Metal conductors are generally neutral: they contain equal amounts of negative and positive charges. Positively charged are the ions in the nodes crystal lattice, negative - electrons moving freely along the conductor. When a conductor is given an excess amount of electrons, it becomes charged negatively, but if a certain number of electrons are “taken” from the conductor, it becomes charged positively.

The excess charge is distributed only over the outer surface of the conductor. If the conductor is hollow, then there are no charges on its internal surfaces. This is used to completely transfer charge from one conductor to another (see Fig. 8).

The absence of a field inside the cavity in the conductor allows for electrostatic protection. A conductor or a fairly dense metal mesh surrounding a certain area on all sides shields it from electric fields created by external charges.

In electrostatics, a stationary, unchanging distribution of charges is considered. The condition for stationarity is the equality of the field strength inside the conductor to zero: E = 0. If the intensity were not equal to zero, this would create electrical forces causing the directed movement of electrons, i.e. electric current.

Excess charges imparted to the conductor are distributed evenly only over the surface of the metal sphere or ball. In all other cases, the charges are distributed unevenly: the greater the surface curvature, the greater the surface charge density on the surface of the conductor. Let's prove it. Let's take two balls of radii R 1 and R 2, charged with charges q 1 and q 2, respectively. Let's connect them with wire. Charges will move from one ball to another until the potential of the entire system becomes the same. We will neglect the influence of the wire.

Table 14

Let's find the field strength of a charged conductor near its surface using Gauss's theorem. The entire conductor represents one equipotential surface. Field lines are perpendicular to equipotential surfaces. Let us choose a very small cylinder as the Gaussian surface S, the generatrices of which are perpendicular to the surface of the conductor (see Fig. 9). Within the cylinder, the surface charge density will be considered constant.

Table 15

Thus, the more curved the surface of a charged conductor is, the more charges accumulate on it and the greater the field strength in this place. The figure shows the field lines and equipotential surfaces of the field of a charged body. The greatest tension is obtained at sharp surface protrusions. This leads to the so-called “draining of charges”. In fact, due to the high voltage near the tip, complex phenomena arise: air molecules can be ionized, dipole molecules are drawn into the region of a stronger field, as a result, the speed of particle flow from the tip is greater, and an “electric wind” is formed. This wind can cause a light wheel located near the tip to rotate. The air becomes a conductive medium, a discharge occurs, and a glow is often observed near the sharp ends. Therefore, all parts in electrical installations under high voltage are given a rounded shape and their surfaces are made smooth.

He will be completely safe inside the metal cabin if he does not try to get out of it until the outer part is discharged or de-energized. Airplane passengers are safe when lightning strikes because the charge is conducted around the outside of the fuselage into the underlying atmosphere. Experiments were carried out in which a potential of 1 million V was applied to the roof of a car driving past a high-voltage generator. Despite the enormous charge between the generator and the car, the driver could repeat the experiment without any damage to yourself, and for the car. These experiments show that the charge is located on the outer surface of the conductor.


Note.

This applies equally to hollow and monolithic conductors, and, of course, to insulators.

If a certain negative charge is placed on a metal sphere located on an insulating stand, as in Figure 1, a, then the negative charges repulse each other and move through the metal. The electrons are distributed until each point on the sphere rises to the same negative potential; charge redistribution then stops. All points on the charged sphere must have the same potential, since if this did not happen, then there would have to be a potential difference between different points on the conductor. This would cause the charges to move until the potentials were equal. A charged conductor, regardless of its shape, must therefore have the same potential at all points both on and inside its surface. The cylindrical conductor in Figure 1, b has a constant positive potential at all points on its surface. In the same way, the negatively charged pear-shaped conductor in Figure 1b has a constant negative potential over its entire surface. So, the charge is distributed in such a way that the potential is uniform throughout the conductor. On bodies of regular shape, such as a sphere, the charge distribution will be uniform or homogeneous. On the bodies irregular shape, such as those shown in Figure 1, b and c, there is no uniform charge distribution over their surface. The charge that accumulates at any given point on a surface depends on the curvature of the surface at that point. The greater the curvature, i.e., the smaller the radius, the greater the charge. Thus, a large concentration of charge is present at the “pointed” end of the pear-shaped conductor in order to maintain the same potential at all points on the surface.


Similar experiments can be carried out to check the distribution of charge over the surfaces of conductors various shapes. You should find that the charged sphere has a uniform charge distribution over its surface.

