FORCE FIELD

FORCE FIELD

A part of space (limited or unlimited), at each point a material object placed there is affected by , the magnitude and direction of which depend either only on the coordinates x, y, z of this point, or on the coordinates and time t. In the first case, S., called. stationary, and in the second - non-stationary. If the force at all points of a linear path has the same value, that is, does not depend on the coordinates, then the force is called. homogeneous.

SP, in which the field forces acting on a material object moving in it, depends only on the initial and final position of the object and does not depend on the type of its trajectory, called. potential. This work can be expressed in terms of the potential energy of the particle P (x, y, z):

A=П(x1, y1, z1)-П(x2, y2, z2),

where x1, y1, z1 and x2, y2, z2 are the coordinates of the initial and final positions of the particle, respectively. When a particle moves in a potential S. space under the influence of only field forces, the law of mechanical conservation takes place. energy, making it possible to establish a relationship between the speed of a particle and its position in the center of space.

Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .

FORCE FIELD

A part of space (limited or unlimited), at each point a material particle placed there is acted upon by a force of a certain numerical value and direction, depending only on the coordinates x, y, z this point. This S. p. is called. stationary; if the field strength also depends on time, then S. p. is called. non-stationary; if the force at all points of a s.p. has the same value, i.e., does not depend on coordinates or time, the s.p. is called. homogeneous.

Stationary S. p. can be specified by equations

Where F x , F y , F z - field strength projections F.

If such a function exists U(x, y, z), called the force function, U(x,y, z), and the force F can be defined through this function by the equalities:

or . The condition for the existence of a power function for a given S. item is that

or . When moving in a potential S. point from a point M 1 (x 1 ,y 1 ,z 1)exactly M 2 (x 2, y 2, z 2) the work of the field forces is determined by equality and does not depend on the type of trajectory along which the point of application of the force moves.

Surfaces U(x, y, z) = const, for which the function maintains a constant state. Examples of potential static fields: a uniform gravitational field, for which U= -mgz, Where T - the mass of a particle moving in the field, g- acceleration of gravity (axis z directed vertically upward); Newtonian flight of gravity, for which U = km/r, where r = - distance from the center of gravity, k - constant coefficient for a given field. potential energy P associated with U addiction P(x,)= = - U(x, y, z). Study of particle motion in potential. p. (in the absence of other forces) is significantly simplified, since in this case the law of conservation of mechanics takes place. energy, which makes it possible to establish a direct relationship between the speed of a particle and its position in the solar system. With. POWER LINES- a family of curves characterizing the spatial distribution of the vector field of forces; the direction of the field vector at each point coincides with the tangent to the line. Thus, level of S. l. arbitrary vector field A (x, y, z) are written in the form:

Density S. l. characterizes the intensity (magnitude) of the force field. Concept of S. l. introduced by M. Faraday during the study of magnetism, and then further developed in the works of J. C. Maxwell on electromagnetism. Maxwell tension tensor el.-magn. fields.

Along with the use of the concept of S. l. more often they simply talk about field lines: electrical intensity. fields E, magnetic induction fields IN etc.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

See what a “FORM FIELD” is in other dictionaries:

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force field- A region of space in which a material point placed there is acted upon by a force that depends on the coordinates of this point in the reference system under consideration and on time... Polytechnic terminological explanatory dictionary

The concept of “field” is encountered very often in physics. From a formal point of view, the definition of a field can be formulated as follows: if at each point in space the value of a certain quantity, scalar or vector, is given, then they say that a scalar or vector field of this quantity is given, respectively .

More specifically, it can be stated that if a particle at every point in space is exposed to the influence of other bodies, then it is in a field of forces or force field .

The force field is called central, if the direction of the force at any point passes through some fixed center, and the magnitude of the force depends only on the distance to this center.

The force field is called homogeneous, if at all points of the field strength, acting on the particle, identical in magnitude and direction.

Stationary called time-invariant field.

If the field is stationary, then it is possible that Job field strength over some particle does not depend on the shape of the path , along which the particle moved and is completely determined by specifying the initial and final position of the particle . Field strengths having this property are called conservative. (Not to be confused with the political orientation of parties...)

The most important property of conservative forces is that their work on arbitrary closed path is zero. Indeed, a closed path can always be arbitrarily divided by two points into some two sections - section I and section II. When moving along the first section in one direction, work is done . When moving along the same section in the opposite direction, work is done - in the formula for work (3.7) each element of movement is replaced by the opposite sign: . Therefore, the integral as a whole changes sign to the opposite.

