In a number of sections of the course of general physics, vector fields are considered (for example, an electrostatic field, a magnetic field).
The concept of vector flow through some surface is often used by the lack of such fields. Ras-look this concept.
Let there be an electric field in some space of space. Choose an elementary platform in this field. ds.. Let Normal to this site n. Forms angle with electric field strength vector (vector module n. = 1).
The stream of the tension of the electric field through this platform is called the value equal to
where dF - elementary stream of tension vector, E. - vector intensity vector within an infinitely small area d.s. .
Composition En It is scalar, therefore the strength vector stream is a scalar value.
Sometimes the work n. ds. Replaced with vector d.s. which is directed perpendicular to the plane of the site; Module vector ds. equal to the area of \u200b\u200bthe elementary platform.
Strength strength through the final area s. Raven
.
Depending on the corner value between the normal to the site and the vector E. The flow can be positive and negative. If the angle between vectors E. and n. Acute, then the flow is positive if stupid is negative.
Note that the direction of the vector n. It is selected before solving the problem arbitrarily (perpendicular to the surface can be sent to two mutually opposite sides). Therefore, the voltage vector flow sign is determined by the choice of vector direction. n.
If the surface is closed, the strength of the voltage vector is equal
,
i.e. integral is taken on a closed surface s..
In this case, the vector is customary n. out from the surface. In this case, the flow through a closed surface is positive if summary chargeembedded by a closed surface is positive.
The dimension of the strength of the tension [F] \u003d c. M \u003d n. m 2 / cl.
1.6. Theorem Gaussa
The Gaussian Theorem is the main theorem of electrostatics. It establishes the relationship between the stream of the voltage vector through the closed surface and the total charge covered by this surface.
Consider this theorem.
Let the electric field created by a positive point charge q..
Find the flow of the electric field strength vector through a closed surface that covers this charge.
As a surface, choose the sphere of radius r.whose center coincides with the charge q..
Since the charge that creates the field is positive and dispensed in the center of the sphere, the angle between the vector E. and vector n. At all points of the surface is zero.
Therefore, the strength of the voltage vector through the elementary surface ds. will be equal EN. ds. = E.cos. ds. = E.cos0. ds. = EDS.
In other words, in the situation under consideration, the scalar product of tension vector electrostatic field On the vector of the elementary surface of the Armenia, the product of the modules of these eyelids.
The intensity of the field created by the Chech charge is equal to 
.
Since the charge is located in the center of the spherical surface, the distance from the charge to the surface in all its points is equally and equal r.. Consequently, the voltage vector module at all points of the spherical surface is the same: E. \u003d const.
The constant can be taken out for the sign of the integral, so the strength of the voltage vector through a closed surface in this case is equal to 
.
Integral from the elementary surface area s.taken over the entire surface is equal to the area of \u200b\u200bthis surface. s.. In this case, the surface is a sphere whose area s. \u003d 4. r. 2 .
Thus, the stream of voltage vector through a closed surface in this case is equal 
.
Substituting the expression to calculate the voltage, we get
.
Can show that field strength vector point charge through a closed surface will be equal 
and in the case when the charge is not in the center of the spherical surface.
Moreover, the flow will be the same, even if the surface will have any form.
If the surface covers several charges q. i. , the stream of each of the charges through the closed surface will be equal to 
. The total flow created by all charges will be equal 
.
Changing the sequence of summation and integration and considering that in accordance with the principle of superposition 
Receive 
where E.
- Vector voltage vector created by all charges covered by a closed surface.
So, the analysis allowed the following ratio:
.
This ratio has a versatile character and is called the Gauss theorem: the flow of the tension of the electric field through a closed surface is equal to the ratio of charges covered with this surface to the electric constant.
Please note: In the expression of the Gauss Theorem, there is no characteristics of the position of charges q. i. .
This means that the strength vector stream does not depend on how charges are covered by closed surface. Moreover, the strength of the voltage vector will not change if the mutual arrangement of charges covered by the surface will change.