If you attach a thinly pointed conductor to a high-voltage power transmission, that is, insert it into the arch of a Van de Graaff generator, you will be able to feel the “electric wind” by holding your hand a few centimeters from the pointed end of the conductor, as in Figure 2, a. The high concentration of positive charge at the tip of the conductor will attract negative charges (electrons) until the charge is neutralized. At the same time, positive ions in the air are repelled by the positive charge on the tip. Among the air molecules in the room there are always positive ions (gas molecules that make up air that have lost one or two electrons) and a certain number of negative ions (“lost” electrons). Figure 2, b shows the movement of charge in the air, i.e. positively charged ions repelled from a positively charged sharp conductor, and negatively charged ions attracted to it. The attraction of negative charges (electrons) to a positively charged tip neutralizes the positive charges on the tip and, therefore, lowers its positive potential. Thus, the charged conductor is discharged in a way known as discharge - the flow of charge from the tip. The positive charges that flow away from a point conductor are positive ions (almost air molecules), and this is what creates air movement, or "wind".

Note.

This process is continuous because charge from the generator is constantly added to the dome of the Van de Graaff generator. This explanation shows that a pointed conductor is very well suited for collecting charge, as well as for maintaining a high concentration of charge.

Lightning rod

An important application of charge drainage from a tip is as a lightning conductor. The movement of clouds in the atmosphere can form a huge static charge on the cloud. This increase in charge can be so great that the potential difference between the cloud and the ground (zero potential) becomes large enough to overcome the insulating properties of the air. When this happens, the air becomes conductive and the charge flows towards the ground in the form of a flash of lightning, striking the nearest or tallest buildings or objects present, i.e. the charge selects shortest path to the ground. Never take shelter under trees during a thunderstorm; lightning can strike a tree and injure or kill you as it travels down the tree to the ground. It is best to kneel in an open place, lowering your head as low as possible and placing your hands on your knees, pointing your fingers towards the ground. If lightning strikes you, it should strike your shoulders, travel down your arms and out of your fingers into the ground. Thus, this position protects your head and vital organs such as the heart.

If a flash of lightning struck a building, a lot of damage could be caused. A lightning rod can protect the building from this. A lightning rod consists of a number of pointed conductors mounted at a high point in the building and connected to a thick copper wire that runs down one of the walls and ends on a metal plate buried in the ground. When a positively charged cloud passes over a building, a separation of equal and opposite charges occurs in copper wire with a high concentration of negative charges at the edges of the conductors and a positive charge that tends to accumulate on the metal plate. The earth, however, has a huge reserve of negative charge, and therefore, as soon as a positive charge is formed on the plate, it is immediately It is gradually neutralized by negative charges (electrons) emanating from the earth. Electrons are also attracted from the ground upward to the pointed ends of the conductor under the influence of a positive potential on the cloud. A very high electrical charge can be concentrated at the tips, and this helps to reduce the positive potential of the cloud, thereby reducing its ability to overcome the insulating properties of the air. Charged ions in the air also move in the “electric wind”; negative charges (electrons) are repelled by the tips and attracted cloud, also helping to reduce its positive potential, i.e. to discharge the cloud. Positive ions in the air are attracted to positively charged pointed conductors, but the enormous reserves of negative charge in the earth can provide unlimited negative charge to the pointed conductors, to neutralize them. If lightning strikes a conductor, then it will send its electrical charge through the conductor and “safely” into the ground.

In conductors, electric charges can move freely under the influence of a field. The forces acting on the free electrons of a metal conductor placed in an external electrostatic field are proportional to the strength of this field. Therefore, under the influence of an external field, the charges in the conductor are redistributed so that the field strength at any point inside the conductor is equal to zero.

On the surface of a charged conductor, the voltage vector must be directed normal to this surface, otherwise, under the action of the vector component tangential to the surface of the conductor, charges would move along the conductor. This contradicts their static distribution. Thus:

1. At all points inside the conductor, and at all points on its surface, .

2. The entire volume of a conductor located in an electrostatic field is equipotential at any point inside the conductor:

The surface of the conductor is also equipotential, since for any line of the surface

3. In a charged conductor, uncompensated charges are located only on the surface of the conductor. Indeed, let us draw an arbitrary closed surface inside the conductor, limiting a certain internal volume of the conductor (Fig. 1.3.1). Then, according to Gauss’s theorem, the total charge of this volume is equal to:

since there is no field at surface points located inside the conductor.

