Capacity of a solitary conductor determination. Electrical capacitance of a solitary conductor

As noted above, the potential electric field  charged conductor has the same value at all points of the conductor. If we take the potential of this electric field at infinity to be zero, then between the charge of a conductor located far enough from other conductors and its potential there is a linear relation

where the coefficient of proportionality, depending on the geometric characteristics of the conductor and the properties of the environment, is called by the capacity of a solitary conductor.

In the case of a conducting ball of radius in a vacuum, its capacitance

If the Earth is considered a conducting ball with a radius, then its capacity will be equal. Experience shows that due to thunderstorm activity the Earth has a negative charge ~, which creates near its surface a constant electric field ~. Due to the conductivity of his body, a person does not feel this field, because the field of induced charges completely compensates the electric field of the Earth at all points of the body. The Earth's atmosphere as a whole has a positive charge, and the potential difference between the upper layers of the atmosphere and the Earth's surface reaches.

If we take two conductors and one of them is charged with a positive charge and the other with a negative charge, then the potential difference of these conductors is related to the charge by a linear relation

where C is the capacity of a given conductor system, which is called capacitor, V is the voltage across the capacitor.

The capacitance of the capacitor depends on the geometry of the conductors, called the capacitor plates, and the characteristics of the medium between these plates. Let us give formulas describing the capacitance of the main types of capacitors for the case when a vacuum is created between their plates.

If the space between the capacitor plates is filled with a dielectric with a relative permittivity, then its capacitance

Thus, the capacitance of the capacitor can be increased by increasing the plate area and the relative permittivity of the medium, as well as reducing the distance between the plates.

The energy of a charged capacitor is equal to the work of forces, which occurs when the charges are separated with opposite signs during the charging process. Let us assume that an infinitesimal charge moves from a negatively charged capacitor plate with a potential to a positively charged plate. With an infinitely slow displacement of such a charge, the work done by an external force,

, (4.13)

where is the charge of the capacitor and C is the capacity of the capacitor.

Full operation of the external force when charging the capacitor

, (4.14)

where - the final charge of the capacitor and - the final voltage on the capacitor. By definition, the energy of a charged capacitor

In the formula (4.14), the charge carriers of the charged capacitor are charges on the capacitor plates, which corresponds to the long-range theory. In this theory, the energy of a charged capacitor is the potential energy of interaction of charges, equal to the work of an external force in the spatial separation of charges with opposite signs.

According to the theory of short-range interaction, the energy carrier of a charged capacitor is the electric field distributed between the capacitor plates. To rewrite formula (4.14) for a new physical interpretation, we rewrite it for the special case flat capacitor  in the following way

where is the volume of the area between the capacitor plates and

Conductors and dielectrics in an electrostatic field.

If you place the conductor in an external electrostatic field or charge it, then the electrostatic field will act on the charges of the conductor, as a result of which they will begin to move.

The charge transfer (current) continues until an equilibrium distribution of charges is established, at which the electrostatic field inside the conductor vanishes. This happens in a very short time. In fact, if the field were not equal to zero, then an ordered movement of charges would arise in the conductor without consuming energy from an external source, which contradicts the law of conservation of energy. So, the field strength at all points inside the conductor is zero:

The absence of a field inside the conductor means, according to (85.2), that the potential at all points inside the conductor is constant (j = const), i.e. The surface of a conductor in an electrostatic field is eq the potential. It follows from this that the field strength vector on the outer surface of the conductor is directed along the normal to each point of its surface.

Consider a solitary explorer,that is, a conductor that is remote from other conductors, bodies and charges. Its potential, according to (84.5), is directly proportional to the charge of the conductor. It follows from experience that different conductors, being equally charged, assume different potentials. Therefore, for a solitary conductor, one can write

The amount

C = Q / j (93.1)

is called electric capacity(or simply capacity)a solitary explorer. The capacity of a solitary conductor is determined by the charge, the message of which to the conductor changes its potential by one. The capacity of the conductor depends on its size and shape, but it does not depend on the material, the aggregate state, the shape and dimensions of the cavities inside the conductor. This is due to the fact that excess charges are distributed on the outer surface of the conductor. The capacity does not depend either on the charge of the conductor or on its potential. This does not contradict formula (93.1), since it only shows that the capacity of a solitary conductor is directly proportional to its charge and is inversely proportional to the potential.

