Using the Gauss theorem, you can easily calculate the tensions in some cases. electric field Around the charged body, if the specified charge distribution has any symmetry and the general structure of the field can be guess in advance.
An example is the task of calculating the field of thin-walled hollow homogeneous charged long cylinder of radius R.. This task has an axial symmetry. For symmetry considerations, the electric field should be directed along the radius. Therefore, to use the Gauss theorem, it is advisable to choose a closed surface S. in the form of a coaxial cylinder of some radius r. and length l.closed with both ends (Fig. 1.3.4).
In the equation, the symbol is equal. To obtain additional intuition by the law of Gauss, let's look at the Gauss law in integral formFor which we take an arbitrary volume having a border. I probably made things less clear, but let it go very quickly. As an example, look at the drawing.
We rewrite equation with a large number of members defined in the equation. An example with a cube in Figure 1 can help make it clear. We can rewrite any field through its tangential and normal components, as shown in the figure. Consequently, the Gauss law is a mathematical statement that a complete electric flow coming out of any volume is equal to total charge Inside. Therefore, if the volume in question does not have a charge inside it, then the net stream of electrical flow from this area is zero if there is positive charge, Then there is a positive value of an electric flux leaving any volume that surrounds the charge.
For r. ≥ R. The entire stream of tension vector will pass through the side surface of the cylinder, the area of \u200b\u200bwhich is 2π rLSince the flow through both bases is zero. The use of the Gauss Theorem gives:
This result does not depend on the radius R. The charged cylinder, so it is applicable to the field of a long uniformly charged thread.
If there is a negative charge in the volume, then there is a negative exit value of the electrical stream. What does it mean? The Gauss law states that the electrical charge acts as a source or an absorber for electric fields. If you use the aqueous analogy again, the positive charge causes a stream from the volume - this means that a positive electrical charge is similar to a source. Conversely, a negative charge leads to a stream in volume - this means that the negative charge acts as a sink.
This gives us a lot of intuition on how fields can physically act in any scenario. For example, impossible situations for the electric field are possible here, as decided by the Universe in the Gauss law. This means that the opposite charges are attracted, and negative charges repel. Thus, the Law Gauss means the following.
To determine the field strength inside the charged cylinder, you need to build a closed surface for the case r. < R.. By virtue of the symmetry of the problem, the stream of tension vector through the side surface of the Gaussian cylinder should be in this case φ \u003d E. 2π. rL. According to the Gauss Theorem, this stream is proportional to the charge, which was inside a closed surface. This charge is zero. It follows that the electric field inside a uniformly charged long hollow cylinder is zero.
If you understand the above approval, you understand the Gauss law, probably better than mathematicians that come up with ultra-sized mathematics to explain physical phenomena! Intuition trumps complication, always. Usually you use Gaussian set to calculate vector fields - for example, a gravitational field or an electric field. It is always relevant, but not always useful. But if you want to calculate the field in which you already know that for any reason - symmetry, then you must shout from the alarms!
For then the Offer Gauss will save you a lot of work. There are mainly three symmetries for which Gaussian theorem is ideal.
- You can use the Gaussian Ball as a volume.
- Spherical symmetry - here you are using the "Gaussian ball".
Potential electrostatic field. The theorem on the circulation of the electrostatic field in the integral form. The theorem on the circulation of the electrostatic field in differential form. Potential of solitary point charge. Potential difference. Equipotential surface. Property of potential additivity.
The potential of the electrostatic field.The field of conservative forces can be described not only by the vector function, but the equivalent description of this field can be obtained by defining a suitable scalar value at each point. For the electrostatic field, such a magnitude is potential of the electrostatic fielddetermined as the ratio of potential energy of the test charge q. to the magnitude of this charge, j \u003d W. P / q.From where it follows that the potential is numerically equal to the potential energy, which has a single positive charge at a given point. The potential measurement unit serves as volts (1 V).
Although there are in fact there are no infinitely extended plates, but you can close the field of a large capacitor capacitor with this calculation if you do not approach too close to the edge of the plate. Cylindrical symmetry is often found in electrodynamics - each round cable, also called a coaxial cable, has such symmetry! This type of symmetry, therefore, is also called axis or rotational symmetry. You always have a spherical symmetry, if it is in any way a spherical problem, and the field strength depends only on the distance to the center of the sphere.
Potential electrostatic field - scalar valueequal to the ratio of the potential energy of charge in the field to this charge:
Energy characteristics Fields at this point. The potential does not depend on the size of the charge placed in this field.
Circulation theorem
Let the field created by a dot charge system. We calculate the integral from tension on a closed trajectory.
Gaussian theorem applied to electrostatics
Also included fields of point objects! For example, you can calculate the gravitational field of the Earth or the electrical electrical field. Gauss Theorem is applicable to an electric field. Thus, you have a somewhat modified Gaussian theorem for calculating electrostatic electric fields. With a famous charge.