The practical value of the Gauss Theorem is that it is considerably simplified by the calculation of fields created by symmetric charge distributions. In this case, you can choose the surface of such a form that 
where S.
- The area of \u200b\u200bthe closed surface, pierced by the electric field.
E R \u003d F s \u003d 4 2 \u003d 2 b m.
Example 2: The site S \u003d 3M2 is located in a uniform field 100 N / CL. How many lines crosses this platform if angle is 30º (Fig. 2.4).
E ┴ \u003d E SIN 300 \u003d 50 N / CL
F \u003d e ┴ · s \u003d 50 · 3 \u003d 150 lines
2.2. Stream vector stream.
So, on the examples, we showed that if the power lines are homogeneous
electric field tension e permeate some playgrounds, then the stream of tension (we used to call the number silest lines through the playground) will be determined by the formula
FR E \u003d ES \u003d ESCOS α \u003d EN S,
where E n is the product of the vector to normal to this site (Fig. 2.5).
And the value of F E here and is called the flow of the vector of electric field strength through the site, i.e. Definition:
The total number of power lines passing through the surface S is called the stream of the tension vector through this surface.
In vector form you can record
F E \u003d (E, S) is a scalar product of two vectors, |
|||
where vector s \u003d n s. | |||
Those. Stream vector |
|||
E is a scalar, which, depending on the magnitude of the angle α |
|||
can be both positive and negative . Consider (Fig. 2.6, 2.7).
For this configuration, the flow through the surface A, negative (calculate the number of power lines).
Fig. 2.6 Fig. 2.7
For Figure 2.6 - Surface A 1 surrounds positive charge And the flow here is directed outward, i.e.F E\u003e 0. Surface 2 - surrounds negative charge And here, E.< 0 направлен внутрь.
For Figure 2.7 - the flow will not be zero if the total charge inside the surface is not zero. Those. The flow depends on the charge. In this sense Theorem Gauss.
2.3. Theorem of Ostrogradsky-Gauss.
So, recall the flow stream of the electric field strength - is equal to the number of tension lines passing through the area S (Fig. 2.8).
dife \u003d EDSCOS α \u003d en ds |
R 2)
q 4 πR 2 \u003d Q
4 πε 0 R 2 2 2 ε 0
Those. in a homogeneous field f e \u003d es in arbitrary electric field
Fe \u003d ∫ EN DS \u003d ∫ E DS | |||
d s r \u003d ds n r - in vector form | |||
The orientation of the DS in the space is set using a single vector R. Those. |
|||
- The direction coincides with the direction of external normal to the surface. |
|||
Calculate the stream of the vector e through | closed surface | S, surrounding |
|
cook charge Q (Fig. 2.9.).
The center of the circle coincides with the charge center. The radius of the sphere S 1 is 1. IN
each point of the surface S 1 projection on the direction of external normal is the same and equal to
E n \u003d. | ||||||||
4 πε. | ||||||||
Then flow through S 1 | ||||||||
Fe \u003d ∫ en ds \u003d | 2 4 πr 1 2 \u003d | |||||||
4 πε0 R1 | ||||||||
Calculate the flow through S 2 (radius
Fe \u003d ∫ 4 πε Q r 2 ds \u003d
S 2 R 0 2
The strength lines e begin and end in infinity) from the continuity of the line it follows that the threadprint surfaces will be equal to the same magnitude:
Fe \u003d ∫ EN DS \u003d | |
The result is valid not only for one charge, but also for any number of arbitrarily spaced charges inside the surface.
Fe \u003d ∫ EN DS \u003d | ΣQ. | - Theorem Gaussa | ||
The stream of the electric field strength through a closed surface in vacuo is equal to the algebraic sum of all charges located inside the surface divided by ε0.
When calculating the flow through a closed surface, the vector of normal N
a positive stream is created by a limited surface, the lines are included in the volume - negative flow.
If there is another S 3 surface between our spheres, not
help charge, then, as can be seen from (Fig. 2.9). Each line e line will cross this surface twice: once from a positive side - will enter
in the surface S 3, another time - from a negative side - will come out of the surfaces 3.
