Electrical consumption of serially connected. Electric capacitance, capacitors. Serial and parallel capacitor connections

The amount of electrical capacity depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are 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 casings. The simplest capacitor is a system of two flat conducting 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 plane capacitor is mainly localized between the plates (Fig. 4.6.1); However, a relatively weak electric field also occurs near the edges of the plates and in the surrounding space, which is called field of scattering. In a whole series of problems, it is possible to neglect approximately 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 tasks, the neglect of the stray field can lead to gross errors, since this violates the potential nature of the electric field (see § 4.4).

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

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

Thus, the electrical 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 capacitance of the capacitor increases by ε times:

Capacitors can be interconnected to form capacitor banks. With parallel connectionthe capacitors (Fig. 4.6.3) have the same voltage on the capacitors: U1 = U2 = U, and the charges are q1 = C1U and q2 = C2U. Such a system can be considered as a single capacitor of electric capacity C, charged by a charge q = q1 + q2 with a voltage between the plates equal to U. Hence,

Electric intensity. Capacitors Lecture №9If the two conductors isolated from each other are charged with charges q 1 and q 2, then a certain potential difference Δφ arises between them, depending on the values of the charges and the geometry of the conductors. The potential difference Δφ between two points in an electric field is often referred to as voltage and is denoted by the letter U. Of greatest practical interest is the case when the charges of the conductors are equal in magnitude and opposite in sign: q 1 = - q 2 = q. In this case, you can introduce the concept of electrical capacitance. The electrical capacitance 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 capacitance depends on the shape and size of the conductors and on the properties of the dielectric separating the conductors. There are 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 conducting plates that are parallel to each other at a small distance compared to the size of the plates and separated by a dielectric layer. Such a capacitor is called flat. The electric field of a plane capacitor is mainly localized between the plates (Fig. 4.6.1); However, a relatively weak electric field also appears near the edges of the plates and in the surrounding space, which is called the stray field. In a whole series of problems, it is possible to neglect approximately 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 tasks, the neglect of the stray field can lead to gross errors, since this violates the potential nature of the electric field (see § 4.4). Each of the charged plates of a flat capacitor creates an electric field near the surface, the strength of which is expressed by the ratio (see § 4.3)

According to the principle of superposition, the field strength 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 in different directions, and therefore E = 0. The surface density σ of the charge of the plates is 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 equal to Ed, where d is the distance between the plates. From these relationships, you can get the formula for the capacitance of a flat capacitor: Examples of capacitors with a different configuration of the plates 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 coaxially conducting cylinders of radii R 1 and R 2 and length L. The capacitances of these capacitors, filled with a dielectric with permittivity ε, are expressed by the formulas:

Capacitors can be interconnected to form capacitor banks. In parallel connection of capacitors (Fig. 4.6.3), the voltages on the capacitors are the same: U 1 = U 2 = U, and the charges are q 1 = C 1 U and q 2 = C 2 U. Such a system can be considered as a single capacitance of electric capacity C charged by a charge q = q 1 + q 2 with a voltage between the plates equal to U. This implies With a series connection (Fig. 4.6.4), the charges of both capacitors are the same: q 1 = q 2 = q, and the voltages on them are equal, and Such a system can be viewed as a single capacitor charged by charge q at a voltage between the plates U = U 1 + U 2. Consequently,

In the case of a series connection of capacitors, the inverse values of the capacities are added. The formulas for parallel and series connection remain valid for any number of capacitors connected to the battery. EnergyelectricfieldsExperience shows that a charged capacitor contains a store of energy. The energy of a charged capacitor is equal to the work of external forces, which must be expended to charge a capacitor. The charging process of a capacitor can be represented as the sequential transfer of sufficiently small portions of charge Δq\u003e 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 one. Since each portion is transferred under conditions when there is already some charge q on the plates, and there is some potential difference between them when transferring each portion Δq, external forces must do work

