Equipotential surfaces and lines of force of the electrostatic field. Equipotential surfaces

Equipotential surfaces These are surfaces, each of the points of which have the same potential. That is, on an equipotential surface, the electric potential has a constant value. Such a surface is the surface of conductors, since their potential is the same.

Let us imagine a surface for which the potential difference will be zero for two points. This will be the equipotential surface. Because the potential on it is the same. If we consider an equipotential surface in two-dimensional space, say in a drawing, then it will have the shape of a line. Work of forces electric field according to the movement of electric charge along this line will be equal to zero.

One of the properties of equipotential surfaces is that they are always perpendicular to the field lines. This property can be formulated vice versa. Any surface that is perpendicular at all points to the electric field lines is called equipotential.

Also, such surfaces never intersect with each other. Since this would mean a difference in potential within one surface, which contradicts the definition. They are also always closed. Surfaces of equal potential cannot begin and go to infinity without having clear boundaries.

As a rule, there is no need to depict entire surfaces in drawings. More often they depict a perpendicular section to equipotential surfaces. Thus they degenerate into a line. This turns out to be quite sufficient to estimate the distribution of this field. When depicting graphically, surfaces are placed at equal intervals. That is, between two adjacent surfaces the same step is observed, say one volt. Then, by the density of the lines formed by the cross-section of equipotential surfaces, one can judge the electric field strength.

For example, consider the field created by a point electric charge. The lines of force of such a field are radial. That is, they start at the center of the charge and point toward infinity if the charge is positive. Or directed towards the charge if it is negative. The equipotential surfaces of such a field will have the shape of spheres centered on the charge and diverging from it. If we depict a two-dimensional section, then the equipotential lines will be in the form of concentric circles, the center of which is also located in the charge.

Figure 1 - equipotential lines of a point charge

For a uniform field such as, for example, the field between the plates of an electric capacitor, surfaces of equal potential will have the shape of planes. These planes are located parallel to each other at the same distance. True, at the edges of the plates the field picture will be distorted due to the edge effect. But we will imagine that the plates are infinitely long.

Figure 2 - equipotential lines uniform field

To depict equipotential lines for a field created by two charges of equal magnitude and opposite in sign, it is not enough to apply the principle of superposition. Since in this case, when two images of point charges are superimposed, there will be points of intersection of the field lines. But this cannot be, since the field cannot be directed in two different directions at once. In this case, the problem must be solved analytically.

Figure 3 - Picture of a field of two electric charges

> Equipotential lines

Characteristics and properties equipotential surface lines: state of the electric potential of the field, static equilibrium, point charge formula.

Equipotential lines fields are one-dimensional regions where the electric potential remains unchanged.

Learning Objective

• Characterize the shape of equipotential lines for several charge configurations.

Main points

• For a particular isolated point charge, the potential is based on the radial distance. Therefore, equipotential lines appear round.
• If several discrete charges come into contact, their fields intersect and exhibit potential. As a result, the equipotential lines become skewed.
• When charges are distributed across two conducting plates in static balance, the equipotential lines are essentially straight.

Terms

• Equipotential - a section where each point has the same potential.
• Static balance – physical state, where all components are at rest and the net force is equal to zero.

Equipotential lines represent one-dimensional regions where the electrical potential remains unchanged. That is, for such a charge (no matter where it is on the equipotential line) it is not necessary to carry out work to move from one point to another within a particular line.

The lines of the equipotential surface can be straight, curved or irregular. All this is based on the distribution of charges. They are located radially around the charged body, so they remain perpendicular to the electric field lines.

Single point charge

For a single point charge, the potential formula is:

Here there is a radial dependence, that is, regardless of the distance to the point charge, the potential remains unchanged. Therefore, the equipotential lines take round shape with a point charge in the center.

Isolated point charge with electric field lines (blue) and equipotential lines (green)

Multiple charges

If several discrete charges are in contact, then we see how their fields overlap. This overlap causes the potential to combine and the equipotential lines to become skewed.

If several charges are present, then equipotential lines are formed irregularly. At the point between the charges, the control is able to feel the effects of both charges.

Continuous charge

If the charges are located on two conducting plates under conditions of static balance, where the charges are not interrupted and are in a straight line, then the equipotential lines are straightened. The fact is that the continuity of charges causes continuous actions at any point.

If the charges are drawn into a line and are not interrupted, then the equipotential lines go directly in front of them. As an exception, we can only remember the bend near the edges of the conductive plates

Continuity is broken closer to the ends of the plates, which is why curvature is created in these areas - the edge effect.

For a visual representation of vector fields, a picture of field lines is used. A line of force is an imaginary mathematical a curve in space, the direction of the tangent to which at each the point through which it passes coincides with the direction of the vector fields at the same point(Fig. 1.17).
Rice. 1.17:
The condition for the parallelism of the vector E → and the tangent can be written as equality to zero vector product E → and arc element d r → force line:

Equipotential is the surface on which for which the electric potential is constantϕ. In the field of a point charge, as shown in Fig. , spherical surfaces with centers at the location of the charge are equipotential; this can be seen from the equation ϕ = q ∕ r = const.