Then work on a closed path

Since, by definition of conservative forces, their work does not depend on the shape of the trajectory, then . Hence

The converse is also true: if the work on a closed path is zero, then the field forces are conservative . Both features can be used to determine conservative forces.

The work done by gravity near the Earth's surface is found by the formula A=mg(h 1 -h 2) and obviously does not depend on the shape of the path. Therefore, gravity can be considered conservative. This is a consequence of the fact that the gravity field within the laboratory can be considered homogeneous with very high accuracy. Has the same property any uniform stationary field, which means the forces of such a field are conservative. As an example, we can recall the electrostatic field in a flat capacitor, which is also a field of conservative forces.

Central field forces Also conservative. Indeed, their work on displacement is calculated as

FORCE FIELD- a part of space (limited or unlimited), at each point a material particle placed there is acted upon by a force determined in numerical magnitude and direction, depending only on the coordinates x, y, z this point. This S. p. is called. stationary; if the field strength also depends on time, then S. p. is called. non-stationary; if the force at all points of a linear force has the same value, that is, it does not depend on coordinates or time, the force is called. homogeneous.

Stationary S. p. can be specified by equations

Where Fx, Fy, Fz- projections of field strength F.

If such a function exists U(x, y, z), called the force function, that the elementary work of the field forces is equal to the total differential of this function, then S. p. is called. potential. In this case, the S. item is specified by one function U(x, y, z), and the force F can be determined through this function by the equalities:

or . The condition for the existence of a power function for a given S. item is that

or . When moving in a potential S. point from a point M 1 (x 1 , y 1 , z 1)exactly M 2 (x 2, y 2, z 2) the work of the field forces is determined by the equality and does not depend on the type of trajectory along which the point of application of the force moves.

Surfaces U(x, y, z) = const, for which the function maintains the posture. meaning, called level surfaces. The force at each point of the field is directed normal to the level surface passing through this point; When moving along the surface of the level, the work done by the field forces is zero.

Examples of potential static fields: a uniform gravitational field, for which U = -mgz, Where T- mass of a particle moving in the field, g- acceleration of gravity (axis z directed vertically upward); Newtonian gravitational field, for which U = km/r, where r = - distance from the center of gravity, k - constant coefficient for a given field. Instead of a power function, one can enter as a characteristic of a potential S. potential energy P associated with U addiction P(x, y, z)= = -U(x, y, z). The study of the motion of a particle in a potential magnetic field (in the absence of other forces) is significantly simplified, since in this case the law of conservation of mechanics holds. energy, which makes it possible to establish a direct relationship between the speed of a particle and its position in the solar system. With. m. Targ. POWER LINES- a family of curves characterizing the spatial distribution of the vector field of forces; the direction of the field vector at each point coincides with the tangent to the line. Thus, level of S. l. arbitrary vector field A (x, y, z) are written in the form:

Density S. l. characterizes the intensity (magnitude) of the force field. An area of ​​space limited by linear lines intersecting lines. closed curve, called power tube. S. l. vortex fields are closed. S. l. potential fields begin at the sources of the field and end at its drains (sources of negative sign).

Concept of S. l. introduced by M. Faraday during the study of magnetism, and then further developed in the works of J. C. Maxwell on electromagnetism. According to the ideas of Faraday and Maxwell, in the space permeated by S. l. electric and mag. fields, there are mechanical stresses corresponding to tension along the S. line. and pressure across them. Mathematically, this concept is expressed as Maxwell stress tensor el-magn. fields.

Along with the use of the concept of S. l. more often they simply talk about field lines: electrical intensity. fields E, magnetic induction fields IN etc., without making special emphasis on the relationship of these zeros to forces.

A force field is a region of space at each point of which a particle placed there is acted upon by a force that varies naturally from point to point, for example, the Earth’s gravity field or the field of resistance forces in a liquid (gas) flow. If the force at each point of the force field does not depend on time, then such a field is called stationary. It is clear that a force field that is stationary in one reference system may turn out to be non-stationary in another frame. In a stationary force field, the force depends only on the position of the particle.

The work that field forces do when moving a particle from a point 1 exactly 2 , depends, generally speaking, on the path. However, among stationary force fields there are those in which this work does not depend on the path between points 1 And 2 . This class of fields, having a number of important properties, occupies a special place in mechanics. We will now move on to studying these properties.

Let us explain this using the example of a tracking force. In Fig. 5.4 shows the body ABCD, at the point ABOUT which force is applied , invariably connected with the body.

