Charge carriers in a conductor are capable of moving under the influence of arbitrarily small forces. Therefore, to balance the charges on the conductor, it is necessary to perform following conditions:

The field strength everywhere inside the conductor must be zero,

In accordance with (8.2), this means that the potential inside the conductor must be constant).

2. The field strength on the surface of the conductor must be directed normal to the surface at each point:

Therefore, in the case of equilibrium of charges, the surface of the conductor will be equipotential.

If a conducting body is given a certain charge q, then it will be distributed so that the equilibrium conditions are met. Let us imagine an arbitrary closed surface completely contained within the body. When charges are in equilibrium, there is no field at each point inside the conductor; so the vector flow electrical displacement through the surface is zero. According to Gauss's theorem, the sum of charges inside the surface will also be equal to zero. This is true for a surface of any size, drawn inside the conductor in an arbitrary manner. Therefore, in equilibrium, there cannot be excess charges anywhere inside the conductor - they will all be distributed over the surface of the conductor with a certain density o.

Since in a state of equilibrium there are no excess charges inside the conductor, removing a substance from a certain volume taken inside the conductor will not in any way affect the equilibrium arrangement of charges. Thus, the excess charge is distributed on a hollow conductor in the same way as on a solid one, i.e., along its outer surface.

Excess charges cannot be located on the surface of a cavity in a state of equilibrium. This conclusion also follows from the fact that the same elementary charges, forming a given charge q, repel each other and, therefore, tend to settle on greatest distance from each other.

Let's imagine a small cylindrical surface, formed by the normals to the surface of the conductor and the bases of the dS value, one of which is located inside and the other outside the conductor (Fig. 24.1). Electrical displacement vector flow through inner part surface is zero, since inside the conductor E, and therefore D, is zero. Outside the conductor, in close proximity to it, the field strength E is directed normal to the surface. Therefore, for the outwardly protruding side surface of the cylinder, and for the outer base (the outer base is assumed to be located very close to the surface of the conductor). Therefore, the displacement flux through the surface under consideration is where D is the magnitude of the displacement in the immediate vicinity of the conductor surface. Inside the cylinder there is a foreign charge (- charge density in this place conductor surface). Applying Gauss's theorem, we obtain: It follows that the field strength near the surface of the conductor is equal to

36) Laplace and Poisson equations. General task electrostatics

Poisson's and Laplace's equations are the basic differential equations of electrostatics. They follow from Gauss's theorem in differential form. Indeed, substituting into the equation

instead of E x values; E y; E z their expressions through potential:

we get the equation

This differential equation is called Poisson equations .

Integral

is a solution to the Poisson equation for the case when the charges are distributed in a finite region of space.

If there are no volumetric electric charges in the region of space under consideration, then Poisson’s equation takes the form

and is called in this particular case Laplace's equation .

Note that in cylindrical and spherical systems coordinates, the Poisson and Laplace equations have a different form of writing. Therefore, these equations are often written in a form that does not depend on the coordinate system.

The main feature of conductors is the presence of free charges (electrons), which participate in thermal movement and can move throughout the entire volume of the conductor. Typical conductors are metals.

In the absence of external field in any element of the volume of the conductor, the negative free charge is compensated positive charge ionic lattice. In a conductor introduced into an electric field, a redistribution of free charges occurs, as a result of which uncompensated positive and negative charges(Fig. 1.5.1). This process is called electrostatic induction, and the charges that appear on the surface of the conductor are called induction charges.

Induction charges create their own field which compensates for the external field throughout the entire volume of the conductor: (inside the conductor).

The total electrostatic field inside the conductor is zero, and the potentials at all points are the same and equal to the potential on the surface of the conductor.

Charge carriers in a conductor are capable of moving under the influence of arbitrarily small forces. Therefore, charge equilibrium on a conductor can be observed only if the following conditions are met:

1. The field strength everywhere inside the conductor must be zero. According to the equation, this means that the potential inside the conductor must be constant, i.e.

2. The field strength on the surface of the conductor must be directed normal to the surface at each point, otherwise a component appears directed along the surface, which will lead to the movement of charges until the component disappears.

Consequently, in the case of equilibrium of charges, the surface of the conductor will be equipotential. If a conducting body is given a certain charge q, then it will be distributed so that the equilibrium conditions are met.