Analyzing the geometry of electrical field lines and equipotential surfaces, we can indicate a number general properties geometry of the electrostatic field.

Firstly, power lines start on charges. They either go to infinity or end on other charges, as in Fig. .

Secondly, in a potential field, field lines cannot be closed. Otherwise, it would be possible to specify such a closed circuit that the work of the electric field when moving a charge along this circuit is not equal to zero.

Thirdly, the lines of force intersect any equipotential normal to it. Really, electric field everywhere is directed towards a rapid decrease in the potential, and on the equipotential surface the potential is constant by definition (Fig. ).
Rice. 1.20:
And finally, the field lines do not intersect anywhere except at points where E → = 0. The intersection of field lines means that the field at the intersection point is an ambiguous function of coordinates, and the vector E → does not have a specific direction. The only vector that has this property is the zero vector. The structure of the electric field near the zero point will be analyzed in problems for ?? .

The field line method is, of course, applicable to the graphical representation of any vector fields. So, in the chapter ?? we will meet the concept of magnetic lines of force. However, the geometry magnetic field completely different from the geometry of the electric field.

Rice. 1.21:
The idea of ​​lines of force is closely related to the concept of a force tube. Let's take any arbitrary closed contour L and draw an electric line of force through each point of it (Fig. ). These lines form the power tube. Let us consider an arbitrary section of the tube with surface S. We draw the positive normal in the same direction in which the field lines are directed. Let N be the flow of the vector E → through the section S. It is easy to see that if there are no electric charges inside the tube, then the flow N remains the same along the entire length of the tube. To prove it, we need to take another cross section S ′. According to Gauss's theorem, the electric field flux through a closed surface limited by the side surface of the tube and sections S, S′ is equal to zero, since there are no electric charges inside the power tube. Flow through lateral surface is equal to zero, since the vector E → touches this surface. Consequently, the flow through the section S ′ is numerically equal to N, but opposite in sign. The outer normal to the closed surface on this section is directed opposite to n →. If the normal is directed in the same direction, then the flows through the sections S and S ′ will coincide in both magnitude and sign. In particular, if the tube is infinitely thin and the sections S and S ′ are normal to it, then

E S = E ′ S ′ .

The result is a complete analogy with the flow of an incompressible fluid. In those places where the tube is thinner, the field E → is stronger. In those places where it is wider, the field E → is stronger. Consequently, the density of the field lines can be used to judge the electric field strength.

Before the invention of computers, to experimentally reproduce force lines, a glass vessel with a flat bottom was taken and a non-conducting liquid, such as castor oil or glycerin, was poured into it. Powdered crystals of gypsum, asbestos or some other oblong particles were evenly stirred into the liquid. Metal electrodes were immersed in the liquid. When connected to sources of electricity, the electrodes excited an electric field. In this field, the particles are electrified and, attracted to each other by oppositely electrified ends, are arranged in the form of chains along the lines of force. The picture of field lines is distorted by fluid flows caused by forces acting on it in a non-uniform electric field.

To Be Done Yet
Rice. 1.22:
The best results are obtained from the method used by Robert W. Pohl (1884-1976). Staniol electrodes are glued onto a glass plate, between which an electrical voltage is created. Then, by lightly tapping it, oblong particles, for example, gypsum crystals, are poured onto the plate. They are located along it along the lines of force. In Fig. ?? The picture of field lines obtained in this way between two oppositely charged circles of staniol is depicted.

▸ Problem 9.1

Write down the equation of field lines in arbitrary orthogonal coordinates

▸ Problem 9.2

Write down the equation of field lines in spherical coordinates.

An electrostatic field can be characterized by a set of force and equipotential lines.

power line - this is a line mentally drawn in the field, starting on a positively charged body and ending on a negatively charged body, drawn in such a way that a tangent to it at any point in the field gives the direction of the tension at that point.

Lines of force close on positive and negative charges and cannot close on themselves.

Under equipotential surface understand a set of field points that have the same potential ().

If you cut the electrostatic field with a secant plane, then traces of the intersection of the plane with equipotential surfaces will be visible in the section. These traces are called equipotential lines.

Equipotential lines are closed to themselves.

Field lines and equipotential lines intersect at right angles.

R
Let's look at the equipotential surface:

(since the points lie on an equipotential surface).

- scalar product

The electrostatic field strength lines penetrate the equipotential surface at an angle of 90 0, then the angle between the vectors
is equal to 90 degrees, and their scalar product is equal to 0.

Equipotential Line Equation

Consider the line of force:

N
the intensity of the electrostatic field is directed tangentially to the line of force (see the definition of a line of force), and the path element is also directed , so the angle between these two vectors is zero.

or

Field line equation

Potential gradient

Potential gradient is the rate of potential increase in the shortest direction between two points.

There is some potential difference between two points. If this difference is divided by the shortest distance between the points taken, then the resulting value will characterize the rate of change of the potential in the direction of the shortest distance between the points.

