called homogeneous. Graph addiction E. (x.) For a plane.

The difference of potentials between two dots located at distances R. 1 I. R. 2 from the charged plane, equal

Example 2. Two uniformly charged planes.
Calculate the voltage of the electric field created by two infinite planes. Electric charge is distributed evenly with surface density and. Field tension will find as the superposition of the tensions of the fields of each of the planes. The electrical field is different from zero only in space between the planes and is equal.

Potential difference between planes where d -distance between planes.
The results obtained can be used for approximate calculation of fields created by flat plates of finite sizes if the distances between them are much less than their linear dimensions. Noticeable errors of such calculations appear when the fields are considered near the edges of the plates. Graph addiction E. (x.) For two planes.

Example 3. Thin charged rod.
To calculate the voltage of the electric field created by a very long charged with a linear density of the charge with a rod, we use the Gauss theorem.
At sufficiently large distances from the ends of the rod line of the voltage of the electric field are directed radially from the axis of the rod and lie in the planes perpendicular to this axis. At all points that are equidistant of the rod axis, the numerical voltage values \u200b\u200bare the same if the rod is in a homogeneous isotropic medium with a relative dielectric
permeability.

To calculate the field strength in an arbitrary point located at a distance r. from the axis of the rod, spend a cylindrical surface through this point
(See Figure). The radius of this cylinder is equal r., and his height h..
Thread streams through the upper and lower base of the cylinder will be zero, since the power lines do not have the components, normal to the surfaces of these bases. At all points of the side surface of the cylinder
E. \u003d const.
Consequently, the full flow of the vector E. through the surface of the cylinder will be equal

,

By the Gauss Theorem, vector stream E. equal to the algebraic amount of electrical charges located inside the surface (in this case of the cylinder) divided into the product of the electrical constant and relative dielectric constant of the medium

where the charge of the part of the rod, which is inside the cylinder. Consequently, the tension of the electric field

The potential difference of the electric field between two dots located at distances R. 1 I. R. 2 From the axis of the rod, we will find using the bond between the tension and the potential of the electric field. Since the intensity of the field varies only in the radial direction, then

Example 4. Charged spherical surface.
The electric field created by the spherical surface, which is evenly distributed by an electrical charge with surface density, has a centrally symmetrical character.

Voltage lines are directed by radius from the center of the sphere, and the vector module E. Depends only on distance r. from the center of the sphere. To calculate the field, choose a closed spherical surface of the radius r..
At r O. E. = 0.
The field strength is zero, since there is no charge inside the sphere.
At R\u003e R (outside the sphere), according to the Gauss Theorem

,

where - relative the dielectric constant Environment surrounding the sphere.

.

The tension decreases by the same law as the intensity of the field of the point charge, i.e. according to the law.
At r O. .
At R\u003e R (outside the sphere) .
Graph addiction E. (r.) For the sphere.

Example 5. Charged in volume ball from dielectric.
If the ball is radius R. From a homogeneous isotropic dielectric with relative constant is evenly charged in volume with a density, the electrical field created by them is also centrally symmetric.
As in the previous case, choose a closed surface to calculate the vector flow E. in the form of a concentric sphere, the radius of which r. It may vary from 0 to.
For r. < R. Stream vector E. Through this surface will be determined by charge

So that

For r. < R. (inside the ball) .
Inside the ball, tensions increase in directly proportional to the distance from the center of the ball. Outside the ball (with r. > R.) in medium with dielectric constant, vector stream E. Through the surface will be determined by charge.
With R O\u003e R O (outside the ball) .
On the border "Ball - environment"The tension of the electric field is changed by a jump, the value of which depends on the ratio of the dielectric permeability of the ball and the environment. Dependency schedule E. (r.) For a ball ().

Outside a bowl ( r. > R.) electric field potential changes by law

.