Therefore, in equilibrium, there cannot be excess charges at any place inside the conductor - they are all located on the surface of the conductor with a certain density. Because in a state of equilibrium inside the conductor there are no excess charges; removing a substance from a certain volume taken inside the conductor will not in any way affect the equilibrium arrangement of charges. Thus, the excess charge is distributed on a hollow conductor in the same way as on a solid one, i.e. along its outer surface. Excess charges cannot be located on the surface of a cavity in a state of equilibrium.

8) Field strength near the surface of a charged conductor.

Let us select a platform on the surface S of the conductor and construct a cylinder on it with generatrices perpendicular to the platform and height :

On the surface of a conductor, the field strength vector and the electric displacement vector are perpendicular to the surface. Therefore the flow through lateral surface equal to zero.

The flux of the electric displacement vector through is also zero, since it lies inside the conductor, where and, therefore, . It follows that flow = through a closed surface equal to the flow through :

Charge carriers in a conductor are capable of moving under the influence of arbitrarily small forces. Therefore, to balance the charges on the conductor, the following conditions must be met:

In accordance with (8.2) this means that the potential inside the conductor must be constant).

2. The field strength on the surface of the conductor must be directed normal to the surface at each point:

Therefore, in the case of equilibrium of charges, the surface of the conductor will be equipotential.

If a conducting body is given a certain charge q, then it will be distributed so that the equilibrium conditions are met. Let us imagine an arbitrary closed surface completely contained within the body. When charges are in equilibrium, there is no field at each point inside the conductor; therefore, the flux of the electrical displacement vector through the surface is zero. According to Gauss's theorem, the sum of charges inside the surface will also be equal to zero. This is true for a surface of any size, drawn inside the conductor in an arbitrary manner. Therefore, in equilibrium, there cannot be excess charges anywhere inside the conductor - they will all be distributed over the surface of the conductor with a certain density o.

Since in a state of equilibrium there are no excess charges inside the conductor, removing a substance from a certain volume taken inside the conductor will not in any way affect the equilibrium arrangement of charges. Thus, the excess charge is distributed on a hollow conductor in the same way as on a solid one, i.e., along its outer surface.

Excess charges cannot be located on the surface of a cavity in a state of equilibrium. This conclusion also follows from the fact that the elementary charges of the same name that form a given charge q repel each other and, therefore, tend to be located at the greatest distance from each other.

Let's imagine a small cylindrical surface formed by normals to the surface of the conductor and bases of magnitude dS, one of which is located inside and the other outside the conductor (Fig. 24.1). The flux of the electric displacement vector through the interior of the surface is zero, since E, and therefore D, is zero inside the conductor. Outside the conductor, in close proximity to it, the field strength E is directed normal to the surface. Therefore, for the outwardly protruding side surface of the cylinder, and for the outer base (the outer base is assumed to be located very close to the surface of the conductor). Therefore, the displacement flux through the surface under consideration is equal to , where D is the magnitude of the displacement in the immediate vicinity of the conductor surface. Inside the cylinder there is a foreign charge ( - charge density at a given location on the surface of the conductor). Applying Gauss's theorem, we obtain: It follows that the field strength near the surface of the conductor is equal to

Where - the dielectric constant environment surrounding the conductor (compare with formula (14.6) obtained for the case)

Let's consider the field created by the one shown in Fig. 24.2 with a charged conductor. On long distances from the conductor equipotential surfaces have a characteristic point charge the shape of a sphere (in the figure due to lack of space spherical surface depicted at a short distance from the conductor; The dotted line shows the field strength lines). As you approach the conductor, the equipotential surfaces become more and more similar to the surface of the conductor, which is equipotential. Near the protrusions, the equipotential surfaces are denser, which means that the field strength is greater here. It follows that the charge density on the protrusions is especially high (see (24.3)). The same conclusion can be reached by taking into account that due to mutual repulsion, charges tend to be located as far away from each other as possible.

Near the depressions in the conductor, equipotential surfaces are less common (see Fig. 24.3). Accordingly, the field strength and charge density in these places will be lower. In general, the charge density at given potential The conductor is determined by the curvature of the surface - it increases with increasing positive curvature (convexity) and decreases with increasing negative curvature (concavity). The charge density at the tips is especially high. Therefore, the field strength near the tips can be so high that ionization of the gas molecules surrounding the conductor occurs.