Unit of electrical capacity - farad(Ф): 1 Ф is the capacity of such a solitary conductor, the potential of which changes by 1 V when the charge is given to it in 1 Cl.

According to (84.5), the potential of a solitary sphere of radius R,located in a homogeneous medium with a permittivity e, is equal to

Using formula (93.1), we obtain that the capacity of the ball

C = 4pe 0 e R. (93.2)

Hence it follows that a capacity of 1 Φ would have a solitary ball, which is in a vacuum and has a radius R =C / (4pe 0) »9 10 6 km, which is approximately 1400 times greater than the Earth's radius (Earth's electric capacity C» 0.7 mF). Consequently, farad is a very large value, therefore in practice, the longest units are used: milli-fad (mF), microfarad (μF), nanofarad (nF), picofarad (pF). It also follows from (93.2) that the unit of the electric constant e 0 is farad per meter (F / m) (see (78.3)).

The dielectric (like any substance) consists of atoms and molecules. Since the positive charge of all

nuclei of a molecule is equal to the total charge of electrons, then the molecule as a whole is electrically neutral. If we replace the positive charges of the nuclei of molecules by the total charge + Q, located at the center of the "gravity" of the positive charges, and the charge of all electrons - by the total negative charge - Q, located in the center of the "gravity" of negative charges, the molecule can be regarded as an electric dipole with an electric moment

The first group of dielectrics (N 2, H 2 , O 2, CO 2, CH 4, ...) are substances whose molecules have a symmetrical structure, that is, centers of "gravity" of positive and

the negative charges in the absence of an external electric field coincide and, consequently,

dipole moment of a molecule r  is equal to zero. Molecules nonpolar.

Under the action of an external electric field, the charges of nonpolar molecules shift in

opposite sides (positive in the field, negative against the field) and the molecule acquires a dipole moment. The second group of dielectrics (H 2 O, NH 3, SO 2, CO, ...) are substances whose molecules have an asymmetric structure, that is, the centers of "gravity" of positive and negative charges do not coincide. Thus, in the absence of an external electric field, these molecules have a dipole moment. Moleculessuch dielectrics are called polar.In the absence of an external field, however, the dipole moments of polar molecules due to thermal motion are randomly oriented in space and their resultant momentum is zero. If such a dielectric is placed in an external field, then the forces of this field will tend to rotate the dipoles along the field and a nonzero resultant moment arises.

Consider   a solitary explorer,  that is, a conductor that is remote from other conductors, bodies and charges. Its potential, according to (84.5), is directly proportional to the charge of the conductor. It follows from experience that different conductors, being equally charged, have different potentials. Therefore, for a solitary conductor, one can write

The amount

is called electric capacity  (or simply capacity) of a solitary conductor. The capacity of a solitary conductor is determined by the charge, the message of which to the conductor changes its potential by one.

The capacity of the conductor depends on its size and shape, but it does not depend on the material, the aggregate state, the shape and dimensions of the cavities inside the conductor. This is due to the fact that excess charges are distributed on the outer surface of the conductor. The capacity does not depend either on the charge of the conductor or on its potential.

Unit of electrical capacity - farad  (Φ): 1 Φ is the capacity of such a solitary conductor, whose potential varies by 1 V when the charge of 1 Cl is reported to it.

According to (84.5), the potential of a solitary sphere of radius R,   located in a homogeneous medium with a permittivity e, is equal to

Using formula (93.1), we obtain that the capacity of the ball

Hence it follows that the capacity 1 Φ would have a solitary ball, which is in a vacuum and has a radius R= C/ (4pe 0) »9 × 10 6 km, which is about 1400 times larger than the Earth's radius (Earth's electrical capacity FROM» 0.7 mF). Consequently, farad is a very large value, therefore in practice, the longest units are used: milli-fad (mF), microfarad (μF), nanofarad (nF), picofarad (pF). It also follows from (93.2) that the unit of the electric constant e 0 is farad per meter (Φ / m) (see (78.3)).

Capacitors

in order for the conductor to have a large capacity, it must be very large. In practice, however, devices that are capable of accumulating significant charges with large sizes and small relative to surrounding bodies are needed, in other words, to have a large capacity. These devices are called   capacitors.

If other bodies are approached to a charged conductor, then induced (on the conductor) or connected (on the dielectric) charges, and the closest to the induced charge Q  there will be charges of the opposite sign. These charges naturally weaken the field created by the charge Q,  ie, lower the potential of the conductor, which leads (see (93.1)) to an increase in its electrical capacity.