Inside the ball ( r. < R.) The potential is described by the expression

In conclusion, we give the expressions to calculate the tensions of fields of charged bodies, of various shapes

Potential difference
Voltage - The difference of potential values \u200b\u200bin the initial and end-screen trajectories. Voltage Numerically equal to the operation of the electrostatic field when moving a single positive charge along the power lines of this field. The potential difference (voltage) does not depend on the choice coordinate systems!
The difference unit of potentials The voltage is 1 V if, when moving a positive charge in 1 CL along the power lines, the field makes a job in 1 J.

Conductor - this is solidin which there are "free electrons" moving within the body.

Metal conductors are generally neutral: they row negative and positive charges. Positively charged - these are ions in nodes crystal lattice, negative - electrons freely moving through the conductor. When the conductor reports an excess amount of electrons, it is charged negatively, if the conductor "select" some number of electrons is charged, it is charged positively.

Excess charge is distributed only on the outer surface of the conductor.

1 . The field strength at any point inside the conductor is zero.

2 . The vector on the surface of the conductor is directed along the normal to each point of the surface of the conductor.

From the fact that the surface of the ExpiPotential Explorer follows that directly in this surface the field is directed along the normal to it at each point (condition 2 ). If it were not so, then under the influence of the tangent component of the charges would be in motion on the surface of the conductor. those. Equilibrium charges on the conductor would be impossible.

Of 1 It follows that since

Inside the exercise of redundant charges no.

Charges are distributed only on the surface of the conductor with some density s. and are in a very thin surface layer (its thickness is about one or two interatomic distances).

Density charge - This is the amount of charge, falling per unit of length, area or volume, thus define linear, surface and volumetric charge density, which are measured in the SI system: in the culons per meter [CL / M], in the quarces per square meter [CL / m² ] And in the cubes on the cubic meter [CL / m³], respectively. Unlike the density of the substance, the charge density can have both positive and negative values, this is due to the fact that there are positive and negative charges.

Total task of electrostatics

Tension vector

according to Theorem Gauss

- Poisson equation.

In the case - there are no charges between the conductors, we get

- laplace equation.

Let the boundary conditions on the surfaces of the conductors: Values ; Then this task has a single decision according to uniqueness theorem.

When solving the problem, the value is determined and then the field between the conductors is determined by the distribution of charges on the conductors (by the vector of tension at the surface).

Consider an example. We find tensions in the empty cavity of the conductor.

The potential in the cavity satisfies the Laplace equation;

potential on the walls of the conductor.

The solution of the Laplace equation in this case is trivial, and on the theorem of the uniqueness of other solutions is not

. There are no fields in the cavity of the conductor.

Poisson equation - Elliptic differential equation in private derivatives, which, among other things, describes

· Electrostatic field,

· Stationary temperature field,

· Pressure field,

· Speed \u200b\u200bpotential in hydrodynamics.

It is named after the famous French physics and mathematics of Simeon Denis Poisson.

This equation has the form:

where is the Laplace or Laplacian operator, and is a real or complex function on a variety.

In a three-dimensional Cartesian coordinate system, the equation takes shape:

In the Cartesian coordinate system, the Laplace operator is written in the form and the Poisson equation takes the form:

If a f. It seeks zero, the Poisson equation turns into the Laplace equation (the Laplace equation is a special case of the Poisson equation):

The Poisson equation can be solved using the Green function; See, for example, an article shielded Poisson equation. there is various methods For numerical solutions. For example, an iterative algorithm is used - the "relaxation method".

