The electrical capacity of the battery. Electrical capacity, capacitors. Series and parallel connection of capacitors

In many cases, to obtain the desired electrical capacity, capacitors come. can be combined into a group called a battery.

Such a connection of capacitors is called sequential, in which the negatively charged plate of the previous capacitor is connected to the positively charged plate of the next (Fig.

15.31). When connected in series, all capacitor plates will have the same charges (Explain why.) Since the charges on the capacitor are in equilibrium, the potentials of the plates connected by conductors will be the same.

Given these circumstances, we derive a formula for calculating the electrical capacity of a battery of series-connected capacitors.

From fig. 15.31 it can be seen that the voltage on the battery is equal to the sum of the voltages on the capacitors connected in series. Really,

Using the ratio we get

After reduction on we will have

From (15.21) it can be seen that when connected in series, the electric capacity of the battery is less than the smallest of the electric capacities of individual capacitors.

The connection of capacitors is called parallel, in which all positively charged plates are connected to one wire, and negatively charged ones to another (Fig. 15.32). In this case, the voltages on all capacitors are the same and equal, and the charge on the battery is equal to the sum of the charges on the individual capacitors:

After reducing for , we obtain the formula for . calculation of the electric capacity of a battery of capacitors connected in parallel:

From (15.22) it can be seen that when connected in parallel, the electric capacity of the battery is greater than the largest of the electric capacities of individual capacitors.

In the manufacture of capacitors of large electrical capacity, a parallel connection is used, shown in Fig. 15.33. This connection method saves in material, since the charges are located on both sides of the capacitor plates (except for the two extreme plates). On fig. 15.33 6 capacitors are connected in parallel, and 7 plates are made. Therefore, in this case, the capacitors connected in parallel are one less than the number of metal sheets in the capacitor bank, i.e.

The magnitude of the electrical capacitance depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are conductor configurations in which electric field turns out to be concentrated (localized) only in a certain region of space. Such systems are called capacitors, and the conductors that make up the capacitor are called facings. The simplest capacitor is a system of two flat conductive plates arranged parallel to each other at a small distance compared to the dimensions of the plates and separated by a dielectric layer. Such a capacitor is called flat. The electric field of a flat capacitor is mainly localized between the plates (Fig. 4.6.1); however, near the edges of the plates and in the surrounding space, a relatively weak electric field also arises, which is called scattering field. In a number of problems, one can approximately neglect the stray field and assume that the electric field of a flat capacitor is entirely concentrated between its plates (Fig. 4.6.2). But in other problems, neglecting the stray field can lead to gross errors, since this violates the potential character electric field(see § 4.4).

Each of the charged plates of a flat capacitor creates an electric field near the surface, the strength modulus of which is expressed by the relation (see § 4.3)

Inside the vector capacitor and are parallel; therefore, the modulus of the total field strength is equal to

Thus, the capacitance of a flat capacitor is directly proportional to the area of the plates (plates) and inversely proportional to the distance between them. If the space between the plates is filled with a dielectric, the electric capacitance of the capacitor increases by ε times:

Capacitors can be interconnected to form capacitor banks. At parallel connection capacitors (Fig. 4.6.3), the voltages on the capacitors are the same: U1 \u003d U2 \u003d U, and the charges are q1 \u003d C1U and q2 \u003d C2U. Such a system can be considered as a single capacitor of electrical capacity C, charged with a charge q = q1 + q2 at a voltage between the plates equal to U. It follows from this

Electrical capacity. Capacitors Lecture #9 If two conductors isolated from each other are given charges q 1 and q 2, then a certain potential difference Δφ arises between them, depending on the magnitude of the charges and the geometry of the conductors. The potential difference Δφ between two points in an electric field is often called voltage and denoted by the letter U. Of greatest practical interest is the case when the charges of the conductors are the same in magnitude and opposite in sign: q 1 = - q 2 = q. In this case, you can introduce the concept of electrical capacity. The electrical capacity of a system of two conductors is a physical quantity defined as the ratio of the charge q of one of the conductors to the potential difference Δφ between them: The magnitude of the electrical capacity depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are such configurations of conductors in which the electric field is concentrated (localized) only in a certain region of space. Such systems are called capacitors, and the conductors that make up the capacitor are called plates. The simplest capacitor is a system of two flat conductive plates located parallel to each other at a distance small compared to the dimensions of the plates and separated by a dielectric layer. Such a capacitor is called flat. The electric field of a flat capacitor is mainly localized between the plates (Fig. 4.6.1); however, a relatively weak electric field also arises near the edges of the plates and in the surrounding space, which is called the stray field. In a number of problems, one can approximately neglect the stray field and assume that the electric field of a flat capacitor is entirely concentrated between its plates (Fig. 4.6.2). But in other problems, neglecting the stray field can lead to gross errors, since the potential nature of the electric field is violated in this case (see § 4.4). Each of the charged plates of a flat capacitor creates an electric field near the surface, the strength modulus of which is expressed by the relation (see § 4.3)

