Conductors are bodies in which electrical charges are capable of moving under the influence of any weak force. electrostatic field.
As a result, the charge imparted to the conductor will be redistributed until at any point inside the conductor the voltage electric field will not be equal to zero.
Thus, the electric field strength inside the conductor must be zero.
Since , then φ=const
The potential inside the conductor must be constant.
2.) On the surface of a charged conductor, the voltage vector E must be directed normal to this surface, otherwise under the influence of a component tangent to the surface (E t). charges would move along the surface of the conductor.
Thus, provided static distribution charge tension on the surface
where E n is the normal component of tension.
This implies, 
that when the charges are in equilibrium, the surface of the conductor is equipotential.
3. In a charged conductor, uncompensated charges are located only on the surface of the conductor.
Let us draw an arbitrary closed surface S inside the conductor, limiting a certain internal volume of the conductor. According to Gauss's theorem, the total charge of this volume is equal to:
Thus, in a state of equilibrium there are no excess charges inside the conductor. Therefore, if we remove a substance from a certain volume taken inside a conductor, this 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 inner surface. This also follows from the fact that like charges repel each other and, therefore, tend to be located at the greatest distance from each other.
Investigating the magnitude of the electric field strength near the surface of charged bodies various shapes One can also judge the distribution of charges over the surface.
Research has shown that the charge density at a given conductor potential is determined by the curvature of the surface - it increases with increasing positive curvature (convexity) and decreases with increasing negative curvature (concavity). The density at the tips is especially high. The field strength near the tips can be so high that ionization of the molecules of the surrounding gas occurs. In this case, the charge of the conductor decreases; it seems to flow off the tip.
If you place an electric charge on the inner surface of a hollow conductor, this charge will transfer to outer surface conductor, increasing the potential of the latter. By repeatedly repeating the transfer to a hollow conductor, its potential can be significantly increased to a value limited by the phenomenon of charges flowing off the conductor. This principle was used by Van der Graaff to build an electrostatic generator. In this device, the charge from an electrostatic machine is transferred to an endless non-conducting tape, carrying it inside a large metal sphere. There the charge is removed and transferred to the outer surface of the conductor, thus it is possible to gradually impart a very large charge to the sphere and achieve a potential difference of several million volts.
Conductors in an external electric field.
Not only charges brought from outside, but also the charges that make up the atoms and molecules of the conductor (electrons and ions) can move freely in conductors. Therefore, when placing an uncharged conductor in an external electric field free charges will move towards its surface, positive charges along the field, and negative charges against the field. As a result, charges of opposite sign arise at the ends of the conductor, called induced charges. This phenomenon, consisting in the electrification of an uncharged conductor in an external electrostatic field by dividing on this conductor the positive and negative electrical charges already present in it in equal quantities, is called electrification through influence or electrostatic induction .
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The movement of charges in a conductor placed in an external electric field E 0 will occur until the additional field E additional created by induction charges compensates for the external field E 0 at all points inside the conductor and the resulting field E inside the conductor becomes equal to zero.
The total field E near the conductor will differ noticeably from its initial value E 0 . The lines E will be perpendicular to the surface of the conductor and will partially end at the induced negative charges and begin again at the induced positive charges.
Charges induced on a conductor disappear when the conductor is removed from the electric field. If you first divert induced charges of one sign to another conductor (for example, into the ground) and turn off the latter, then the first conductor will remain charged with electricity of the opposite sign.
The absence of a field inside a conductor placed in an electric field is widely used in technology for electrostatic protection from external electric fields (shielding) of various electrical devices and wires. When they want to protect a device from external fields, it is surrounded by a conductive case (screen). Such a screen also works well if it is made not continuous, but in the form of a dense mesh.
The ideal physical model of charge in electrostatics is a point charge.
Spot A charge is a charge concentrated on a body, the dimensions of which can be neglected in comparison with the distance to other bodies or to the field point in question. In other words, a point charge is material point, which has an electric charge.
If the charged body is so large that it cannot be considered as a point charge, then in this case it is necessary to know distribution charges inside the body.