We will consider a secluded conductor, i.e. the conductor, significantly removed from other conductors, bodies and charges. Its potential, as is known, is directly proportional to the charge of the conductor. From experience it is known that different conductors, while being equally charged, have different potentials. Therefore, for a solitary conductor, it can be recorded (1) called electrical capacity (or simply capacity) of a secluded conductor. The capacity of the secluded conductor is set by charge, the message of which the conductor changes its potential per unit. The capacity of the secluded conductor depends on its size and shape, but does not depend on the material, the shape and size of the cavities inside the conductor, as well as its aggregate state. The reason for this is the fact that excess charges are distributed on the external surface of the conductor. Capacity also does not depend on the charge of the conductor or on its potential. The unit of electrical capacity - Farad (F): 1 F - the capacity of such a secluded conductor, in which the potential changes to 1 V when the charge of 1 CL is reported. According to the formula for the potential of a point charge, the potential of a solitary bowl of R radius R, which is located in a homogeneous medium with a dielectric constant ε, is equal to formula (1), we obtain that the container of the ball (2) follows from this that the capacity of 1 f would have a secluded ball, Located in vacuo and having a radius R \u003d C / (4πε 0) ≈9 10 6 km, which is approximately 1400 times more than the radius of the Earth (the electrical capacity of the Earth is C≈0.7 MF). Consequently, Farad is quite large, therefore, in practice, dolly units are used - millifarara (MF), microfarad (ICF), nanofarad (NF), picophadera (PF). From formula (2), it is also necessary that the unit of electrical constant ε 0 - pharand per meter (F / M) (see (see (78.3)).

Capacitor (from lat. cONDENSARE - "Seal", "thicken") - a two-solid, with a certain value of tank and low ohmic conductivity; Device for accumulation and electric field energy. The condenser is a passive electronic component. It usually consists of two electrodes in the form of plates (called planmarks) separated by a dielectric whose thickness is made compared to the size of the plates.

Poker

The main characteristic of the capacitor is its capacitycharacterizing the ability of the capacitor to accumulate an electrical charge. In the designation of the capacitor, the value of the rated capacity appears, while the real container can vary significantly depending on many factors. The actual capacitance of the capacitor determines its electrical properties. So, by definition of the container, the charge on the plug is proportional to the voltage between the plates ( q \u003d Cu.). Typical capacitors capacitance values \u200b\u200bare from picophade units to thousands of microfrades. However, there are capacitors (ionistors) with a capacity to dozen Farad.

Capacity of a flat capacitor consisting of two parallel metal plates area S. Each distance d. From each other, in the SI system is expressed by the formula: where -to-smelly dielectric permeability of the medium that fills the space between the plates (in the vacuum is equal to one), - electrical constant, numerically equal to 8,854187817 · 10 -12 f / m. This formula is valid only when d. Many less linear dimensions of the plates.

For large capacities, capacitors are connected in parallel. In this case, the voltage between the plates of all capacitors is equally. Total battery capacity parallel The connected capacitors are equal to the sum of the containers of all capacitors included in the battery.

If all of all parallel condensers are the distance between the plates and the properties of the dielectric are the same, then these capacitors can be represented as a large capacitor separated into fragments of a smaller area.

With a consistent connection of the capacitors of the charges of all capacitors are the same, since they come from the power source only to external electrodes, and on the internal electrodes they are obtained only due to the separation of charges that previously neutralized each other. Total battery capacity sequence The connected condensers are equal

Or

This container is always less than the minimum capacitance of the capacitor included in the battery. However, with a sequential connection, the possibility of a breakdown of capacitors decreases, since only a capacitor accounts for only a part of the difference in voltage source potentials.

If the plated area of \u200b\u200ball capacitors connected in series is the same, then these capacitors can be represented as a large capacitor, between the plates of which there is a stack from the dielectric plates of all components of its capacitors.

[edit] Specific tank

Capacitors are also characterized by a specific capacity - the ratio of the container to the volume (or mass) of the dielectric. The maximum value of the specific capacity is achieved with a minimum thickness of the dielectric, however, its breakdown voltage decreases.

IN electrical chains Different are applied methods of connecting condensers. Compound condenser Can be made: sequence, parallel and seriously parallel (The latter is sometimes called the mixed connection of capacitors). The existing types of condensers are shown in Figure 1.

Figure 1. Methods for connecting condensers.