According to the principle of superposition, the strength of the field created by both plates is equal to the sum of the strengths and fields of each of the plates: Outside the vector plates and directed towards different sides, and therefore E = 0. The surface charge density σ of the plates is equal to q / S, where q is the charge and S is the area of each plate. The potential difference Δφ between the plates in a uniform electric field is Ed, where d is the distance between the plates. From these relationships, you can get a formula for the electric capacitance of a flat capacitor: Examples of capacitors with a different plate configuration are spherical and cylindrical capacitors. A spherical capacitor is a system of two concentric conducting spheres of radii R 1 and R 2 . A cylindrical capacitor is a system of two coaxial conductive cylinders of radii R 1 and R 2 and length L. The capacitances of these capacitors filled with a dielectric with permittivityε are expressed by the formulas:

Capacitors can be interconnected to form capacitor banks. When capacitors are connected in parallel (Fig. 4.6.3), the voltages on the capacitors are the same: U 1 \u003d U 2 \u003d U, and the charges are equal to q 1 \u003d C 1 U and q 2 \u003d C 2 U. Such a system can be considered as a single capacitor of electrical capacity C , charged with a charge q \u003d q 1 + q 2 at a voltage between the plates equal to U. It follows from this When connected in series (Fig. 4.6.4), the charges of both capacitors turn out to be the same: q 1 \u003d q 2 \u003d q, and the voltages on them are equal and Such a system can be considered as a single capacitor charged with a charge q at a voltage between the plates U \u003d U 1 + U 2 . Consequently,

When capacitors are connected in series, the reciprocals of the capacitances are added. Formulas for parallel and serial connection remain valid for any number of capacitors connected in a battery. Energyelectricfields Experience shows that a charged capacitor contains a reserve of energy. The energy of a charged capacitor is equal to the work of external forces that must be expended to charge the capacitor. The process of charging a capacitor can be represented as a sequential transfer of sufficiently small portions of charge Δq> 0 from one plate to another (Fig. 4.7 .one). In this case, one plate is gradually charged with a positive charge, and the other with a negative charge. Since each portion is transferred under conditions when there is already a certain charge q on the plates, and there is a certain potential difference between them, when transferring each portion Δq, external forces must do the work

The energy W e of a capacitance C charged with a charge Q can be found by integrating this expression between 0 and Q: The electrical energy W e should be considered as the potential energy stored in a charged capacitor. The formulas for W e are similar to the formulas for the potential energy E p of a deformed spring (see § 2.4)

where k is the stiffness of the spring, x is the deformation, F = kx is the external force. According to modern concepts, Electric Energy capacitor is localized in the space between the capacitor plates, that is, in the electric field. Therefore, it is called the energy of the electric field. This can be easily illustrated by the example of a charged flat capacitor. The intensity of a uniform field in flat capacitor is equal to E = U/d, and its capacitance Therefore is the electrical (potential) energy of a unit volume of the space in which the electric field is created. It is called the volume density of electric energy. The energy of the field created by any distribution of electric charges in space can be found by integrating the volume density w e over the entire volume in which the electric field is created. Electrodynamics

Constantelectriccurrent

Electriccurrent.LawOhmaLecture № 10 If an insulated conductor is placed in an electric field, then a force will act on the free charges q in the conductor. As a result, a short-term movement of free charges occurs in the conductor. This process will end when the own electric field of the charges that have arisen on the surface of the conductor does not completely compensate for the external field. The resulting electrostatic field inside the conductor is zero (see § 4.5). However, in conductors, under certain conditions, a continuous ordered movement of free carriers of electric charge can occur. This movement is called electric current. The direction of movement of positive free charges is taken as the direction of the electric current. For the existence of an electric current in a conductor, it is necessary to create an electric field in it. The quantitative measure of the electric current is the current strength I - a scalar physical quantity equal to the ratio of the charge Δq transferred through the cross section of the conductor (Fig. 4.8.1) over the time interval Δt, to this time interval: In the International System of Units SI, the current strength is measured in amperes (BUT). The current unit 1 A is set according to magnetic interaction two parallel current-carrying conductors (see § 4.16). Constant electricity can only be created in a closed circuit in which free charge carriers circulate along closed paths. The electric field at different points in such a circuit is constant over time. Therefore, the electric field in the circuit direct current has the character of a frozen electrostatic field. But when moving an electric charge in an electrostatic field along a closed path, the work of electric forces is zero (see § 4.4). Therefore, for the existence of direct current, it is necessary to have in electrical circuit a device capable of creating and maintaining potential differences in circuit sections due to the work of forces of non-electrostatic origin. Such devices are called direct current sources. Forces of non-electrostatic origin acting on free charge carriers from current sources are called external forces. The nature of external forces can be different. In galvanic cells or batteries, they arise as a result of electrochemical processes, in DC generators, external forces arise when conductors move in a magnetic field. The current source in the electrical circuit plays the same role as the pump, which is necessary for pumping fluid in a closed hydraulic system. Under the action of external forces, electric charges move inside the current source against the forces of the electrostatic field, due to which a constant electric current can be maintained in a closed circuit. When electric charges move along the DC circuit, external forces acting inside the current sources do work. Physical quantity, equal to the ratio of work A st external forces when moving charge q from the negative pole of the current source to the positive to the value of this charge, is called electromotive force source (EMF):

