Force and equipotential field lines. §9 Lines of force and equipotentials

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

Imagine such a surface, for two points of which the potential difference will be equal to zero. This will be the equipotential surface. Because it has the same potential. If we consider the equipotential surface in two-dimensional space, let's say in the drawing, then it will have the shape of a line. The work of the forces of the electric field to move the 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 and vice versa. Any surface that is perpendicular at all points to the electric field lines is called an equipotential surface.

Also, such surfaces never intersect with each other. Since this would mean a difference in potential within the same 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, the drawings do not need to depict the entire surface. More often depict a perpendicular section to equipotential surfaces. Thus, they degenerate into lines. This turns out to be quite sufficient for estimating the distribution of this field. When depicted graphically, the surfaces are placed at the same interval. That is, between two adjacent surfaces, the same step is observed, let's say one volt. Then, according to the density of lines formed by the section of equipotential surfaces, one can judge the strength of the electric field.

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 go to 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 form of spheres centered in 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, the surfaces of equal potential will have the form of planes. These planes are parallel to each other at the same distance. True, at the edges of the plates, the field pattern will be distorted due to the edge effect. But we imagine that the plates are infinitely long.

Figure 2 - uniform field equipotential lines

To depict the equipotential lines for a field created by two charges equal in 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 the field of two electric charges

> Equipotential lines

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

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

Learning task

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

Key Points

• For a particular isolated point charge, the potential is based on the radial distance. Therefore, the equipotential lines are round.
• If several discrete charges are in contact, then their fields intersect and show a potential. As a result, the equipotential lines are skewed.
• When charges are distributed across two conductive plates in static balance, the equipotential lines are nearly straight.

Terms

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

Equipotential lines display one-dimensional areas where the electric potential remains unchanged. That is, for such a charge (wherever it is on the equipotential line) it is not necessary to do work to move from one point to another within a particular line.

The lines of an 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, a radial dependence is observed, that is, regardless of the distance to the point charge, the potential remains unchanged. Therefore, equipotential lines take on a circular 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 skew.

If multiple charges are present, then the equipotential lines form irregularly. At the point between charges, the control is able to sense the effects of both charges.

continuous charge

If the charges are located on two conducting plates in a static balance, where the charges are not interrupted and are on 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 devoid of interruption, then the equipotential lines go right in front of them. As an exception, we can only remember the bend near the edges of the conductive plates

The continuity is broken closer to the ends of the plates, due to which curvature is created in these areas - the edge effect.

THEORETICAL FOUNDATIONS OF THE WORK.

There is an integral and differential relationship between the strength of the electric fraction and the electric potential:

j 1 - j 2 = ∫ E dl (1)

E=-grad j (2)

The electric field can be represented graphically in two ways, complementing each other: using equipotential surfaces and lines of tension (lines of force).

A surface all of whose points have the same potential is called an equipotential surface. The line of its intersection with the plane of the drawing is called the equipotential. Lines of force - lines, tangents to which at each point coincide with the direction of the vector E . In Figure 1, the dotted lines show the equipotentials, the solid lines show the lines of force of the electric field.

Fig.1

The potential difference between points 1 and 2 is 0, since they are on the same equipotential. In this case, from (1):

∫E dl = 0 or ∫E dlcos ( Edl ) = 0 (3)

Insofar as E and dl in expression (3) are not equal to 0, then cos ( Edl ) = 0 . Therefore, the angle between the equipotential and the field line is p/2.

It follows from the differential relation (2) that the lines of force are always directed in the direction of decreasing potential.

The magnitude of the electric field strength is determined by the "thickness" of the lines of force. The thicker the lines of force, the smaller the distance between the equipotentials, so that the lines of force and equipotentials form "curvilinear squares". Based on these principles, it is possible to construct a picture of lines of force, having a picture of equipotentials, and vice versa.

A sufficiently complete picture of the field equipotentials allows us to calculate at different points the value of the projection of the intensity vector E to the chosen direction X , averaged over a certain interval of the coordinate ∆х :

E cf. ∆х = - ∆ j /∆х,

where ∆х - coordinate increment when moving from one equipotential to another,

∆ j - the corresponding increase in potential,

E cf. ∆х - mean E x between two potentials.

DESCRIPTION OF THE INSTALLATION AND MEASUREMENT TECHNIQUE.

To model the electric field, it is convenient to use the analogy that exists between the electric field created by charged bodies and the electric field of direct current flowing through a conductive film with uniform conductivity. In this case, the location of the lines of force of the electric field turns out to be similar to the location of the lines of electric currents.

The same statement is true for potentials. The distribution of field potentials in a conducting film is the same as in an electric field in vacuum.

As a conductive film, electrically conductive paper with the same conductivity in all directions is used in the work.

The electrodes are placed on the paper so that there is good contact between each electrode and the conductive paper.

The operating scheme of the installation is shown in Figure 2. The installation consists of module II, external element I, indicator III, power supply IV. The module is used to connect all used devices. The remote element is a dielectric panel 1, on which a sheet of white paper 2 is placed, a sheet of carbon paper 3 is placed on top of it, then a sheet of conductive paper 4, on which electrodes 5 are attached. Voltage is supplied to the electrodes from module II using connecting wires. Indicator III and probe 6 are used to determine the potentials of points on the surface of electrically conductive paper.