Annotation: main course of electricity and magnetism. The electrostatic field in Vacuum is the main concepts and laws. The law of the coulon and its application, the principle of superposition. Description of the electrostatic field, intensity, email. potential. Gauss and its application. Potential energy Charge in El. field, electric potential, EL. potential. Electric field from charged wire, distribution of charge on the surface of the charged wire, electrostatic induction. Capacity insulated conductor. Condenters, capacitors.
This statement is the essence of the circulation theorem. In mathematics, a similar integral is called circulation.
Concept of circulation
Let a vector field exist in some space space.
Here is a single vector tangent to the contour at this point, directed towards the positive circuit bypass.
Concept of circulation
Electrostatic field in dielectrics - dielectric polarization, polarization vector, dielectric susceptibility and relative resolution. Vector electric induction, Generalized Gaussian theorem. Dielectric materials and their use. Energy electrostatic field. Electrostatic counters. Stable electricity - Types of current, current size, current density. Ohm law, wire resistance, resistance of the resistor. Operation and power of electric current.
The dependence of temperature resistance, superconductivity, nonlinear conductors. Circhhoff law, electrical network solutions. Main measurement electrical quantities. Contact potential difference, thermoelectric phenomena. Current conductor in semiconductors, vacuum, gases and electrolytes. The stationary magnetic field is the main magnetic phenomena, the law Bio-Savart - Laplace, the power of Lorentz. Movement of charged particles in magnets. and he. field. Magnetic Induction Flow, Law full current Ampere.
There is an agreement that the positive direction of circuit bypass (direction) is chosen so that the area covered by the contour remains when around bypass on the left.
We will remind, briefly, how can I "construct" a curvilinear integral. To do this, select the point on the circuit, show in it the vector, at the same point show a piece vector tangent, calculate the scalar product, split the outline into small elements, the length of the element to designate, calculate the product; do it for all elements of the contour; Conduct the summation of the results, ascertaining the element of the contour length to zero - go from the summation to integration.
Just like the flow, circulation is another characteristic of the properties of the vector field. Namely, circulation characterizes the degree of vulnence of the vector field.
Example:if you can take a turbine as a "meter" of circulation of the fluid velocity field, then if it rotates, circulation is not zero.
Circulation - This is the integral characteristics of the field.
The concept of rotor
The field in its structure can be quite inhomogeneous. Circulation does not give a detailed field characteristic. Therefore, let's start pulling the integration contour to any point M. (Reduce the turbine). The circulation will strive for zero, but also the area covered by the contour will also strive for zero. And their ratio gives a finite number.
Turbine can be oriented in space in three independent ways. Therefore, in this way, 3 independent numbers can be obtained, and three numbers are a vector, therefore, a vector characteristic of the field, which is called the rotor.
Rotor- This is a local or differential characteristic.
The surface, in all points of which the potential of the electric field has the same values, is called equipotential surfaceor the surface of equal potential .
Power lines The electrostatic fields are always perpendicular to the equipotential surfaces.
Equipotential surfaces of the Coulomb field of point charge are concentric spheres. In fig. 1.4.3 Presents patterns of power lines and the equipotential surfaces of some simple electrostatic fields.
When uniform field equipotential surfaces represent a system of parallel planes.
If a trial charge q. performed small moving along force line From point (1) to point (2), then you can write:
This ratio in the scalar form expresses the relationship between field strength and potential. Here l. - Coordinate, counted along the power line.
Of the principle of superposition of fields of fields created by electric chargesThe principle of superposition for potentials is followed:
| φ \u003d φ 1 + φ 2 + φ 3 + ... |
Potential difference (voltage)between the two points is equal to the function of the field when the charge is moved from the starting point to the final to the module of this charge:
U. \u003d φ 1 - φ 2 \u003d -Δφ \u003d A / Q,
A \u003d - (W p2 - w p1) \u003d -q (φ 2 - φ 1) \u003d -qδφ
The difference in potentials is measured in volts (B \u003d J / CL)
The relationship between the voltage of the electrostatic field and the difference of potentials:
E. X. = Δφ / Δ x.
The voltage of the electrostatic field is directed toward the decrease in the potential.
In physics additivity The values \u200b\u200bmean that the value relating to the system as a whole is equal to the amount of quantities relating to its components. Such values \u200b\u200bare also called extensive, Unlike intensive (for example, temperature, density, etc.).
Obviously, at the same distance r. From thread meanings E. will be the same, so 
According to Theorem Gauss
where 
- The charge enclosed inside the Gaussian cylinder. Then
and 
- Field tension of the charged thread at a distance r.
from her.