Let us determine the field strength of a charged conductor. To do this, we select an arbitrary small area on its surface and construct a cylinder of height on it with a generatrix perpendicular to the area, with bases and parallel to . On the surface of the conductor and near it, the vectors and are perpendicular to this surface, and the vector flux through lateral surface cylinder is zero. The flow of electric displacement through is also zero, since it lies inside the conductor, and at all its points.

The displacement flux through the entire closed surface of the cylinder is equal to the flux through the upper base:

According to Gauss's theorem, this flow equal to the sum charges covered by the surface:

,

where is the surface charge density on the conductor surface element. Then

And, since.

Thus, if an electrostatic field is created by a charged conductor, then the strength of this field on the surface of the conductor is directly proportional to the surface density of the charges contained in it.

Studies of the distribution of charges on conductors of various shapes located in a homogeneous dielectric far from other bodies have shown that the distribution of charges in the outer surface of a conductor depends only on its shape: the greater the curvature of the surface, the greater the charge density; there are no excess charges on the internal surfaces of closed hollow conductors and.

A large field strength near a sharp protrusion on a charged conductor results in electric wind. In a strong electric field near the tip, the positive ions present in the air move with high speed, colliding with air molecules and ionizing them. Everything arises larger number moving ions that form electric wind. Due to the strong ionization of the air near the tip, it quickly loses its electrical charge. Therefore, to preserve the charge on the conductors, they strive to ensure that their surfaces do not have sharp protrusions.

1.3.2.CONDUCTOR IN AN EXTERNAL ELECTRIC FIELD

If an uncharged conductor is introduced into an external electrostatic field, then, under the influence of electrical forces, free electrons will move in it in the direction opposite to the direction of the field strength. As a result, opposite charges will appear at the two opposite ends of the conductor: negative at the end where there are extra electrons, and positive at the end where there are not enough electrons. These charges are called induced. A phenomenon consisting of the electrification of an uncharged conductor in an external electric field by dividing on this conductor the positive and negative already present in it in equal quantities electric charges, is called electrification through influence or electrostatic induction. If the conductor is removed from the field, the induced charges disappear.

The induced charges are distributed over the outer surface of the conductor. If there is a cavity inside the conductor, then with a uniform distribution of induced charges, the field inside it is zero. Electrostatic protection is based on this. When they want to protect (shield) a device from external fields, it is surrounded by a conductive screen. The external field is compensated inside the screen by induced charges arising on its surface.

1.3.3. ELECTRIC CAPACITY OF A SOLE CONDUCTOR

Consider a conductor located in a homogeneous medium far from other conductors. Such a conductor is called solitary. When this conductor receives electricity, its charges are redistributed. The nature of this redistribution depends on the shape of the conductor. Each new part charges are distributed over the surface of the conductor similar to the previous one, thus, with an increase in the charge of the conductor by a factor, the surface charge density at any point on its surface increases by the same amount, where is a certain function of the coordinates of the surface point under consideration.

We divide the surface of the conductor into infinitesimal elements, the charge of each such element is equal, and it can be considered point-like. The charge field potential at a point distant from it is equal to:

Potential at an arbitrary point electrostatic field, formed by the closed surface of the conductor, is equal to the integral:

(1.3.1)

For a point lying on the surface of a conductor, is a function of the coordinates of this point and element. In this case, the integral depends only on the size and shape of the conductor surface. In this case, the potential is the same for all points of the conductor, therefore the values ​​are the same.

It is believed that the potential of an uncharged solitary conductor is zero.

From formula (1.3.1) it is clear that the potential of a solitary conductor is directly proportional to its charge. The ratio is called electrical capacitance

. (1.3.2)

The electrical capacity of an isolated conductor is numerically equal to the electric charge that must be imparted to this conductor in order for the potential of the conductor to change by one. The electrical capacity of a conductor depends on its shape and size, and geometrically similar conductors have proportional capacities, since the distribution of charges on them is also similar, and the distances from similar charges to the corresponding points of the field are directly proportional to the linear dimensions of the conductors.

The potential of the electrostatic field created by each point charge is inversely proportional to the distance from this charge. Thus, the potentials of equally charged and geometrically similar conductors change in inverse proportion to their linear dimensions, and the capacitance of these conductors changes in direct proportion.

From expression (1.3.2) it is clear that the capacitance is directly proportional to the dielectric constant of the medium. Neither from the material of the conductor, nor from its state of aggregation, nor does its capacity depend on the shape and size of possible cavities inside the conductor. This is due to the fact that excess charges are distributed only on the outer surface of the conductor. does not also depend on and .

Capacity units: - farad, its derivatives ; .

The capacity of the Earth as a conducting ball () is equal to .