Thus, the EMF is determined by the work done by external forces when moving a single positive charge. The electromotive force, as well as the potential difference, is measured in volts (V). When a single positive charge moves along a closed DC circuit, the work of external forces is equal to the sum of the EMF acting in this circuit, and the work of the electrostatic field is zero. The DC circuit can be divided into certain sections. Those sections on which external forces do not act (i.e., sections that do not contain current sources) are called homogeneous. Sections that include current sources are called heterogeneous. When a single positive charge moves along a certain section of the circuit, both electrostatic (Coulomb) and external forces do work. The work of electrostatic forces is equal to the potential difference Δφ 12 \u003d φ 1 - φ 2 between the initial (1) and final (2) points of the inhomogeneous section. The work of external forces is, by definition, the electromotive force 12 acting in this area. So the total work is The German physicist G. Ohm in 1826 experimentally established that the strength of the current I flowing through a homogeneous metal conductor (i.e., a conductor in which no external forces act) is proportional to the voltage U at the ends of the conductor:

where R = const. The value of R is usually called electrical resistance. A conductor with electrical resistance is called a resistor. This ratio expresses Ohm's law for a homogeneous section of the circuit: the current strength in the conductor is directly proportional to the applied voltage and inversely proportional to the resistance of the conductor. In SI, the unit of electrical resistance of conductors is the ohm (Ohm). A section of the circuit has a resistance of 1 ohm, in which, at a voltage of 1 V, a current of 1 A arises. Conductors that obey Ohm's law are called linear. The graphical dependence of the current strength I on the voltage U (such graphs are called current-voltage characteristics, abbreviated as CVC) is depicted by a straight line passing through the origin. It should be noted that there are many materials and devices that do not obey Ohm's law, for example, semiconductor diode or gas lamp. Even metal conductors at sufficiently high currents, a deviation from the linear Ohm's law is observed, since electrical resistance metal conductors increases with increasing temperature. For a circuit section containing EMF, Ohm's law is written in the following form:

According to Ohm's law, Adding both equalities, we get:

I(R + r) = Δφ cd + Δφ ab + .

But Δφ cd = Δφ ba = – Δφ ab. That's why

This formula expresses Ohm's law for a complete circuit: the current strength in a complete circuit is equal to the electromotive force of the source divided by the sum of the resistances of the homogeneous and inhomogeneous sections of the circuit. The resistance r of the inhomogeneous section in fig. 4.8.2 can be seen as internal resistance current source. In this case, section (ab) in Fig. 4.8.2 is the internal section of the source. If points a and b are closed with a conductor whose resistance is small compared to the internal resistance of the source (R<< r), тогда в цепи потечет ток короткого замыкания

Short circuit current is the maximum current that can be obtained from a given source with electromotive force and internal resistance r. For sources with low internal resistance, the short-circuit current can be very large and cause the destruction of the electrical circuit or source. For example, lead-acid batteries used in automobiles can have a short circuit current of several hundred amperes. Particularly dangerous are short circuits in lighting networks powered by substations (thousands of amperes). To avoid the destructive effect of such high currents, fuses or special circuit breakers are included in the circuit. In some cases, some external ballast resistance is connected to the source to prevent dangerous short-circuit currents. Then the resistance r is equal to the sum of the internal resistance of the source and the external ballast resistance. If the external circuit is open, then Δφ ba \u003d - Δφ ab \u003d, i.e., the potential difference at the poles of an open battery is equal to its EMF. If the external load resistance R is turned on through the battery current I flows, the potential difference at its poles becomes equal to

Δφ ba = – Ir.

On fig. 4.8.3 is a schematic representation of a DC source with equal EMF and internal resistance r in three modes: "idling", work on load and short circuit mode (short circuit). The strength of the electric field inside the battery and the forces acting on positive charges are indicated: – electric force and – third-party force. In short circuit mode, the electric field inside the battery disappears. To measure voltages and currents in DC electrical circuits, special devices are used - voltmeters and ammeters. The voltmeter is designed to measure the potential difference applied to its terminals. It is connected in parallel with the section of the circuit on which the potential difference is measured. Any voltmeter has some internal resistance R B . In order for the voltmeter not to introduce a noticeable redistribution of currents when connected to the measured circuit, its internal resistance must be large compared to the resistance of the section of the circuit to which it is connected. For the circuit shown in Fig. 4.8.4, this condition is written as:

R B >> R 1 .

This condition means that the current I B \u003d Δφ cd / R B flowing through the voltmeter is much less than the current I \u003d Δφ cd / R 1, which flows through the measured section of the circuit. Since no external forces act inside the voltmeter, the potential difference at its terminals coincides in definition with stress. Therefore, we can say that the voltmeter measures voltage. The ammeter is designed to measure the current strength in the circuit. The ammeter is connected in series to the break in the electrical circuit so that the entire measured current passes through it. The ammeter also has some internal resistance R A . Unlike a voltmeter, the internal resistance of an ammeter must be sufficiently small compared to the total resistance of the entire circuit. For the circuit in fig. 4.8.4 the resistance of the ammeter must satisfy the condition