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

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

Constantelectriccurrent

Electriccurrent.LawOmaghLecture № 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 arising on the surface of the conductor does not compensate for the completely external field. The resulting electrostatic field inside the conductor is zero (see § 4.5). However, in certain conductors, under certain conditions, a continuous ordered movement of free charge carriers can occur. Such a movement is called an electric shock. 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 I, a scalar physical quantity equal to the ratio of the charge Δq transferred through the conductor cross-section (Fig. 4.8.1) over the time interval Δt to this time interval: In the International System of Units SI the current is measured in amperes (BUT). The unit of current measurement of 1 A is set by the magnetic interaction of two parallel conductors with current (see § 4.16). A constant electric current can only be created in a closed circuit in which free charge carriers circulate along closed trajectories. The electric field at different points in such a circuit is constant in time. Consequently, the electric field in the DC circuit has the character of a frozen electrostatic field. But when moving an electric charge in an electrostatic field along a closed trajectory, the work of the electric forces is zero (see § 4.4). Therefore, for the existence of direct current, it is necessary to have a device in the electrical circuit capable of creating and maintaining potential differences in the circuit sections due to the work of forces of non-electrostatic origin. Such devices are called DC sources. Forces of non-electrostatic origin acting on free charge carriers from current sources are called external forces. The nature of external forces may be different. In galvanic cells or batteries, they occur 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, so that 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 the work. A physical quantity equal to the ratio of the work A st of external forces when the charge q moves from the negative pole of the current source to the positive value of this charge is called the source electromotive force (EMF):

Thus, the EMF is determined by the work done by outside 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. DC circuit can be divided into specific sections. Those areas that are not affected by external forces (that is, areas that do not contain current sources) are called homogeneous. Sections that include current sources are called non-uniform. When moving a single positive charge along a certain part of the circuit, both electrostatic (Coulomb) and third-party forces do the work. The operation of electrostatic forces is equal to the potential difference Δφ 12 = φ 1 - φ 2 between the initial (1) and final (2) points of the non-uniform area. The work of external forces is equal, by definition, the electromotive force 12 acting on this site. Therefore, the total work is The German physicist G. Ohm experimentally established in 1826 that the strength of current I flowing through a uniform metal conductor (that is, a conductor in which external forces do not act) is proportional to the voltage U at the ends of the conductor:

where R = const. The value of R is called electric resistance. A conductor with electrical resistance is called a resistor. This relation expresses Ohm’s law for a homogeneous section of the circuit: the current 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 ohm (ohm). A resistance of 1 Ohm has such a part of the circuit in which, at a voltage of 1 V, a current of 1 A arises. Conductors obeying Ohm's law are called linear. The graphical dependence of the current I on the voltage U (such graphs are called voltage-current characteristics, abbreviated VAC) is represented 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, a semiconductor diode or a discharge lamp. Even with metal conductors at sufficiently high currents, there is a deviation from Ohm’s linear law, since the electrical resistance of metal conductors increases with increasing temperature. For a circuit section containing an emf, Ohm’s law is written in the following form:

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

I (R + r) = Δφ cd + Δφ ab +.

But Δφ cd = Δφ ba = - Δφ ab. therefore

This formula expresses Ohm’s law for a complete circuit: the current in a complete circuit is equal to the source's electromotive force divided by the sum of the resistances of the uniform and non-uniform sections of the circuit. The resistance r of the inhomogeneous section in fig. 4.8.2 can be considered as the internal resistance of the current source. In this case the section (ab) in fig. 4.8.2 is the inside of the source. If a and b points 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 this source with electromotive force and internal resistance r. For sources with low internal resistance, the short-circuit current can be very high and cause destruction of the electrical circuit or source. For example, in lead batteries used in automobiles, the short-circuit current may be several hundred amperes. Especially dangerous are short circuits in lighting networks powered by substations (thousands of amperes). To avoid the destructive effect of such large currents, fuses or special circuit breakers are included in the circuit. In some cases, to prevent dangerous values of short circuit current, some external ballast resistance is connected to the source. 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 = - Δφ ab =, i.e. the potential difference at the poles of the open battery is equal to its EMF. If the external load resistance R is switched on and through the battery current I flows, the potential difference at its poles becomes equal

Δφ ba = - Ir.

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

R B \u003e\u003e R 1.

This condition means that the current IB = Δφ cd / RB flowing through the voltmeter is much less than the current I = Δφ cd / R 1 that flows through the measured section of the circuit. Because external forces do not act inside the voltmeter, the potential difference across its terminals matches definition with stress. Therefore, we can say that the voltmeter measures the voltage. Ammeter is designed to measure the current in the circuit. The ammeter is connected in series to an open 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

In many cases, to obtain the desired capacity capacitors come. It is supposed to be connected in 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 one (Fig.

15.31). With a series connection on all plates of the capacitors will be the same in magnitude charges (Explain why.) Since the charges on the capacitor are in equilibrium, the potentials of the plates interconnected 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 is seen that the voltage on the battery is equal to the sum of the voltages on the series-connected capacitors. Really,

Using the ratio we get

After reduction we will have

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

Parallel is the connection of capacitors, in which all positively charged plates are connected to one wire, and negatively charged - to another (Fig. 15.32). In this case, the voltages on all the 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 to get the formula for. calculation of the electrical capacity of the battery parallel connected capacitors:

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

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