Stokes' theorem. Field of a uniformly charged infinite plate

Knowing the rotor of vector a at each point of some (not necessarily flat) surface S, one can calculate the circulation of this vector along the contour Г bounding S (the contour can also be non-flat). To do this, we divide the surface into very small elements. Due to their smallness, these elements can be considered flat.

Therefore, in accordance with (11.23), the circulation of vector a along the boundary contour can be represented in the form

where is the positive normal to the surface element

In accordance with formula (11.21), summing expression (11.29) over all , we obtain the circulation of vector a along the contour Г, limiting

Having carried out the passage to the limit, in which all AS tend to zero (their number grows without limit), we arrive at the formula

(11.30)

Relation (11.30) is called Stokes' theorem. Its meaning is that the circulation of the vector a along an arbitrary contour Г is equal to the flow of the vector rota through an arbitrary surface S limited by a given contour.

Observatory operator Writing vector analysis formulas is greatly simplified and facilitated if you introduce a vector differential operator, denoted by a symbol and called the Nabla operator or Hamilton operator. This operator means a vector with components Therefore,

By itself, this vector has no meaning. It takes on meaning when combined with a scalar or vector function by which it is symbolically multiplied. So, if you multiply the vector y by a scalar, you get the vector

which is the gradient of the function (see (11.1)).

If the vector y is multiplied scalarly by the vector a, the result is a scalar

which is nothing more than the divergence of the vector a (see (11.14)).

Finally, if you multiply y by a vectorially, you get a vector with components: etc., which coincide with the components rota (see (11.25) - (11.27)).

Therefore, using the notation vector product using the determinant, we can write

(11-34)

Thus, there are two ways of notating gradient, divergence and rotor:

Notation using y has a number of advantages. Therefore, in what follows we will use such notation. You should accustom yourself to identify the symbol with the words “gradient” (i.e., say not “nabla” but “gradient phi”), the symbol with the words “divergence a” and, finally, the symbol with the words “rotor a”.

When using the vector y, you need to remember that it is differential operator, acting on all functions to the right of it. Therefore, when converting expressions that include y, you need to take into account both the rules vector algebra, so are the rules differential calculus. For example, the derivative of the product of functions is equal to

In accordance with this

Likewise

The gradient of some function is a vector function. Therefore, divergence and rotor operations can be applied to it.