Let us select a small volume inside the charged body and denote by the electric charge located in this volume. The limit of the ratio, when the volume decreases without limit, is called volumetric density of electric charge at a given point. It is designated by the letter:
The SI unit of volumetric charge density is the coulomb per cubic meter(C/m 3).
In the case of an unevenly charged body, the density is different at different points. The charge distribution in the volume of the body is specified if known as a function of coordinates.
In metallic bodies, charges are distributed only within a thin layer adjacent to the surface. In this case it is convenient to use surface charge density, which represents the limit of the ratio of charge to the surface area over which this charge is distributed:
where is the charge located on a surface area of .
Consequently, surface charge density is measured by the charge per unit surface area of the body. The distribution of charges over the surface is described by the dependence of the surface density (x, y, z) on the coordinates of surface points.
The SI unit of surface charge density is the coulomb per square meter(C/m2).
In the event that the charged body is shaped like a thread (diameter cross section body is much smaller than its length, it is convenient to use linear charge density
where is the charge located along the length of the body.
The SI unit of linear charge density is coulomb per meter (C/m).
If the distribution of charges inside a body is known, then the strength of the electrostatic field created by this body can be calculated. To do this, a charged body is mentally divided into infinitesimal parts and, considering them as point charges, the field strength created by in separate parts bodies. The total field strength is then found by summing the fields created by individual parts of the body, i.e.
We have seen that the surface of a conductor, whether neutral or charged, is an equipotential surface (§ 24) and inside the conductor the field strength is zero (§ 16). The same applies to a hollow conductor: its surface is an equipotential surface and the field inside the cavity is zero, no matter how strongly the conductor is charged, unless, of course, inside the cavity there are no charged bodies isolated from the conductor.
This conclusion was clearly demonstrated by the English physicist Michael Faraday (1791-1861), who enriched science with a number of major discoveries. His experience was as follows. A large wooden cage was covered with sheets of staniol (tin paper), insulated from the Earth and highly charged with electric machine. Faraday himself was placed in the cage with a very sensitive electroscope. Despite the fact that sparks flew from the outer surface of the cell when bodies connected to the Earth approached it, indicating a large potential difference between the cell and the Earth, the electroscope inside the cell did not show any deviation (Fig. 53).
Rice. 53. Faraday's experiment
A modification of this experiment is shown in Fig. 54. If we make a closed cavity out of a metal mesh and hang pieces of paper on the inside and outside of the cavity, we will find that only the outer sheets are deflected. This shows that the electric field exists only in the space between the cell and the objects surrounding it, that is, outside the cell; There is no field inside the cell.
Rice. 54. Modification of Faraday's experiment. The metal cage is charged. The pieces of paper on the outside are deflected, indicating the presence of charge on the outer surfaces of the cage walls. There is no charge inside the cell, the pieces of paper do not deviate
When charging any conductor, the charges are distributed in it so that the electric field inside it disappears, and the potential difference between any points becomes zero. Let's see how the charges should be placed for this.
Let's charge a hollow conductor, for example, a hollow insulated ball 1 (Fig. 55), which has a small hole. Let's take a small metal plate 2 mounted on an insulating handle (“test plate”), touch it to some place on the outer surface of the ball and then bring it into contact with the electroscope. The sheets of the electroscope will diverge at a certain angle, indicating that the test plate has become charged upon contact with the ball. If, however, we touch the inner surface of the ball with the test plate, the plate will remain uncharged, no matter how strongly the ball is charged. Charges can only be drawn from the outer surface of the conductor, but this turns out to be impossible from the inner surface. Moreover, if we pre-charge the test plate and touch it to the inner surface of the conductor, then all the charge will transfer to this conductor. This happens regardless of what charge was already on the conductor. In § 19 we explained this phenomenon in detail. So, in a state of equilibrium, charges are distributed only on the outer surface of the conductor. Of course, if we repeated the experiment depicted in Fig. 45, touching the conductor with the end of the wire leading to the electrometer, you would be convinced that the entire surface of the conductor, both external and internal, is the surface of the same potential: the distribution of charges over the external surface of the conductor is the result of the action of the electric field. Only when the entire charge is transferred to the surface of the conductor will equilibrium be established, i.e., inside the conductor the field strength will become zero and all points of the conductor (outer surface, inner surface and points in the thickness of the metal) will have the same potential.
