How are charges distributed along a conductor? Distribution of charges in a conductor

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 coulomb per cubic meter (C/m3).

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 coulomb per square meter (C/m2).

In the event that the charged body is shaped like a thread (the cross-sectional diameter of the body is much less than its length, it is convenient to use the 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 individual parts of the body is calculated. The total field strength is then found by summing the fields created by individual parts of the body, i.e.

CONDUCTORS IN AN ELECTROSTATIC FIELD

§1 Charge distribution in a conductor.

Relationship between field strength at the surface of a conductor and surface charge density

Consequently, the surface of a conductor when charges are in equilibrium is equipotential.

When charges are in equilibrium, there can be no excess charges anywhere inside the conductor - they are all distributed over the surface of the conductor with a certain density σ.

Let us consider a closed surface in the shape of a cylinder, the generatrices of which are perpendicular to the surface of the conductor. On the surface of the conductor there are free charges with surface density σ.

Because There are no charges inside the conductor, then the flux through the surface of the cylinder inside the conductor is zero. The flux through the upper part of the cylinder outside the conductor, according to Gauss's theorem, is equal to

those. the electric displacement vector is equal to the surface density of free charges of the conductor or

2. When an uncharged conductor is introduced into an external electrostatic field, free charges will begin to move: positive charges along the field, negative charges against the field. Then positive charges will accumulate on one side of the conductor and negative charges on the other. These charges are called INDUCED. The process of charge redistribution will occur until the tension inside the conductor becomes equal to zero, and the tension lines outside the conductor are perpendicular to its surface. Induced charges appear on the conductor due to displacement, i.e. are the surface density of displaced charges, etc. that is why it was called the electric displacement vector.

§2 Electrical capacity of conductors.

Capacitors

SOLITUDEcalled a conductor that is distant from other conductors, bodies, charges. The potential of such a conductor is directly proportional to the charge on it

From experience it follows that different conductors, being equally chargedQ 1 = Q 2 acquires different potentials φ 1 ¹ φ 2due to the different shape, size and environment surrounding the conductor (ε). Therefore, for a solitary conductor the formula is valid

Where - capacity of a solitary conductor. The capacity of an isolated conductor is equal to the charge ratioq, the message of which to the conductor changes its potential by 1 Volt.

In the SI system Capacitance is measured in Farads

Ball capacity

Let's calculate the capacitance of a flat capacitor with the area of the platesS, surface charge density σ, dielectric constant ε of the dielectric between the plates, distance between the platesd. The field strength is

Using the Δφ relationship and E, we find

Capacitance of a parallel plate capacitor.

For a cylindrical capacitor:

For a spherical capacitor

Because at certain voltage values, breakdown occurs in the dielectric (electric discharge through the dielectric layer), then for capacitors there is a breakdown voltage. The breakdown voltage depends on the shape of the plates, the properties of the dielectric and its thickness.

Capacitance for parallel and series connection of capacitors

a) parallel connection

According to the law of conservation of charge

b) serial connection

According to the law of conservation of charge

§3 Electrostatic field energy

Energy of a system of stationary point charges

The electrostatic field is potential. The forces acting between charges are conservative forces. A system of stationary point charges must have potential energy. Let's find the potential energy of two stationary point chargesq 1 And q 2 , located at a distancer from each other.

Potential charge energyq 2 in the field created

charge q 1 , is equal

Similarly, the potential energy of the chargeq 1 in the field created by the chargeq 2 , is equal

It's clear that W 1 = W 2 , then denoting the potential energy of the system of chargesq 1 And q 2 through W, we can write

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 the side surface of the 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 flux is equal to the sum of the 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 the strong electric field near the tip, the positive ions present in the air move at high speed, colliding with air molecules and ionizing them. An increasing number of moving ions appear, forming an 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 electrical forces free electrons will move in it in the direction opposite direction 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. The phenomenon of electrification of an uncharged conductor in an external electric 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. 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 of the charges is 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:

The potential at an arbitrary point of the electrostatic field formed by a closed surface of a 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, its capacity does not 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 those closest to the guiding charge are charges of the 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, the absolute value of the 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, then the potential difference increases and the mutual capacitance decreases, tending to the value of the capacitance of the isolated conductor.

Let's consider two differently charged conductors whose shape and mutual arrangement 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 is the dielectric constant of the medium 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 inside this ball electric field. 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.