The potential gradient shows the direction of the greatest increase in potential, is numerically equal to the voltage modulus and is negatively directed relative to it.

In defining the gradient, two provisions are essential:

The direction in which two nearby points are taken should be such that the rate of change is maximum.

The direction is that scalar function is increasing in this direction.

For a Cartesian coordinate system:

Rate of potential change in the direction of the X, Y, Z axis:

;
;

Two vectors are equal only if their projections are equal to each other. Projection of the tension vector onto the axis X equal to the projection of the rate of change of potential along the axis X, taken with the opposite sign. Same for axes Y And Z.

;
;
.

In a cylindrical coordinate system, the expression for the potential gradient will have the following form.

The relationship between tension and potential.

For a potential field, between the potential (conservative) force and potential energy there is a connection

where ("nabla") is the Hamiltonian operator.

Because the That

The minus sign shows that vector E is directed towards decreasing potential.

To graphically display the potential distribution, equipotential surfaces are used - surfaces at all points of which the potential has the same value.

Equipotential surfaces are usually drawn so that the potential differences between two adjacent equipotential surfaces are the same. Then the density of equipotential surfaces clearly characterizes the field strength in different points. Where these surfaces are denser, the field strength is greater. In the figure, the dotted lines show the lines of force, the solid lines show sections of equipotential surfaces for: a positive point charge (a), a dipole (b), two charges of the same name (c), a charged metal conductor complex configuration(G).

For a point charge the potential therefore equipotential surfaces are concentric spheres. On the other hand, tension lines are radial straight lines. Consequently, the tension lines are perpendicular to the equipotential surfaces.

It can be shown that in all cases the vector E is perpendicular to the equipotential surfaces and is always directed in the direction of decreasing potential.

Examples of calculations of the most important symmetrical electrostatic fields in vacuum.

1. Electrostatic field of an electric dipole in a vacuum.

An electric dipole (or double electric pole) is a system of two equal in magnitude opposite point charges (+q,-q), the distance l between which is significantly less than the distance to the field points under consideration (l<< r).

The dipole arm l is a vector directed along the dipole axis from the negative to the positive charge and equal to the distance between them.

The electric moment of the dipole re is a vector coinciding in direction with the dipole arm and equal to the product of the charge modulus |q| on shoulder I:

Let r be the distance to point A from the middle of the dipole axis. Then, given that

2) Field strength at point B on the perpendicular restored to the dipole axis from its center at

Point B is equidistant from the +q and -q charges of the dipole, so the field potential at point B is zero. Vector Ёв is directed opposite to vector l.

3) In an external electric field, a pair of forces acts on the ends of the dipole, which tends to rotate the dipole in such a way that the electric moment re of the dipole turns along the direction of the field E (Fig. (a)).

In an external uniform field, the moment of a pair of forces is equal to M = qElsin a or In an external inhomogeneous field (Fig. (c)), the forces acting on the ends of the dipole are not identical and their resultant tends to move the dipole to a field region with higher intensity - the dipole is pulled into a region of a stronger field.

2. Field of a uniformly charged infinite plane.

An infinite plane is charged with constant surface density The tension lines are perpendicular to the plane under consideration and directed from it in both directions.

As a Gaussian surface, we take the surface of a cylinder, the generators of which are perpendicular to the charged plane, and the bases are parallel to the charged plane and lie on opposite sides of it at equal distances.

Since the generators of the cylinder are parallel to the tension lines, the flux of the tension vector through the side surface of the cylinder is zero, and the total flux through the cylinder is equal to the sum of the fluxes through its bases 2ES. The charge contained inside the cylinder is equal to . By Gauss's theorem where:

E does not depend on the length of the cylinder, i.e. The field strength at any distance is the same in magnitude. Such a field is called homogeneous.

The potential difference between points lying at distances x1 and x2 from the plane is equal to

3. The field of two infinite parallel oppositely charged planes with equal absolute value surface charge densities σ>0 and - σ.

From the previous example it follows that the tension vectors E 1 and E 2 of the first and second planes are equal in magnitude and are everywhere directed perpendicular to the planes. Therefore, in the space outside the planes they compensate each other, and in the space between the planes the total tension . Therefore, between the planes

(in dielectric.).

The field between the planes is uniform. Potential difference between planes.
(in dielectric ).

4.Field of a uniformly charged spherical surface.

A spherical surface of radius R with a total charge q is charged uniformly with surface density

Since the system of charges and, consequently, the field itself is centrally symmetrical relative to the center of the sphere, the lines of tension are directed radially.

As a Gaussian surface, we choose a sphere of radius r that has a common center with the charged sphere. If r>R, then the entire charge q gets inside the surface. By Gauss's theorem, whence

At r<=R замкнутая поверхность не содержит внутри зарядов, поэтому внутри равномерно заряженной сферы Е = 0.

Potential difference between two points lying at distances r 1 and r 2 from the center of the sphere

(r1 >R,r2 >R), is equal to

Outside the charged sphere, the field is the same as the field of a point charge q located at the center of the sphere. There is no field inside the charged sphere, so the potential is the same everywhere and the same as on the surface