A wire with a plug at the end is used as a probe. Potential j probe is equal to the potential of the point on the surface of the electrically conductive paper, which it touches. The set of field points with the same potential is the image of the field equipotential. The power supply unit IV is used as a power supply unit TES - 42, which is connected to the module using a plug connector on the rear wall of the module. A voltmeter V7 - 38 is used as an indicator Ш.

ORDER OF PERFORMANCE OF WORK.

1. Place a sheet of white paper on the panel 1 2. Place carbon paper 3 and a sheet of conductive paper 4 on it (Fig. 2).

2. Install electrodes 5 on electrically conductive paper and secure with nuts.

3. Connect the power supply unit IV (TEC-42) to the module using the plug connector on the rear wall of the module.

4. Using two wires, connect indicator III (V7-38 voltmeter) to the "PV" sockets on the front panel of the module. Press the corresponding button on the voltmeter to measure the DC voltage (Fig. 2).

5. Using two conductors, connect electrodes 5 to module P.

6. Connect the probe (wire with two plugs) to the socket on the front panel of the module.

7. Connect the stand to the 220 V network. Turn on the general power supply of the stand.

Relationship between tension and potential.

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

where ("nabla") is the Hamilton operator.

Insofar as then

The minus sign shows that the vector E is directed in the direction of decreasing potential.

For a graphical representation of the potential distribution, equipotential surfaces are used - surfaces at all points of which the potential has the same value.

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

For a point charge, the potential therefore equipotential surfaces are concentric spheres. On the other hand, the lines of tension are radial straight lines. Therefore, the lines of tension 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 calculation of the most important symmetric electrostatic fields in vacuum.

1. Electrostatic field of an electric dipole in vacuum.

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

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

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

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

2) The field strength at point B on the perpendicular restored to the axis of the dipole from its middle at

Point B is equidistant from the charges +q and -q of the dipole, so the field potential at point B is zero. The vector Yb is directed opposite to the 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 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 the same and their resultant tends to move the dipole into the region of the field with a greater intensity - the dipole is drawn into the region of a stronger field.

2. The field of a uniformly charged infinite plane.

Infinite plane charged with constant surface density The tension lines are perpendicular to the considered plane 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 generatrices of the cylinder are parallel to the lines of tension, the flow of the tension vector through the side surface of the cylinder is equal to zero, and the total flow through the cylinder is equal to the sum of the flows through its bases 2ES. The charge inside the cylinder is . According to the Gauss theorem where:

E does not depend on the length of the cylinder, i.e. the field strength at any distance is the same in absolute value. 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 in absolute value surface charge densities σ>0 and - σ.

It follows from the previous example that the intensity vectors E 1 and E 2 of the first and second planes are equal in absolute value and directed everywhere 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 uniformly charged with surface density

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

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

For 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

Let's find the relationship between the strength of the electrostatic field, which is its power feature, and potential - energy characteristic of the field. Relocation work single point positive charge from one point of the field to another along the axis X provided that the points are infinitely close to each other and x 1 – x 2 = dx , equals E x dx . The same work is equal to j 1 -j 2 = dj . Equating both expressions, we can write

where the partial derivative symbol emphasizes that differentiation is made only with respect to X. Repeating similar reasoning for the y and z axes , we can find the vector E:

where i, j, k - unit vectors of coordinate axes x, y, z.

From the definition of the gradient (12.4) and (12.6). follows that

i.e., the field strength E is equal to the potential gradient with a minus sign. The minus sign is determined by the fact that the field strength vector E is directed to downward direction potential.

For a graphical representation of the distribution of the potential of an electrostatic field, as in the case of a gravitational field (see § 25), equipotential surfaces are used - surfaces at all points of which the potential j has the same value.

If the field is created by a point charge, then its potential, according to (84.5),

Thus, the equipotential surfaces in this case are concentric spheres. On the other hand, the lines of tension in the case of a point charge are radial straight lines. Therefore, the lines of tension in the case of a point charge perpendicular equipotential surfaces.

Tension lines always normal to equipotential surfaces. Indeed, all points of the equipotential surface have the same potential, so the work of moving the charge along this surface is zero, i.e., the electrostatic forces acting on the charge, always directed along the normals to the equipotential surfaces. Therefore, the vector E is always normal to equipotential surfaces, and therefore the lines of the vector E are orthogonal to these surfaces.

There are an infinite number of equipotential surfaces around each charge and each system of charges. However, they are usually carried out so that the potential differences between any two adjacent equipotential surfaces are the same. Then the density of equipotential surfaces clearly characterizes the field strength at different points. Where these surfaces are denser, the field strength is greater.

So, knowing the location of the electrostatic field strength lines, it is possible to construct equipotential surfaces and, conversely, from the known location of the equipotential surfaces, it is possible to determine the modulus and direction of the field strength at each point of the field. On fig. 133 for example shows the view of the lines of tension (dashed lines) and equipotential surfaces (solid lines) of the fields of a positive point charge (a) and a charged metal cylinder having a protrusion at one end and a depression at the other (b).