IN result the algebraic amount of tension lines passing through
the closed surface S 3 will be zero, i.e. The total flow passing throughs 3 is zero.
Thus, for a point charge Q, a full flow through any closed surface S will be equal to:
Fe \u003d 0 - if the charge is located outside the closed surface and this result
it does not depend on the shape of the surface and the flow sign coincides with the charging sign.
In general electric charges There may be "smeared" with some bulk density ρ different in different places of space. Recall another concept - the volumetric density of the charge
ρ = | |||||
where DV is infinitely small volume.
Under a physically infinitely small amount of DV, it is necessary to understand this volume that, on the one hand, is sufficiently small, so that within its density charge is considered the same, and on the other - it is large enough so that the discreteness of the charge cannot be manifested. The fact that any charge is kathed in an integer number of elementary charge or orp + (proton). Then total charge
Σ qi \u003d ∫ ρDV | |||
Then, from the Gauss Theorem (2.3.7.) We write | |||
ε 0V ∫ | |||
This is another form of the Gaussian Theorem, if the charge is continuous.
It is necessary to pay attention to the following circumstance: while R
the field E itself depends on the configuration of all charges, the flux e through an arbitrary closed surface is determined only by the algebraic sum of charges inside the surfaces. This means that if you move the charges, the TE will change throughout, and on the surfaces, but the flow of the vector through this surface will remain the same. Amazing typical property of tension.
2.4. Differential form of the Gaussian Theorem.
IN it establishes communication between the volumetric charge densityρ and change in the vicinity of this point of space
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Home\u003e Abstract\u003e Physics
Introduction ..................................................................... ..... 3
Electric field strength. Gaussian theorem in an integrated form ............................................................ 4
The emergence and development of the theory of the electromagnetic field ......... .. 8
Conclusion ........................................................................ fifteen
List of references used ....................................... sixteen
Introduction
According to modern ideas, electrical charges do not act directly. Each charged body creates an electric field in the surrounding space, which has a power action on other charged bodies.
The main property of the electric field is an effect on electrical charges with some force. Thus, the interaction of charged bodies is carried out not directly by their impact on each other, but through electrical fields surrounding charged bodies.
For the quantitative determination of the electric field, the power characteristic is introduced - the electric field strength.
The tension of the electric field is called a physical value equal to the ratio of strength with which the field acts on a positive test charge, placed at this point of space, to the magnitude of this charge:
Electric field strength - vector physical value. The direction of the vector E coincides at every point of space with the direction of force acting on a positive trial charge.
The tension of the electric field created by the charge system at this point of space is equal to the vector sum of the electric fields of the electric fields created at the same point by charges separately:
This property of the electric field means that the field is subordinate to the superposition principle.
Electric field strength. Gauss theorem in integral form
Let N be a single normal to the DS platform (small enough to neglect the change in electrical tension E within the site). The DF e-electrical stream stream through this platform is defined as a product of the normal component E and DS:
The flow sign of the DF e is obviously depends on the mutual orientation of normal and tension. If these two vectors form a sharp angle, the flow is positive if stupid is negative.
The flow DF e via the pad inclined to the power line (that is, the vector E) is also equal to the flow through the projection of this site to the plane perpendicular to the power line (see Fig. 1.1.2):
This equality (1.1.1) follows from the definition (1.1.1) for DF E and the theorems on angles with mutually perpendicular parties.
Flow F e electrical tension E through a closed surface S (Fig. 1.3.3) is defined as the sum of the elementary streams through all surface platforms. In the limit, when the number of sites n tends to infinity, the amount of streams through the platforms goes into the surface integral from the normal component of the e n intensity:
K. Gauss in 1844 is proved by theorem (Gaussian theorem in integral form), which establishes the connection of field sources and stream of tension through an arbitrary surface surrounding sources.
For the proof, withdraw auxiliary formula. Flow from a point charge through an arbitrary surrounding scope.
.