Let's move the body from position I to position II two ways. Let us first choose a point as a pole ABOUT(Fig. 5.4a)) and rotate the body around the pole by an angle π/2 opposite to the direction of clockwise rotation. The body will take a position A"B"C"D". Let us now impart to the body a translational movement in the vertical direction by the amount OO". The body will take a position II (A"B"C"D"). The work done by a force on the perfect movement of a body from a position I to position II equal to zero. The pole displacement vector is represented by the segment OO".

In the second method, we select the point as the pole K rice. 5.4b) and rotate the body around the pole by an angle π/2 counterclockwise. The body will take a position A"B"C"D"(Fig. 5.4b). Now let's move the body vertically upward with the pole displacement vector KK", after which we give the body a horizontal movement to the left by the amount K"K". As a result, the body will take the position II, the same as in position, Fig. 5.4 A)Figure 5.4. However, now the vector of movement of the pole will be different than in the first method, and the work of force in the second method of moving the body from position I to position II equal to A = F K "K", i.e., different from zero.

Definition: a stationary force field in which the work of the field force on the path between any two points does not depend on the shape of the path, but depends only on the position of these points, is called potential, and the forces themselves are conservative.

Potential such forces ( potential energy) is the work done by them to move the body from the final position to the initial one, and the initial position can be chosen arbitrarily. This means that potential energy is determined to within a constant.

If this condition is not met, then the force field is not potential, and the field forces are called non-conservative.

In real mechanical systems there are always forces whose work during actual motion of the system is negative (for example, friction forces). Such forces are called dissipative. They are a special type of non-conservative forces.

Conservative forces have a number of remarkable properties, to identify which we introduce the concept of a force field. Space is called a force field(or part thereof), in which a certain force acts on a material point placed at each point of this field.

Let us show that in a potential field, the work of field forces on any closed path is equal to zero. Indeed, any closed path (Fig. 5.5) can be arbitrarily divided into two parts, 1a2 And 2b1. Since the field is potential, then, by condition, . On the other hand, it is obvious that . That's why

Q.E.D.

Conversely, if the work of field forces on any closed path is zero, then the work of these forces on the path between arbitrary points 1 And 2 does not depend on the shape of the path, i.e. the field is potential. To prove it, let's take two arbitrary paths 1a2 And 1b2(see Fig. 5.5). Let's make a closed path from them 1a2b1. The work on this closed path is equal to zero by condition, i.e. . From here. But, therefore

Thus, the equality of the work of the field forces to zero on any closed path is a necessary and sufficient condition for the independence of the work from the shape of the path, and can be considered a distinctive feature of any potential field of forces.

Field of central forces. Any force field is caused by the action of certain bodies. Force acting on a particle A in such a field is due to the interaction of this particle with these bodies. Forces that depend only on the distance between interacting particles and are directed along a straight line connecting these particles are called central. An example of the latter are gravitational, Coulomb and elastic forces.

Central force acting on a particle A from the particle side IN, can be represented in general view:

Where f(r) is a function that, for a given nature of interaction, depends only on r- distances between particles; - unit vector specifying the direction of the radius vector of the particle A relative to the particle IN(Fig. 5.6).

Let's prove that every stationary field of central forces is potentially.

To do this, let us first consider the work of central forces in the case when the force field is caused by the presence of one stationary particle IN. The elementary work of force (5.8) on displacement is . Since is the projection of the vector onto the vector, or onto the corresponding radius vector (Fig. 5.6), then . The work of this force along an arbitrary path from the point 1 to the point 2

The resulting expression depends only on the type of function f(r), i.e., on the nature of the interaction, and on the meanings r 1 And r 2 initial and final distances between particles A And IN. It does not depend in any way on the shape of the path. This means that this force field is potential.

Let us generalize the obtained result to a stationary force field caused by the presence of a set of stationary particles acting on the particle A with forces, each of which is central. In this case, the work of the resulting force when moving a particle A from one point to another is equal to the algebraic sum of the work done by individual forces. And since the work of each of these forces does not depend on the shape of the path, then the work of the resulting force also does not depend on it.

Thus, indeed, any stationary field of central forces is potential.

Potential energy of a particle. The fact that the work of potential field forces depends only on the initial and final positions of the particle makes it possible to introduce the extremely important concept of potential energy.

Let's imagine that we move a particle in a potential force field from different points P i to a fixed point ABOUT. Since the work of field forces does not depend on the shape of the path, it remains dependent only on the position of the point R(at a fixed point ABOUT). This means that this work will be some function of the radius vector of the point R. Having denoted this function, we write

The function is called the potential energy of a particle in a given field.

Now let's find the work done by field forces when a particle moves from a point 1 exactly 2 (Fig. 5.7). Since the work does not depend on the path, we take the path passing through point 0. Then the work is on the path 1 02 can be represented in the form

or taking into account (5.9)

The expression on the right is the decrease* in potential energy, i.e., the difference in the values ​​of the potential energy of a particle at the initial and final points of the path.