Rice. 55. Study of charge distribution in conductor 1 using test plate 2. There is no charge inside the cavity of the conductor
Thus, a conducting surface completely protects the area it surrounds from the action of the electric field created by charges located on or outside this surface. The external field lines end on this surface; they cannot pass through the conducting layer, and the internal cavity is free from the field. Therefore such metal surfaces called electrostatic protection. It is interesting to note that even a surface made of metal mesh can serve as protection, as long as the mesh is thick enough.
31.1. There is a charge in the center of a hollow, insulated metal ball. Will a charged weight suspended on a silk thread and placed outside the ball be deflected? Analyze in detail what happens. What happens if the ball is grounded?
31.2. Why are powder warehouses surrounded on all sides by a grounded metal mesh to protect them from lightning strikes? Why brought into such a building water pipes must also be well grounded?
The fact that charges are distributed on the outer surface of a conductor is often used in practice. When they want to completely transfer the charge of some conductor to an electroscope (or electrometer), then a closed metal cavity is connected to the electroscope, if possible, and a charged conductor is introduced into this cavity. The conductor is completely discharged, and all its charge is transferred to the electroscope. This device is called a “Faraday cylinder” in honor of Faraday, since in practice this cavity is most often made in the form of a metal cylinder. We have already used this property of a Faraday cylinder (glass) in the experiment shown in Fig. 9, and explained it in detail in § 19.
Van de Graaff proposed using the properties of a Faraday cup to obtain very high voltages. The operating principle of its generator is shown in Fig. 56. An endless tape 1 made of some insulating material, for example silk, moves with the help of a motor on two rollers and one end goes inside a hollow metal ball 2, isolated from the Earth. Outside the ball, the tape is charged with a brush 3 by some source , for example, a battery or an electric machine 4, up to a voltage of 30-50 kV relative to the Earth, if the second pole of the battery or machine is grounded. Inside the ball, 2 charged sections of the tape touch the brush 5 and completely transfer their charge to the ball, which is immediately redistributed over the outer surface of the ball. Thanks to this, nothing prevents the continuous transfer of charge to the ball. The voltage between ball 2 and the Earth continuously increases. In this way, voltages of several million volts can be achieved. Similar machines were used in experiments on splitting atomic nuclei.
Rice. 56. The principle of the Van de Graaff generator
31.3. Could the Van de Graaff generator described above work if the ball were made of an insulating material or if the conveyor belt in it were conductive (metal)?
One of the general problems of electrostatics is to determine the electric field or potential for a given surface charge distribution. Gauss's theorem (1.11) allows us to immediately write some particular relation for the electric field. If on a surface S with a unit normal the charge is distributed with surface density , and the electric field on both sides of the surface is equal, respectively (Fig. 1.4), then, according to Gauss’s theorem,
This relationship does not yet determine the fields themselves, the only exception being those cases when there are no other sources of the field other than surface charges with a density and the distribution has a particularly simple form. Relation (1.22) only shows that when moving from the “inner” side of the surface, on which the surface charge a is located, to the “outer” side, the normal component of the electric field experiences a jump
Using relation (1.21) for the linear integral of E over closed loop, it can be shown that the tangential component of the electric field is continuous when passing through the surface.
Fig. 1.4. A jump in the normal component of the electric field when crossing the surface distribution of charges.
The general expression for the potential created by the surface charge distribution at an arbitrary point in space (including on the very surface S on which the charges are located) can be found from (1.17), replacing by
The expression for the electric field can be obtained from here by differentiation.
Also of interest is the problem of the potential created by a double layer, i.e., the distribution of dipoles over the surface
Fig. 1.5. Transition to the limit during the formation of a double layer.
A double layer can be imagined as follows: let the charge be located on the surface S with a certain density , and on the surface S close to S, the surface density at the corresponding (adjacent) points is , i.e. equal in value and opposite in sign (Fig. 1.5). Double layer, i.e. dipole distribution with moment per unit surface
turns out to be a limiting transition, in which S approaches infinitely close to S, and the surface density tends to infinity so that the product by the distance between at the corresponding point tends to the limit
The dipole moment of the layer is perpendicular to the surface S and is directed from the negative to the positive charge.