(1.1.4)
Power lines of a point charge perpendicular to the surface of the concentric sphere (see Figure 1.1.4). Taking into account this fact, formula (1.1.4) is derived from an expression for a point-charge field. As can be seen, in this case the flow F e does not depend on the radius of the sphere, and depends only on q.
From (1.1.2) and (1.1.4) it follows that the flow of a point charge field through any surface surrounding the charge is flowing through the sphere of an arbitrary radius, a concentric charge. Indeed, the flow of a point charge field through any platform DS, carved by a corporal angle D from an arbitrary surface, is the same as the flow through the area of \u200b\u200bthe sphere carved by the same bodily corner. Field stream F e via the sphere, as already noted, does not depend on its radius. Therefore, the stream of strength of the point charge field through the surface S (see Fig. 1.3.5) is given by formula (1.3.4). From formula (1.3.4) and the superposition principle follows the Gaussian theorem in the integral form: the full flow F e of the electric field strength through an arbitrary closed surface, inside which is like a distributed (volume, surface, etc.) charge Q, calculated by Formula
When the Gaussian Theorem is applied to solve problems, it is necessary to remember that in equation (1.1.5) q - the sum of all charges inside the mental surface, through which the flow is calculated, including charges belonging to atoms and molecules of the medium (the so-called associated charges).
Field strength e via any closed surface, inside which the full charge is zero, is also zero.
The emergence and development of the theory of the electromagnetic field
In the 17-18th centuries, electromagnetic processes were increasingly penetrated into science: physics and chemistry. The era of the electromagnetic picture of the world, which changed the mechanical one.
Maxwell clearly saw the fundamental meaning of electromagnetic laws, having carried out the grand synthesis of optics and electricity. It was he who managed to reduce the optics to electromagnetism, creating an electromagnetic theory of light and having laughed in the more new ways not only in theoretical physics, but also in the technique, having prepared the soil for radio engineering.
Faradays appeared in a new way to the study of electricity and magnetic phenomena, pointing to the role of the environment and introducing the concept of the field described by it using power lines. Maxwell gave the ideas of mathematical conversion, introduced the exact term "electromagnetic field", which was not yet in Faraday, formulated the mathematical laws of this field. Galilee and Newton laid the foundations of the mechanical picture of the world, Faraday and Maxwell are the foundations of the electromagnetic picture of the world.
Maxwell's electromagnetic theory develops in the works "On physical lines of force" (1861-1862) and "Dynamic field theory" (1864-1865). He wrote these works no longer in Aberdeen, but in London, where he received a professor in King - College. Here Maxwell met with Faraday, who was old and sick. Maxwell, having received data confirming the electromagnetic nature of the world, sent them to Faraday. Maxwell wrote: "The electromagnetic theory of the world proposed by them (Faraday) in" Thoughts about radial vibrations "(Phil. Mag., May 1846) or" experimental studies "(EXP. REC, p. 447) - this is essentially The same thing I began to develop in this article ("Dynamic field theory" -Phil. Mag., 1865), except that in 1846 there was no data to calculate the speed of distribution. J.K.M. ".
In 1873, Maxwell's chief work was released "Treatise on electricity and magnetism". He began to write a popular statement of his theory "Electricity in elementary presentation", but it did not have time to finish it.
Maxwell was a versatile scientist: theorist, experimenter, technician. But in the history of physics, his name is primarily associated with the theory of the electromagnetic field created by it, which is also called the Maxwell theory or Maxwell electrodynamics. It entered the history of science along with such fundamental generalizations as Newtonian mechanics, relativistic mechanics, quantum mechanics, and marked the beginning of a new stage in physics. In accordance with the Law of the Development of Science, Formulated by Aristotle, she raised the knowledge of nature to the new, the highest level and at the same time was more incomprehensible, abstract than previous theories, "less explicit for us", according to Aristotle.