1.3.4. MUTUAL ELECTRIC CAPACITY. CAPACITORS

Consider a conductor near which there are other conductors. This conductor can no longer be considered solitary; its capacity will be greater than the capacity of a solitary conductor. This is due to the fact that when a charge is imparted to a conductor, the conductors surrounding it are charged through influence, and the charges closest to the guiding charge are opposite sign. These charges somewhat weaken the field created by the charge. Thus, they lower the potential of the conductor and increase its electrical capacity (1.3.2).

Let us consider a system composed of closely spaced conductors whose charges are numerically equal but opposite in sign. Let us denote the potential difference between the conductors, absolute value charges is equal to . If the conductors are located away from other charged bodies, then

where is the mutual electrical capacitance of two conductors:

- it is numerically equal to the charge that must be transferred from one conductor to another to change the potential difference between them by one.

The mutual electrical capacitance of two conductors depends on their shape, size and relative position, as well as on the dielectric constant of the medium. For a homogeneous environment.

If one of the conductors is removed, the potential difference increases, and mutual capacitance decreases, tending to the value of the capacitance of a solitary conductor.

Let's consider two differently charged conductors whose shape and relative position are such that the field they create is concentrated in a limited area of ​​space. Such a system is called a capacitor.

1. A flat capacitor has two parallel metal plates of area , located at a distance from one another (1.3.3). Charges of plates and . If the linear dimensions of the plates are large compared to the distance , then the electrostatic field between the plates can be considered equivalent to the field between two infinite planes charged oppositely with the surface charge densities and , field strength , potential difference between the plates , then , where - permittivity environment filling the capacitor.

2. A spherical capacitor consists of a metal ball of radius , surrounded by a concentric hollow metal ball of radius , (Fig. 1.3.4). Outside the capacitor, the fields created by the inner and outer plates cancel each other out. The field between the plates is created only by the charge of the ball, since the charge of the ball does not create an electric field inside this ball. Therefore, the potential difference between the plates is: , Then

The inner lining of a spherical capacitor can be considered as a solitary sphere. In this case, and.

Conductors are bodies in which electric charges can move under the influence of an arbitrarily weak electrostatic field.

As a result, the charge imparted to the conductor will be redistributed until at any point inside the conductor the electric field strength becomes 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, under the condition of a static charge distribution, the tension on the surface

where E n is the normal component of tension.

It follows from this, 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 and, therefore, tend to be located at the greatest distance from each other.

By examining the magnitude of the electric field strength near the surface of charged bodies of 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 the outer surface of the 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 an uncharged conductor is placed in an external electric field, free charges will move to 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.

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 inductive charges compensates external field E 0 at all points inside the conductor and the resulting field E inside the conductor will become zero.

The total field E near the conductor will differ noticeably from its original 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.

In the case of equilibrium distribution, the charges of the conductor are distributed in a thin surface layer. So, for example, if a conductor is given a negative charge, then due to the presence of repulsive forces between the elements of this charge, they will be dispersed over the entire surface of the conductor.

Examination using a test plate

In order to experimentally study how charges are distributed on the outer surface of a conductor, a so-called test plate is used. This plate is so small that when it comes into contact with the conductor, it can be considered as part of the surface of the conductor. If this plate is applied to a charged conductor, then part of the charge ($\triangle q$) will transfer to it and the magnitude of this charge will be equal to the charge that was on the surface of the conductor in area equal area plates ($\triangle S$).

Then the value is equal to:

\[\sigma=\frac(\triangle q)(\triangle S)(1)\]

is called the surface charge distribution density at a given point.

By discharging a test plate through an electrometer, one can judge the value of the surface charge density. So, for example, if you charge a conducting ball, you can see, using the above method, that in a state of equilibrium the surface charge density on the ball is the same at all its points. That is, the charge is distributed evenly over the surface of the ball. For conductors with more complex shapes, the charge distribution is more complex.