To find the potential created by a double layer, one can first consider an individual dipole and then move on to the distribution of dipoles over the surface. The same result can be reached if we start from the potential (1.23) for the surface charge distribution and then carry out the passage to the limit described above. The first method of calculation is perhaps simpler, but the second is a useful exercise in vector analysis, so we prefer the second one here.
Fig. 1.6. Double layer geometry.
Let the unit normal vector be directed from S to S (Fig. 1.6). Then the potential due to two close surfaces S and S is equal to
For small d we can expand the expression into a series. Let us consider the general expression in which In this case
Obviously, this is simply a Taylor series expansion in the three-dimensional case. Thus, passing to the limit (1.24), we obtain the expression for the potential
Relation (1.25) can be very simply interpreted geometrically. notice, that
where is the solid angle element at which the area element is visible from the observation point (Fig. 1.7). The value is positive if the angle is acute, i.e., the “inner” side of the double layer is visible from the observation point.
Fig. 1.7. Towards the conclusion of the double layer potential. The potential at point P, created by an area element of a double layer with a unit surface moment D, is equal to the product of the moment D taken with the opposite sign and the solid angle at which the area element from point P is visible.
The expression for the double layer potential can be written as
If the surface density of the dipole moment D is constant, then the potential is simply equal to the product of the dipole moment taken with the opposite sign and the solid angle at which the entire surface is visible from the observation point, regardless of its shape.
When crossing the double layer, the potential undergoes a jump equal to the surface dipole moment density times . This is easy to verify if we consider an observation point approaching infinitely close to the surface S with inside. Then, according to (1.26), the potential on the internal
side will be equal
since almost the entire solid angle rests on a small portion of the surface S near the observation point. Similarly, if you approach the surface S from outside, then the potential becomes equal
the sign is reversed due to a change in the sign of the solid angle. Thus, the potential jump when crossing the double layer is equal to
This relationship is an analogue of formula (1.22) for the jump in the normal component of the electric field when crossing a “simple” layer, i.e., the surface charge distribution. Relationship (1.27) can be physically interpreted as a drop in potential “inside” the double layer. This potential drop can be calculated (before going to the limit) as the product of the field strength between both layers carrying a surface charge and the distance between them.
Let's show that ~
Topic 4. Question 3.
Distribution of charges in conductors.
Conductors in an electrostatic field.
When an uncharged conductor is introduced into an external electrostatic field, charges appear on its surface. The phenomenon of charge redistribution in a conductor when it is introduced into an external electrostatic field is called electrostatic induction ( induction of charges, electrification by induction).
1) If an uncharged metal conductor from two contacting parts is introduced into the field, induced charges will appear on their surfaces. If these parts are separated using insulating handles, then each part will be charged with the corresponding charge (see figure). In this case, the field strength inside the conductors is always zero.
2) An uncharged conductor introduced into an electrostatic field distorts the field (see Fig. - lines with arrows - power lines external uniform field; the lines perpendicular to them are equipotential surfaces; ± - induced charges are indicated).
3) The magnitude of the induced charge is always less than the magnitude of the induced charge. Only in the case when the induced charge is located inside a metal cavity, the induced charge turns out to be the same in magnitude, but at the same time the surface charge density turns out to be different. In the figure: a point charge is surrounded by an uncharged metal hollow body. Both the inner and outer surfaces are spherical, but their centers are shifted. The induced charge is distributed uniformly on the outer surface, but in a complex manner on the inner surface.
4) Induced charges affect the electric field of the induced charges.
5). An induced charge also occurs on an already charged body. If there are two positive charges nearby + Q and + q, they must push off. But the induced negative charge on one of the charges may turn out to be greater than its own charge, and the charges will be attracted to each other.
Electrostatic protection: Conductor or thick enough metal grid, surrounding a certain area on all sides, shielding it from electric fields created by external charges.
Topic 5. Question 1.
Electrical capacity.