Maxwell began to develop his theory in 1854
Maxwell characterizes an electrotonic state with three functions that calls electrotonic functions or components of an electrotonic state. In modern designations, this vector function corresponds to the potential vector. The curvilinear integral of this vector along the closed line Maxwell calls "full electrotonic intensity along a closed curve." For this value, it finds the first effect of the electrotonic state: "Complete electrotonic intensity along the boundary of the surface element serves as a measure of the amount of magnetic induction passing through this element, or, in other words, a measure of the number of magnetic power lines that permeate this element." In modern designations, this law can be expressed by the formula:
where A is the component of the potential vector in the direction of the element of the DL curve, Bn ~ the normal component of the induction vector in the direction of the normal to the DS surface element.
binding magnetic induction in with tension vector magnetic field N.
The third law binds the tension of the magnetic field H with the force of its current current I. Maxwell formulates it like this: "Full magnetic intensity along a line bounding some part of the surface serves as a measure of the amount of electric current flowing through this surface." In modern designations, this proposal is described by the formula
which is now called the first Maxwell equation in integral form. It reflects the experimental fact, Outdoor Erstended: Current is surrounded by a magnetic field.
The fourth law is the Ohm Law:
To characterize the power interactions of the currents of Maxwell introduces the value called them magnetic potential. This value is subject to the fifth law: "The complete electromagnetic potential of the closed current is measured by the amount of current on the total electrotonic intensity along the circuit, considered in the direction of the current:
The sixth law of Maxwell refers to electromagnetic induction: "The electromotive force acting on the conductor element is measured by the time derivative from electrotonic intensity, regardless of whether this derivative is due to a change in the value or direction of an electrotonic state." In modern designations, this law is expressed by the formula:
This is the second equation of Maxwell in an integral form. Note that Maxwell's electromotive force calls the circulation of the electric field strength vector. Maxwell summarizes the law of induction of Faraday - Lenzhana, considering that the change in the time of the magnetic flux (electrotonic state) generates a vortex electric field, which exists regardless of whether there are closed conductions in which this field excites current or not. Generalizes the law of Ersteda Maxwell does not yet give.
Another important news is to introduce the concepts of offset and offset currents. Displacement, according to Maxwell, is the characteristic of the dielectric states in the electric field. The full flow of displacement through a closed surface is equal to the algebraic amount of charges inside the surface. This introduces the fundamental concept of the offset current. This current, as well as conductivity current, creates a magnetic field. Therefore, Maxwell summarizes the equation that is currently called the first Maxwell equation, and introduces the shift current into the first part. In modern designations, this Maxwell equation has the form:
Finally, Maxwell finds that in its elastic medium, transverse waves are spread at the speed of light. This fundamental result leads it to an important conclusion: "The speed of transverse wave oscillations in our hypothetical environment, calculated from the electromagnetic experiments of Kolrauha and Weber, so exactly coincides with the speed of light calculated from the optical experiments of the FIZO, that we can hardly refuse the withdrawal that The light consists of transverse oscillations of the same medium that causes electrical and magnetic phenomena. Thus, at the beginning of the 60s of the XIX century. Maxwell has already found the foundations of its theory of electricity and magnetism and made an important conclusion that the light is an electromagnetic phenomenon.
In the theory of Maxwell, the magnitude of the "electromagnetic moment" is associated with the magnetic flow. The circulation of the vector-potential on a closed contour is equal to a magnetic stream through the surface covered by the contour. The magnetic flow has inertial properties, and the electromotive force of the induction according to the regulation of the Lenz is proportional to the rate of change of the magnetic flux taken with the opposite sign. Hence the tension of the induction electric field:
Maxwell considers this expression similar expression for the power of inertia in mechanics:
Mechanical impulse, or the number of movement. This analogy explains the term introduced by Maxwell for the vector potential. Equations themselves electromagnetic field In Maxwell theory, they have a view other than modern.