Ions of a different sign than q are attracted to the conductor and neutralize its charge. Ions of the same sign as q begin to move away from the conductor, carrying with them neutral gas molecules. The result is a tangible movement of gas called electric wind. The charge of the conductor decreases; it flows off the tip and is carried away by the wind. Therefore, this phenomenon is called the outflow of charge from the tip.

Within the framework of electrostatics, we consider problems in which the charge distribution is different static . In other words, such states of bodies that are realized after the bodies of the systems under consideration came into equilibrium after some influences, for example, a charge message, placement in an electric field, etc. Conductors , unlike dielectrics, contain free charge carriers , which can move throughout the volume of the conductor. In the case of metals, such charge carriers are electrons. The speed of their movement through the metal is very high, so the metals come to equilibrium in very small fractions of a second. In the case of other materials, it may turn out that the transition to equilibrium occurs much more slowly, but we will now consider situations where equilibrium has been achieved.

IN state of equilibrium the following conditions are met:

1. The field strength inside the conductor was zero: .

2. On the surface (near, in the immediate vicinity...) of the conductor, the tension electric field perpendicular to the surface.

These conditions are consequences of the presence in the conductor free media charge. Indeed, in equilibrium there should be no movement of charges, and, therefore, the field strength inside the conductor should be equal to zero. A consequence of this condition is the statement that all points of the conductor must have the same potential, and the surface of the conductor is equipotential .

Since there cannot be uncompensated charges inside a conductor in equilibrium (they would create a non-zero field inside the conductor), then the charge imparted to the conductor is located very thin layer conductor near the surface, i.e. on the surface of the conductor .

On the surface of the conductor at the electric field strength vector there must be no tangential (component directed tangentially to the surface) component . If it were present, there would have to be a movement of charges along the surface, which cannot happen in equilibrium. This statement is true for any direction, therefore the tension vector must be perpendicular to the surface .

The charge imparted to the conductor is located on its surface with density . The flow of the electrical induction vector through the surface of the cylinder shown in Figure 16.1, according to Gauss’s theorem, should be equal to the value free charge contained inside the surface - . However, there is no flux through the side surface, since the voltage vector (and therefore the induction vector) is parallel to it, there is no flux through the base inside the conductor - there is no electric field, and the flux through the external base is equal to . That's why

Let's imagine a solitary conductor to which a certain charge is imparted. At a large distance from it compared to the size of the conductor, regardless of the shape of the conductor, it can be considered point charged body . The equipotential surfaces of a point charge are spheres. Near the conductor, the equipotential surfaces should approximately follow its shape. As a result, near the ends of the conductor, the equipotential surfaces become denser. This means that the potential at these points in space changes quickly, and the field strength, accordingly, reaches large values. Due to the high field strength near the sharp ends of the conductors, it is possible that gas discharge, accompanied by the flow of charge from the conductor. For this reason, elements of high-voltage power lines must be made with rounded surfaces.

When a conductor is placed in an external field, the free charges of the conductor are displaced until the equilibrium conditions are met. In this case, charges arise in different parts of the conductor, distributed over its surface with a certain density so that the equilibrium conditions are satisfied. These charges are called induced, and the very phenomenon of their occurrence is electrical induction (not to be confused with the vector of electrical induction!).

Conditions for equilibrium of charges in a conductor. Electric field inside conductors

Conductors– bodies containing great amount free electrically charged particles. These particles can move inside the conductor under the influence of arbitrarily small forces.

To balance the charges in a conductor, the following conditions must be met:

But, therefore

The potential inside the conductor must be constant.

2. The tension on the surface of the conductor must be directed normal to the surface at each point.

If a conductor is given a certain charge, it will be distributed over the surface so that these equilibrium conditions are again observed.

If an uncharged conductor is introduced into an external electric field, then the charge carriers in the conductor will begin to move - the electrons will begin to move against the direction of the voltage vector. As a result, charges will arise at the ends of the conductor opposite sign. These are induced charges. Inside the conductor, its own electric field is formed, directed against the external one, it weakens the external field by superimposing on it. Redistribution of charges occurs until the conditions for equilibrium of charges in the conductor are met, those. the tension inside will not be equal to zero, and the lines will not be outside perpendicular to the surface( And, ). Thus, a conductor introduced into the field breaks the lines of tension. They end in negative

induced charges, but begin on positive

induced charges. The induced charges are distributed over outer surface conductor. If there is a cavity inside the conductor, then with an equilibrium distribution of charges there is no field inside the cavity. Electrostatic protection is based on this.