The capacitor consists of two conductors (plates) separated by a dielectric. The capacitance of the capacitor should not be affected by the surrounding bodies, so the conductors are shaped so that the field created by the accumulated charges is concentrated in the narrow gap between the capacitor plates. This condition is satisfied (see § 82): 1) two flat plates; 2) two coaxial cylinders; 3) two concentric spheres. Therefore, depending on the shape of the plates, the capacitors are divided into   flat, cylindrical  and spherical.

Since the field is concentrated inside the capacitor, the lines of tension begin on one plate and end on the other, so the free charges that occur on different plates are equal in absolute value to opposite charges. Under capacitance of the capacitor  means a physical quantity equal to the ratio of the charge Q, accumulated in the capacitor, to the potential difference (j 1 - j 2) between its plates:

(94.1)

We calculate the capacitance of a flat capacitor consisting of two parallel metal plates with an area of S  each located at a distance d  from each other and having charges + Q and – Q.   If the distance between the plates is small in comparison with their linear dimensions, then the edge effects can be neglected and the field between the plates considered homogeneous. It can be calculated using formulas (86.1) and (94.1). In the presence of a dielectric between the plates, the potential difference between them, according to (86.1),

(94.2)

where e is the permittivity. Then from formula (94.1), replacing Q= sS, with allowance for (94.2), we obtain an expression for the capacitance of a flat capacitor:

(94.3)

To determine the capacity of a cylindrical capacitor consisting of two hollow coaxial cylinders with radii r  1 and r 2 (r 2 > r  1), inserted one into the other, again neglecting the edge effects, we consider the field to be radially symmetric and concentrated between the cylindrical plates. We shall calculate the potential difference between the plates by formula (86.3) for the field of a uniformly charged infinite cylinder with linear density t = Q/ l (l- the length of the plates). If there is a dielectric between the plates, the potential difference

(94.4)

Substituting (94.4) into (94.1), we obtain an expression for the capacitance of a cylindrical capacitor:

(94.5)

To determine the capacitance of a spherical capacitor consisting of two concentric plates separated by a spherical dielectric layer, we use the formula (86.2) for the potential difference between two points lying at distances r  1 and r 2 (r 2 > r  1) from the center of the charged spherical surface. If there is a dielectric between the plates, the potential difference

(94.6)

Substituting (94.6) into (94.1), we obtain

If d= r 2 - r 1<<r 1 ,   then r  2 » r  1 " r  and C =4pe 0 e r 2 / d.  Since 4p r  2 is the area of ​​the spherical shell, we obtain the formula (94.3). Thus, for a small gap in comparison with the radius of the sphere, the expressions for the capacitance of the spherical a plane capacitors coincide. This conclusion is also valid for a cylindrical capacitor: for a small gap between the cylinders, in comparison with their radii, in formula (94.5), ln ( r 2 /r  1) can be expanded in a series, limited only to a first-order term. As a result, we again arrive at the formula (94.3).

It follows from (94.3), (94.5), and (94.7) that the capacitance of capacitors of any shape is directly proportional to the permittivity of the dielectric filling the space between the plates. Therefore, the use of ferroelectrics as a layer significantly increases the capacitance of capacitors.

Capacitors are characterized by breakdown voltage  - the potential difference between the capacitor plates, at which the breakdown  - electric discharge through the dielectric layer in the capacitor. The breakdown voltage depends on the shape of the plates, the properties of the dielectric and its thickness.

To increase capacitance and vary its possible values, capacitors are connected to batteries, while using their parallel and serial connections.

1. Parallel connection of capacitors  (Figure 144). In parallel-connected capacitors, the potential difference across the capacitor plates is the same and equal to j A – j B  . If the capacitance of individual capacitors FROM 1 , FROM 2 , ..., FROM n ,   then, according to (94.1), their charges are equal to

and the charge of the capacitor bank

Full battery capacity

ie, when the capacitors are connected in parallel, it is equal to the sum of the capacitances of the individual capacitors.

2. Serial connection of capacitors  (Figure 145). In series-connected capacitors, the charges of all the plates are equal in modulus, and the potential difference at the battery terminals

where for any of the capacitors D j i = Q/FROM i  . On the other hand,

that is, when the capacitors are connected in series, the quantities inverse to the capacitances are added up. Thus, with a successive coupling of capacitors, the resultant capacitance FROM  is always less than the smallest capacity used in the battery.