Let a continuous vector field a) k and a closed oriented contour L be given in some domain G. Definition 1. The circulation of a vector a along a closed contour L is called line integral 2nd kind from the vector a along the contour L Here dr is a vector whose length is equal to the differential of the arc L, and the direction coincides with the direction of the tangent to L, op- Fig. 31 determined by the orientation of the contour (Fig. 31); the symbol f means that the integral is taken over an alternate contour L. Example 1. calculate the circulation of a vector field along the ellipse L: By definition of circulation we have Parametric equations of this ellipse have the form: , and, therefore, . Substituting these expressions into formula (2), we find the Circulation of the vector field. Rotor of a vector Stokes theorem Rotor (vortex) of a vector field Invariant definition rotor field Physical meaning field rotor Rules for calculating the rotor 8.1. Rotor (vortex) of a vector field Consider the field of a vector P, Q, R of which are continuous and have continuous partial derivatives of the first order with respect to all their arguments. Definition 2. The rotor of the vector "(M) is a vector denoted by the symbol rot a and defined by the equality or, in a symbolic, easy-to-remember form, This determinant is expanded by the elements of the first row, while the operations of multiplying the elements of the second row by the elements of the third row are understood as differentiation operations, for example, Definition 3. If in some domain G we have rot a = 0, then the field of the vector a in the domain G is called irrotational. Example 2. Find the rotor of vector 4 According to formula (3) we have Since rot a is a vector, we can consider a vector field - the field of the rotor of vector a. Assuming that the coordinates of the vector a have continuous partial derivatives of the second order, we calculate the divergence of the vector rot a. We obtain Thus, the field of the vector rota is solenoidal. Theorem 7 (Stokes). The circulation of the vector a along an oriented closed contour L is equal to the rotor flux of this vector through any surface E spanned by the contour L. It is assumed that the coordinates of the vector a have continuous partial derivatives in some region G of space containing the surface E, and that the orientation of the unit vector of the normal n° to the surface EC G is coordinated with the orientation of the contour L so that from the end of the norm, the circuit around the contour in a given direction is seen to be taking place counterclockwise. Considering that, and using the definition of a rotor (3), we rewrite formula (4) in the following form: Let us first consider the case when a smooth surface E and its contour L are uniquely projected onto the region D of the xOy plane and its boundary - contour A, respectively (Fig. 32). The orientation of the contour L gives rise to a certain orientation of the contour A. For definiteness, we will assume that the contour L is oriented so that the surface E remains to the left, so that the normal vector n to the surface E is the axis Oz acute angle 7 (cos 7 >0). Let be the equation of the surface E and the function φ(x)y) be continuous and have continuous partial derivatives gf and ^ in closed area D. Consider the integral Line L lies on the surface E. Therefore, using the equation of this surface, we can replace r under the integral sign with ^(x, y). The coordinates of the variable point of curve A are equal to the coordinates of the corresponding point on curve L, and therefore integration over L can be replaced by integration over A. Let us apply Green's formula to the integral on the right. We now move from the integral over the region D to the integral over the surface E. Since dS = cos 7 da, then from formula (8) we obtain that the normal vector n° to the surface E is determined by the expression k. From here it is clear that. Therefore, equality (9) can be rewritten as follows: Considering E a smooth surface that uniquely projects onto all three coordinate planes, similarly we are convinced of the validity of the formulas Circulation of a vector field. Rotor of a vector Stokes' theorem Rotor (vortex) of a vector field Invariant definition of a rotor of a field Physical meaning of a rotor of a field Rules for calculating the rotor By adding the equalities term by term, we obtain the Stokes formula (5), or, in short, Remark 1. We have shown that the field of the vector rote is solenoidal, and therefore, the flow of the vector rota does not depend on the type of surface E spanned by the contour L. Remark 2. Formula (4) was derived under the assumption that the surface £ is uniquely projected onto all three coordinate planes. If this condition is not met, then we divide £ into parts so that each part specified condition satisfied, and then we use the additivity of the integrals. Example 3. Calculate the circulation of a vector along a line 1) using the definition; 2) according to Stokes' theorem. 4 1) Let's define the line L parametrically: Then 2) Find rota: Let's stretch a piece of plane onto the contour L Then. Invariant definition of the field rotor From Stokes' theorem, one can obtain an invariant definition of the field rotor, not related to the choice of coordinate system. Theorem 8. The projection of the rotor a to any direction does not depend on the choice of coordinate system and is equal to surface density circulation of vector a along the contour of the platform, perpendicular to this direction, Here (E) is a flat platform, perpendicular to the vector l; 5 - area of this site; L - the contour of the site, oriented so that the circuit bypass is visible from the end of the vector n counterclockwise; (E) M means that the area (E) contracts to the point M, at which the vector rot a is considered, and the normal vector n to this area remains the same all the time (Fig. 33). 4 Let us first apply the Stokes theorem to the circulation (a,dr) of the vector a, and then to the resulting double integral- the mean value theorem: where (the scalar product is taken at some midpoint Mf of the site (E)). As the area (E) attracts to point M, the average point A/c also tends to point M and, due to the assumed continuity of partial derivatives of the coordinates of the vector a (and hence the continuity of rot a), we obtain Since the projection of the vector rot a to an arbitrary direction is not depends on the choice of coordinate system, then the vector rota itself is invariant with respect to this choice. From here we obtain the following invariant definition of the field rotor: the field rotor is a vector whose length is equal to the greatest surface circulation density at a given point, directed perpendicular to the area on which this highest density circulation is achieved; in this case, the orientation of the rota vector is consistent with the orientation of the contour, at which the circulation is positive, according to the right screw rule. 8.3. The physical meaning of a field rotor Let a rigid body rotate around fixed axis I with angular velocity and. Without loss of generality, we can assume that the I axis coincides with the Oz axis (Fig. 34). Let M(r) be the body point being studied, where Vector angular velocity in our case is equal to from = wk, calculate the vector v linear speed points M, Hence the circulation of the vector field. Rotor of a vector Stokes theorem Rotor (vortex) of a vector field Invariant definition of a rotor of a field Physical meaning of a rotor of a field Rules for calculating the rotor So, the vortex of a rotating velocity field solid is the same at all points of the field, parallel to the axis of rotation and equal to twice the angular velocity of rotation. 8.4. Rules for calculating the rotor 1. Rotor constant vector c is equal to the zero vector, 2. The rotor has the property of linearity of constant numbers. 3. Product rotor scalar function u(M) to vector a(M) is calculated by the formula

This theorem allows you to calculate the circulation of a vector along a contour of finite length using the rotor of this vector.

Circulation vector field along a closed positively oriented contour L equal to rotor flow this field through any smooth surface S , based on this contour:

. (2.12)

To prove the theorem, consider a contour with the area it covers (Fig. 2.6). The entire contour is divided into elementary contours of the same orientation (Fig. 2.10).

The circulation along the elementary circuit is equal to
.

All adjacent contours ( 1 And 2 in Fig. 2.10) have the following feature: on a common boundary with the same field value, the contribution to the circulation along each of the adjacent contours will occur with a change in sign (for the contour 1 -a  b , and for 2 - b  a ). As a result, the contribution to the circulation of all internal sections of the circuits is mutually compensated, and only the sections belonging to the circuit will remain uncompensated L , which ultimately gives (2.12) .