The study of the electrostatics of conductors is complicated by the fact that the distribution of electric charge over the outer surface of the same conducting body under different conditions can turn out to be completely different. An exception is the case of distribution of electric charge over the surface of a solitary conductor in an infinite homogeneous isotropic space. This distribution depends only on the shape of the boundary surface of the conductor. Below, for simplicity of presentation, we will consider solitary conductors in a vacuum. Mathematicians call the problem of the distribution of electric charge over the surface of a conductor the “Robin problem.” A distinction is made between the volumetric (three-dimensional) case and the two-dimensional case of the Robin problem. In the two-dimensional case, an infinite cylinder of arbitrary cross-section is considered as a conductor. Outside the conductor, the potential of the electrostatic field satisfies the Laplace equation, on the surface of the conductor the potential becomes zero, and the integral over the surface of the conductor from the normal derivative of the potential is proportional to the value of the total electric charge. In the flat (two-dimensional) case, methods from the theory of functions of a complex variable, in particular, the conformal mapping method, are effective for solving the Robin problem.

Let us assume that the conductor is an ellipsoid, the equation of the boundary surface of which is described in the Cartesian coordinate system by the equation

It is known (F. Frank, R. Mises. Differential and integral equations of mathematical physics. - L.-M.: ONTI. Chief editor of general technical literature. - 1937.-998 pp., p. 706) the distribution of the surface density of electric charge over the surface conducting ellipsoid:

. (2)

From this relation follows the estimate

where i.e. surface electric charge densities at the points of intersection of the ellipsoid axes with the surface. If size A very large and the dimensions b And c small, it becomes very large. Let us remember that this value is proportional to the normal component of the electrostatic field strength near the surface of the conductor. Electrical breakdown depends on the strength of the electrostatic field. It turns out that the breakdown occurs in the vicinity of the “sharp” end of an ellipsoid elongated in one direction.

For a conducting ball we have

, , (4)

the distribution of surface electric charge density is uniform.

The uneven distribution of electric charge over the surface of an arbitrary conductor is the cause of an error that arises, for example, in an elementary, simplified calculation of the capacitance of a capacitor of finite dimensions. Strictly taking into account “edge effects” is sometimes a rather difficult task. In particular, the derivation of relation (2) requires the introduction of ellipsoidal coordinates, the ability to write the Laplace equation in these coordinates, construct a solution to the resulting partial differential equation with variable coefficients (i.e., obtain the distribution of the electrostatic field potential outside the conducting ellipsoid), and calculate the strength of the electrostatic field in the vicinity of the boundary surface of the ellipsoid and, finally, calculate the value of the surface density of the electric charge on the surface of the conducting ellipsoid. Only in rare exceptional cases can the solution to problems of the type under consideration be obtained in a closed analytical form; in other cases, the solution is obtained using numerical methods using special software on modern computers.

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

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

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 flux is equal to the sum of the charges covered by the surface:

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

And, since.

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. The phenomenon of electrification of an uncharged conductor in an external electric 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. If the conductor is removed from the field, the induced charges disappear.

1.3.3. ELECTRIC CAPACITY OF A SOLE CONDUCTOR

The potential at an arbitrary point of the electrostatic field formed by a closed surface of a conductor is equal to the integral:

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

From expression (1.3.2) it is clear that the capacitance is directly proportional to the dielectric constant of the medium. Its capacity does not depend either on the material of the conductor, or on its state of aggregation, or 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 .

Units of capacitance: - farad, its derivatives; .

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

1.3.4. MUTUAL ELECTRIC CAPACITY. CAPACITORS

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, then the potential difference increases and the mutual capacitance decreases, tending to the value of the capacitance of the isolated conductor.

Let's consider two oppositely 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.

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: , then

An example of a cylindrical capacitor is a Leyden jar. If the gap between the capacitor plates is small, then and , where - lateral area linings.

Thus, the electrical capacity of any capacitor is proportional to the dielectric constant of the substance filling the gap between the plates.

In addition to electrical capacity, a capacitor is characterized by breakdown voltage. This is the potential difference between the plates at which breakdown can occur.

1.3.5. CAPACITOR CONNECTIONS

1. Parallel connection. Let's consider a battery of capacitors connected by plates of the same name (Fig. 1.3.6). The capacitances of the capacitors are respectively equal. The potential differences for all capacitors are the same, so the charges on the plates are always less than the minimum electrical capacity included in the battery.