All conductors have the property of accumulating electrical charges. This property is called electrical capacitance. The quantitative characteristic of this property is also called electrical capacity and is denoted WITH. A distinction is made between the electrical capacitance of a solitary conductor (its own capacitance), located far from other conductors, and the mutual capacitance of a system of two or more conductors.
The farad, the SI unit of capacitance, is an extremely large unit. Thus, the capacity of the globe is approximately 7 × 10 - 4 F, so micro-, nano- and picofarads are usually used.
Intrinsic capacitance depends only on the shape and size of the conductor and on the dielectric properties environment(vacuum, air, kerosene,...) and does not depend on the material of the conductor (Fe, Cu, Al,...), nor on whether it is charged or not. Each isolated conductor has “its own” capacitance; if, for example, you bend a piece of wire or make a dent in a ball, their capacitance will change.
Calculating capacitance is a complex mathematical problem, and if the conductor has a complex configuration, then this problem cannot be solved analytically.
Let's calculate electrical capacity of a solitary sphere (ball).
Topic 5. Question 2.
Electrical capacity.
Let's calculate parallel plate capacitor capacitance– these are two metal parallel plates (plates) same sizes, separated by a dielectric layer (vacuum, air, etc.). If the distance between the plates is significant smaller sizes plates: d<<L, H, the field between the plates can be considered uniform. In fact, near the edges of the plates the field is inhomogeneous (see the figure, which shows half of a flat capacitor, lines with arrows are lines of force, without arrows are equipotential surfaces). It is difficult to take these edge effects into account.
Topic 5. Question 3.
Electrical capacity.
Mutual capacitance also depends on the shape and size of the conductors and, in addition, on their relative position. A system of two conductors is called a capacitor when the distance between them is sufficiently small that the electric field (when they are charged) is concentrated mainly between the conductors. The conductors themselves are called plates. The capacity of such a system can be calculated for plates of simple shapes: flat, spherical and cylindrical (without taking into account edge effects).
Cylindrical capacitor. These are two coaxial metal cylinders, with a dielectric (vacuum, air, etc.) in between. Length of cylinder linings l, radii R And r(see picture). If you give the inner lining a charge + q, charges are induced on the outer plate - q and + q, the positive charge from the outer surface of the outer lining is transferred to the ground. The capacitor field is mainly concentrated between the plates if the distance between them ( R-r) << l. We do not take edge effects into account.
Topic 5. Question 4.
Electrical capacity.
Mutual capacitance also depends on the shape and size of the conductors and, in addition, on their relative position. A system of two conductors is called a capacitor when the distance between them is sufficiently small that the electric field (when they are charged) is concentrated mainly between the conductors. The conductors themselves are called plates. The capacity of such a system can be calculated for plates of simple shapes: flat, spherical and cylindrical (without taking into account edge effects)
Spherical capacitor. These are two metal concentric spheres separated by a spherical dielectric layer. If the inner lining is charged + q, a charge is induced on the inner surface of the outer plate - q, and on its outer surface + q. This charge is discharged into the ground due to grounding (see figure). The field of such a capacitor is concentrated only between the plates.
Topic 5. Question 5.
Electrical capacity.
Capacitor connections.
Capacitors can be connected in parallel or in series, or in a mixed way: some in parallel, some in series. With a parallel connection, the system capacity increases and becomes equal to the sum of the capacities. With a series connection, the system capacity always decreases. A series connection is used not to reduce the capacitance, but mainly to reduce the potential difference across each capacitor so that there is no breakdown of the capacitor.
| Let us introduce a simpler notation for the potential difference. Sometimes U called voltage, this is an outdated term. Voltage U = IR- this is the product of current and resistance (see below - current), and no current should flow through the capacitor. If dielectric breakdown occurs, the capacitor must be thrown away. | ||
 | Let's write the formula for each capacitor and for the entire system (replacing D j® U); substituting q into the last formula, we get: C parallel = C 1 + C 2 Let's generalize to the case of 3 or more capacitors |  parallel connection |
 | system capacitance when connecting capacitors in parallel( i=1,2,…,n) n- number of capacitors |
Topic 6. Question 1.