2
. Field of an infinite homogeneous charged plane. Surface density charge 
in all points plane the same . The intensity of the field is perpendicular to the plane. In symmetrical relative to the plane points, the field strength is the same in size and the opposite direction. We construct a cylindrical surface with forming, perpendicular to the plane and bases 
(Fig.1.1.11). By the symmetry 
.
P 
saws through the side surface is zero, since the vector 
perpendicular to this surface, so the total flow through the surface of the cylinder is equal 
, I. 
.
3. Match the electric field created by two variemlessly charged planes with surface charge densities 
and 
. Obviously, the tensions of the planes of the planes are directed in one direction (from a positive plane to negative, Fig.1.1.12), and 
esulting tension 
where 
- The field strength of one charged plane. Finally get
4. Extract the electric field strength created by the charged sphere of the radius R.. Pack of sphere q., its surface density 
To determine the tension, we construct a Gaussian surface in the form of a radius sphere r., the center of which coincides with the center of the charged sphere (Fig.1.1.13).
For r.≤
R.
There is no charge inside the Gaussian surface, since the entire charge is distributed over the surface of the sphere. According to Theorem Gauss 
or 
, hence, 
- The tension of the electric field inside the charged sphere is zero.
For
inside the Gaussian surface hits the whole charge q. spheres. Due to the central symmetry field, tension at a distance r. from the center of the sphere everywhere the same, and 
or 
wherein 
, then, and 
With growth r. Values E.decrease proportionally 
(Fig.1.1.14). On the surface of the sphere, tensions are rapid 
5. Create an electric field created by a volumetric radius ball R. . Bowl Charge Density ρ. Gaussian surface construct in the form of a sphere, the center of which coincides with the center of the ball, and the radius is equal r. (Fig.1.1.15) .
For 
inside the Gaussian surface gets charge
then on the Gaussian theorem 
, I. 
. On the surface of the ball with r.=
R. Tension 
.
For 
inside the Gaussian surface hits the whole charge 
, I. 
From here 
On the surface of the sphere 
those. 
and the jump intensity does not occur. Addiction 
presented in Fig.1.1.16.
Lecture 4.
1.1.9. The factory nature of the electrostatic field. The work of the field forces when moving charges. Circulation and tension vector rotor
Work performed by the power of the electrostatic field when charging charge 
on cut 
equal to:
Work on the movement of a single positive charge is numerically equal to
Work performed when moving a single positive charge on the final way 
equal
.
(1.1.2)
Here 
- Cool strength, which is central. From the mechanics it is known that the field of central forces is conservative. Consequently, the operation of the electrostatic field to move the charge does not depend on the trajectory, but is determined only by the initial and finite points. Work on the closed path is zero. The field with such properties is called potential. Then from (1.1.2) we have:
(1.1.3)
- circulation vector along the closed path is zero.The field with such properties is called potential.
We prove the potential nature of the electrostatic field.
Consider first work electric forces In the field of elementary point charge . The work of these forces with infinitely small movement 
trial single positive charge is equal to:
,
where 
- Projection of the Projection of the Trial Charge on the radius-vector 
conducted from the exciting charge field . Fig.1.1.17 shows that 
- This is the increment of the numerical value of the radius-vector , that is, an increase in the distance of the trial charge 
from charge . Therefore, work 
it can be represented as a complete differential of the scalar point function 
:
where 
- numerical value of the radius vector . Then work on moving a single positive charge from point 
exactly 
on the final path equal to:
g. 
de 
and 
- distances of the initial and endpoint paths from charge . Thus, the work of electric forces on an arbitrary path in the stationary field
the elementary point charge does depends on the positions of the initial and endpoints of this path and does not depend on the shape of the path. Figure 1.1.18 Work on the way equal to work on the way 
: redundant work performed on the way 
when moving the trial charge outside the range of radius 
is compensated by negative work performed upon subsequent approximation of the trial charge to the charging at the last section of the road 
. Thus, the field of a still point charge is a potential field.
Obviously, the amount of potential fields is also a potential field (as if the work of the considerable forces does not depend on the form of the path, then the work is independent of it that does not depend on it). The field of an arbitrary charge system can be considered as the sum of the fields of each of the point charges, so every electrostatic field has a potential field.
By definition, projection 
on arbitrary direction field 
equal
,
where 
- Infinitely small platform passing through the point 
perpendicular to the vector .
Since the circulation of the vector 
on a closed contour is equal 
,
T.
, or 
.
(1.1.4)
Since the direction selected arbitrarily, then projection 
on any directions is 0, therefore, from (1.1.4) 
in all points of the electrostatic field, that is, the electrostatic field is
bereviezhered. This result can be obtained from the Stokes Theorem. Expressions (1.1.3) and (1.1.4) are equivalent.