Surface density of conductor

The surface of any conductor is equipotential, but in general the charge distribution density can vary greatly at different points. The surface charge distribution density depends on the curvature of the surface. In the section that was devoted to describing the state of conductors in an electrostatic field, we established that the field strength near the surface of the conductor is perpendicular to the surface of the conductor at any point and is equal in magnitude:

where $(\varepsilon )_0$ is the electric constant, $\varepsilon $ is the dielectric constant of the medium. Hence,

\[\sigma=E\varepsilon (\varepsilon )_0\ \left(3\right).\]

The greater the curvature of the surface, the greater the field strength. Consequently, the charge density on the protrusions is especially high. Near the depressions in the conductor, equipotential surfaces are located less frequently. Consequently, the field strength and charge density in these places are lower. The charge density at a given conductor potential is determined by the curvature of the surface. It increases with increasing convexity and decreases with increasing concavity. The charge density is especially high at the edges of the conductors. Thus, the field strength at the tip can be so high that ionization of the gas molecules that surrounds the conductor can occur. Gas ions of the opposite sign of charge (relative to the charge of the conductor) are attracted to the conductor and neutralize its charge. Ions of the same sign are repelled from the conductor, “pulling” neutral gas molecules with them. This phenomenon is called electric wind. The charge of the conductor decreases as a result of the neutralization process; it seems to flow off the tip. This phenomenon is called the outflow of charge from the tip.

We have already said that when we introduce a conductor into an electric field, a separation of positive charges (nuclei) and negative charges (electrons) occurs. This phenomenon is called electrostatic induction. The charges that appear as a result are called induced. Induced charges create an additional electric field.

The field of induced charges is directed towards opposite direction external field. Therefore, the charges that accumulate on the conductor weaken the external field.

Charge redistribution continues until the charge equilibrium conditions for conductors are met. Such as: zero field strength everywhere inside the conductor and perpendicularity of the intensity vector of the charged surface of the conductor. If there is a cavity in the conductor, then with an equilibrium distribution of the induced charge, the field inside the cavity is zero. Electrostatic protection is based on this phenomenon. If they want to protect a device from external fields, it is surrounded by a conductive screen. In this case, the external field is compensated inside the screen by induced charges arising on its surface. This may not necessarily be continuous, but also in the form of a dense mesh.

Assignment: An infinitely long thread, charged with linear density $\tau$, is located perpendicular to an infinitely large conducting plane. Distance from the thread to the plane $l$. If we continue the thread until it intersects with the plane, then at the intersection we will obtain a certain point A. Write a formula for the dependence of the surface density $\sigma \left(r\right)\ $of induced charges on the plane on the distance to point A.

Let's consider some point B on the plane. An infinitely long charged thread at point B creates an electrostatic field; a conducting plane is in the field; induced charges are formed on the plane, which in turn create a field that weakens the external field of the thread. The normal component of the plane field (induced charges) at point B will be equal to the normal component of the thread field at the same point if the system is in equilibrium. Let us isolate an elementary charge on the thread ($dq=\tau dx,\ where\ dx-elementary\ piece\ of the thread\ $), and find at point B the tension created by this charge ($dE$):

Let's find the normal component of the filament field strength element at point B:

where $cos\alpha $ can be expressed as:

Let us express the distance $a$ using the Pythagorean theorem as:

Substituting (1.3) and (1.4) into (1.2), we get:

Let us find the integral from (1.5) where the limits of integration are from $l\ (distance\ to\ the nearest\ end\ of\ the thread\ from\ the\ plane)\ to\ \infty $:

On the other hand, we know that the field of a uniformly charged plane is equal to:

Let us equate (1.6) and (1.7) and express the surface charge density:

\[\frac(1)(2)\cdot \frac(\sigma)(\varepsilon (\varepsilon )_0)=\frac(\tau )(4\pi (\varepsilon )_0\varepsilon )\cdot \frac (1)((\left(r^2+x^2\right))^((1)/(2)))\to \sigma=\frac(\tau )(2\cdot \pi (\left (r^2+x^2\right))^((1)/(2))).\]

Answer: $\sigma=\frac(\tau )(2\cdot \pi (\left(r^2+x^2\right))^((1)/(2))).$

Example 2

Assignment: Calculate the surface charge density that is created near the Earth's surface if the Earth's field strength is 200$\ \frac(V)(m)$.

We will assume that the dielectric conductivity of air is $\varepsilon =1$ like that of a vacuum. As a basis for solving the problem, we will take the formula for calculating the voltage of a charged conductor:

Let us express the surface charge density and obtain:

\[\sigma=E(\varepsilon )_0\varepsilon \ \left(2.2\right),\]

where the electric constant is known to us and is equal in SI $(\varepsilon )_0=8.85\cdot (10)^(-12)\frac(F)(m).$

Let's carry out the calculations:

\[\sigma=200\cdot 8.85\cdot (10)^(-12)=1.77\cdot (10)^(-9)\frac(Cl)(m^2).\]

Answer: The surface charge distribution density of the Earth's surface is equal to $1.77\cdot (10)^(-9)\frac(C)(m^2)$.