_________________

* Changing any value X can be characterized either by its increase or decrease. Increment of value X is called the difference of the finite ( X 2) and initial ( X 1) values ​​of this quantity:

increment Δ X = X 2 - X 1.

Decrease in value X is called the difference of its initial ( X 1) and final ( X 2) values:

decline X 1 - X 2 = -Δ X,

i.e., a decrease in value X equal to its increment taken with the opposite sign.

Increase and decrease are algebraic quantities: if X 2 > X 1, then the increase is positive and the decrease is negative, and vice versa.

Thus, the work of the field forces on the path 1 - 2 is equal to the decrease in the potential energy of the particle.

Obviously, a particle located at point 0 of the field can always be assigned any pre-selected value of potential energy. This corresponds to the fact that by measuring work, only the difference in potential energies at two points of the field can be determined, but not its absolute value. However, once the value is fixed

potential energy at any point, its values ​​at all other points of the field are uniquely determined by formula (5.10).

Formula (5.10) makes it possible to find an expression for any potential force field. To do this, it is enough to calculate the work done by the field forces on any path between two points, and present it in the form of a decrease in a certain function, which is potential energy.

This is exactly what was done when calculating work in fields of elastic and gravitational (Coulomb) forces, as well as in a uniform gravitational field [see. formulas (5.3) - (5.5)]. From these formulas it is immediately clear that the potential energy of a particle in these force fields has the following form:

1) in the field of elastic force

2) in the field of a point mass (charge)

3) in a uniform gravity field

Let us emphasize once again that potential energy U is a function that is determined up to the addition of some arbitrary constant. This circumstance, however, is completely unimportant, since all formulas include only the difference in values U in two particle positions. Therefore, an arbitrary constant, the same for all points of the field, drops out. In this regard, it is usually omitted, which is what was done in the three previous expressions.

And one more important circumstance that should not be forgotten. Potential energy, strictly speaking, should be attributed not to a particle, but to a system of particles and bodies interacting with each other, causing a force field. With this type of interaction, the potential energy of interaction of a particle with these bodies depends only on the position of the particle relative to these bodies.

Relationship between potential energy and force. According to (5.10), the work done by the potential field force is equal to the decrease in the potential energy of the particle, i.e. A 12 = U 1 - U 2 = - (U 2 - U 1). For elementary displacement, the last expression has the form dA = - dU, or

F l dl= - dU. (5.14)

that is, the projection of the field force at a given point onto the direction of movement is equal, with the opposite sign, to the partial derivative of the potential energy in a given direction.

, then using formula (5.16) we have the opportunity to restore the field of forces.

The geometric location of points in space at which potential energy U has the same value and defines the equipotential surface. It is clear that each value U corresponds to its own equipotential surface.

From formula (5.15) it follows that the projection of the vector onto any direction tangent to the equipotential surface at a given point is equal to zero. This means that the vector is normal to the equipotential surface at a given point. In addition, the minus sign in (5.15) means that the vector is directed towards decreasing potential energy. This is illustrated by Fig. 5.8, relating to the two-dimensional case; here is a system of equipotentials, and U 1 < U 2 < U 3 < … .

Force field is a physical space that satisfies the condition that the points of a mechanical system located in this space are acted upon by forces that depend on the position of these points or on the position of the points and time (but not on their velocities).

Force field, whose forces do not depend on time is called stationary(examples of a force field are the gravity field, the electrostatic field, the elastic force field).

Potential force field.

Stationary force field called potential, if the work of field forces acting on a mechanical system does not depend on the shape of the trajectories of its points and is determined only by their initial and final positions. These forces are called potential forces or conservative forces.

Let us prove that the above condition is satisfied if there is a unique coordinate function:

called the field force function, the partial derivatives of which with respect to the coordinates of any point M i (i=1, 2...n) are equal to the projection tions of the force applied to this point on the corresponding axes, i.e.

The elementary work of force applied to each point can be determined by the formula:

The elementary work of forces applied to all points of the system is equal to:

Using the formulas we get:

As can be seen from this formula, the elementary work of the potential field forces is equal to the total differential of the force function. The work of the field forces on the final displacement of the mechanical system is equal to:

that is, the work of forces acting on the points of a mechanical system in a potential field is equal to the difference in the values ​​of the force function in the final and initial positions of the system and does not depend on the shape of the trajectories of the points of this system. positions of the system and does not depend on the shape of the trajectories of the points of this system. It follows from this that the force field for which the force function exists is indeed potential.