In modern form, the Maxwell's system of equations is as follows:
The relationship between the displacement d and the electric field in the electric field e Maxwell is expressed by the equation:
Then discharges the equation divd \u003d p and equation where
as well as the border condition:
Such is the system of Maxwell equations. The most important conclusion from these equations is the existence of transverse electromagnetic waves propagating in the magnetized dielectric at speed: where
This conclusion was obtained by him in the last section of the "dynamic field theory", which is called the "electromagnetic theory of light". "... Science of electromagnetism," Maxwell writes here, - leads to completely the same conclusions as optics in relation to the direction of perturbations that can spread through the field; Both of these sciences approve the crossiness of these oscillations, and both give the same speed of distribution. " On the air this speed C - the speed of light (Maxwell denotes it v), in a dielectric it is smaller where
Thus, the refractive index N, according to Maxwell, is determined by the electrical and magnetic properties of the medium. In a non-magnetic dielectric where
This is the famous ratio of Maxwell.
V.Tomson In 1853, he investigated the discharge of the conductor of a given container through the conductor of this shape and resistance. Applying the energy conservation law to the discharge process, it derived the equation of the discharge process in the following form:
where q is the amount of electricity on the dischargeable conductor at a given moment T, the capacity of the conductor, K is the galvanic resistance of the arrester, and the "constant, which can be called the electrodynamic capacity of the discharge" and which we now call the self-induction coefficient or inductance. Thomson, analyzing the solution to this equation at various roots of the characteristic equation, finds that when the value
it has a valid value (1 / Ca\u003e 4 * (k / a) 2), then the solution shows "that the chief conductor loses its charge, charges less than the electricity of the reverse sign, is again discharged, again it turns out to be charged by even smaller number of initial sign electricity, And this phenomenon repeats the infinite number of times until the equilibrium is established. " Cyclic frequency of these decaying oscillations:
Thus, the oscillation period can be submitted by the formula:
With small resistance values, we get the famous Thomson formula:
Conclusion
The electric field is a special form of a field that exists around the bodies or particles with an electric charge, as well as in free form in electromagnetic waves. The electric field is directly invisible, but it can be observed in its action and with the help of devices. The main action of the electric field is the acceleration of bodies or particles with an electric charge.
The electrical field can be considered as a mathematical model describing the value of the magnitude of the electric field at this point of space. Douglas Jancoli wrote like this: "It should be emphasized that the field is not a kind of substance; it is more correct to say, this is an extremely useful concept ... The question of" reality "and the existence of an electric field is actually a philosophical, rather even a metaphysical question. In physics, the idea of The field turned out to be extremely useful - this is one of the greatest achievements of the human mind. "
The electric field is one of the components of a single electromagnetic field and manifestation of electromagnetic interaction.
List of used literature
Dmitrieva V.F., Prokofiev V.L. Basics of physics. - M.: Higher School, 2003
Kalashnikov N. P., Smondyrev M. A. Basics of physics. - M.: Drop, 2003
Makarov E. F, Ozers R. P. Physics. - M.: Scientific World, 2002
Savelyev I.V. Course of general physics: studies. Manual: For universities. In 5 kN. KN.2. Electricity and magnetism - 4th ed., Pererab.- M.: Science, Fizmatlit, 2003, SS. 9-30, 41-71
Trofimova T.I. Course of physics: studies. Manual: For universities. - 5th ed., Sure.- M.: Higher. Shk., 2003, ss. 148-164
Detlaf A. A., Yavorsky B. M. Physics course: studies. manual for universities. - 2nd ed., University. and add-m.: Higher. Shk., 20049, ss. 182-190, 193-202.
Irodov I. E. Electromagnetism. Basic laws. - 3rd ed., Upar.-M.: Laboratory of Basic Knowledge, 2000, SS. 6-34.
Fields. Theorem Gaussa in integral form 4. Divergence of vector field. Theorem Gaussa In Differential form Conclusion List ... en:. (1.3.3.) K. Gauss In 1844 proved theorem (theorem Gaussa in integral form), establishing communication sources ...
Study of the electrostatic field
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Mechanics. Molecular physics
Abstract \u003e\u003e PhysicsOstrogradsky theorem, you can formulate theorem Gaussa For B. integral form: Vector stream through any ... zero:. - theorem Gaussa. Using the Ostrogradsky theorem, we get theorem Gaussa For vector in differential form