A special case of (2.12) in the case of a contour located on a plane is the formula of D. Green (M. Ostrogradsky-D. Green):

. (2.13)

Formulas (2.12) and (2.13) allow us to reduce the calculation of a curvilinear integral of the second kind to the calculation of a double integral over the region S .

The reverse transition according to (2.12) is carried out similarly to (2.8).

2.4. Observer operator and Laplace operator

Writing vector analysis formulas is simplified when using radar operator (operator W. Hamilton), which is a vector
. By itself, this vector has no meaning, but it allows us to compactly write formulas (2.3), (2.5) and (2.9):

;
;
. (2.14)

In addition, the nabla operator makes it possible to simplify the calculation of higher order differential operators.

It should be noted that with  should be handled with care, and when using it, you should remember that this operator is not only vector , but also differential .

For example, let's find
. Using we get
. According to the rules differentiation product operator acts first on first multiplier and then by second: . As a result we get. The calculation procedure through vector coordinates would require an order of magnitude more operations.

Try to obtain on your own the formula for the expansion not included in (2.15)
. The correct answer is given at the end applications 1 .

Some identities and second order operations.

;
;

Laplace operator ( , Laplacian ) is a second order operator.

Like  ,  applies to both scalar and vector.

. (2.17)

In case Cartesian system coordinates (2.18) is simplified:

Information about curvilinear coordinate systems often used in EMF theory ( cylindrical And spherical ) and vector operations in them are given in Appendix 2 .

2.5. Classification of vector fields

Vector field is given uniquely if its rotor and divergence are known as functions of spatial coordinates.

Depending on the values of these functions, there are potential , vortex (solenoidal ) field and generic field .

Vector field potentially , if there is some scalar function U , which is associated with as follows:
. Function U called scalar field potential .

Necessary and sufficient condition potentiality is rotor equal to zero (
).

Solenoidal (vortex ) is called a vector field , at each point of which
(necessary and sufficient condition),
.

Solenoidal vector field can be represented as
. In this case, the vector quantity called vector field potential (
).

Field name of this type can be explained by the fact that it was discovered in solenoid , – an inductor (it can be either with or without a core), the length of which significantly exceeds the diameter.

If the vector field
And
, then this is - generic field .

An arbitrary vector field of general type can be represented as the sum of the potential and vortex parts:
, – where in included field sources (
), and in –field vortices (
).

Now, after studying integral and differential operations and the basic theorems of vector analysis, we can begin to study the basis of the theory of EMF - Maxwell's system of equations .

Knowing at every point S, you can find the circulation by G around S. Let's break it down S on  S:

And

- normal to surface element  S.

Let everything  S 0 , Then:

Stokes' theorem:

Circulation vector along an arbitrary contour G equal to the flux of the vector
through an arbitrary surface S, limited by this contour.

3.7 Circulation and rotor of the electrostatic field

The work of electrostatic forces along any closed circuit is zero.

those. circulation of the electrostatic field along any circuit is zero.

Let's take any surface S, based on the contour G.

According to Stokes' theorem:

;

since this is for any surface S, That

There is an identity:

those. electrostatic field lines do not circulate in space.

3.8 Gauss's theorem

We'll find
electrostatic field. For a point charge, the line density is numerically equal to

Flow through any closed surface is equal to the number of lines going out, i.e. starting with the charge “+” and ending with the charge “-“:

The sign of the flow matches the sign q, the dimensions are the same.

Let there be N point charges q i .

The flow of the electrostatic field strength vector through a closed surface is equal to the algebraic sum of the charges contained inside this surface, divided by  0.

4 Calculating fields using Gauss's theorem

4.1 Field of a uniformly charged infinite plate.

4.2 Field of a uniformly charged spherical surface.

4.3 Field of two infinite parallel oppositely charged planes

4.4 Field of a volumetrically charged ball

4.1 Field of a uniformly charged infinite plate

IN introduce the concept of surface density

- charge per unit surface.

An infinite plate charged with constant surface density +  . The tension lines are perpendicular to the considered plane and directed from it in both directions.

As a closed surface, we will construct a cylinder, the bases of which are parallel to the plane, and the axis is perpendicular to it, because the generatrices of the cylinder are parallel E, That cos =0 and the flux through the side surface is 0, and full flow through a cylinder is equal to the sum of the flows through its base.

E'=E''=E,

That F= 2E S;

q = S

It follows that E does not depend on the length of the cylinder, i.e. The field surface at any distance is the same in absolute value, i.e. The field of a uniformly charged plate is uniform.

4.2 Field of a uniformly charged spherical surface

WITH spherical surface radius R with common charge q.

Because the charge is distributed uniformly, then the field has spherical symmetry, i.e. the plane lines are directed radially.

Let's mentally construct a sphere of radius r R. Because r R, then the entire charge falls inside the surface, according to Gauss’s theorem:

At r R the field decreases with distance r according to the same law as that of a point charge.

If r' R, then the closed surface does not contain charges inside, it follows that there is no electrostatic field inside a uniformly charged spherical surface E=0.