PhysMat: Electric capacitance of a conductor. Capacitor. Capacitance of a parallel plate capacitor. Connection of capacitors. Energy stored in a capacitor. Electric field energy. Electric field energy density. Potential energy of a charged sphere

When charging a capacitor, an external source expends energy to separate the charges into positive and negative. Which will be located on the capacitor plates. Therefore, based on the law of conservation of energy, it does not disappear anywhere, but remains in the capacitor. The energy in the capacitor is stored in the form of the interaction force between the positive and negative charges located on its plates. That is, in the form of an electric field. Which is concentrated between the plates. This interaction tends to attract one plate to another, since, as is known, unlike charges attract.

As is known from mechanics F=mg, similar in electrical F=qE, the role of mass is played by charge, and the role of attractive force is played by field strength.

The work of moving a charge in an electric field looks like this :A=qEd1-qEd2=qEd

On the other hand, work is also equal to the difference in potential energies A=W1-W2=W.

Thus, using these two expressions, we can conclude that the potential energy accumulated in the capacitor is equal to:

Formula 1 - Energy of a charged capacitor

It is not difficult to notice that the formula is very similar to potential energy from mechanics W=mgh.

If we draw an analogy with mechanics: Imagine a stone located on the roof of a building. Here the mass of the earth interacts with the mass of the stone through gravity, and the building is tall h counteracts the force of gravity. If the building removes the stone, it falls, therefore, potential energy will turn into kinetic energy.

In electrostatics, there are two opposite charges tending to be attracted to each other; they are opposed by a dielectric of thickness d located between the plates. If the plates are closed together, the potential energy of the charge will turn into kinetic energy, that is, into heat.

In electrical engineering, the formula for energy in this form is not used. It is convenient to express it in terms of the capacitance of the capacitor and the voltage to which it is charged.

Since the charge of the capacitor is determined by the charge of one of its plates, the field strength created by it will be equal to E/2. Since the total field consists of the fields created by both plates when charged equally, but with the opposite sign.

Therefore, the energy of the capacitor will be: W=q(E/2)d

Since the beginning of the study of electricity, the issue of its accumulation and conservation was solved only in 1745 by Ewald von Kleist and Pieter van Musschenbroeck. The device created in Leiden, Holland, made it possible to accumulate and use it when necessary.

The Leyden jar is a prototype of a capacitor. Its use in physical experiments advanced the study of electricity far forward and made it possible to create a prototype of electric current.

What is a capacitor

Collecting electricity is the main purpose of a capacitor. Usually this is a system of two insulated conductors located as far as possible closer friend to friend. The space between the conductors is filled with a dielectric. The charge accumulated on the conductors is chosen to be opposite. The property of opposite charges to attract each other contributes to its greater accumulation. The dielectric has a dual role: the greater the dielectric constant, the greater the electrical capacity; charges cannot overcome the barrier and be neutralized.

Electrical capacity is the main physical quantity characterizing the ability of a capacitor to accumulate charge. The conductors are called plates; the electric field of the capacitor is concentrated between them.

The energy of a charged capacitor, apparently, should depend on its capacitance.

Electrical capacity

The energy potential makes it possible to use (high electrical capacity) capacitors. The energy of a charged capacitor is used when it is necessary to apply a short-term current pulse.

What quantities does electrical capacity depend on? The process of charging a capacitor begins with connecting its plates to the poles of a current source. The charge accumulated on one plate (the value of which is q) is taken as the charge of the capacitor. The electric field concentrated between the plates has a potential difference U.

Electrical capacity (C) depends on the amount of electricity concentrated on one conductor and the field voltage: C = q/U.

This value is measured in F (farads).

The capacity of the entire Earth cannot be compared with the size of which is approximately the size of a notebook. The accumulated powerful charge can be used in technology.

However, it is not possible to accumulate an unlimited amount of electricity on the plates. When the voltage increases to the maximum value, breakdown of the capacitor may occur. The plates are neutralized, which can lead to damage to the device. The energy of the charged capacitor is completely used to heat it.

Energy value

Heating of the capacitor occurs due to the conversion of the electric field energy into internal energy. The ability of a capacitor to do work to move a charge indicates the presence of a sufficient supply of electricity. To determine how great the energy of a charged capacitor is, consider the process of discharging it. Under the influence of an electric field of voltage U, a charge of magnitude q flows from one plate to another. By definition, the work of the field is equal to the product of the potential difference and the amount of charge: A=qU. This relationship is valid only for a constant voltage value, but during the discharge process on the capacitor plates it gradually decreases to zero. To avoid inaccuracies, let's take its average value U/2.

From the formula for electrical capacity we have: q=CU.

From here, the energy of a charged capacitor can be determined by the formula:

We see that its value is greater, the higher the electrical capacity and voltage. To answer the question of what the energy of a charged capacitor is, let us turn to their varieties.

Types of capacitors

Since the energy of the electric field concentrated inside the capacitor is directly related to its capacitance, and the operation of capacitors depends on their design features, they use Various types drives.

According to the shape of the plates: flat, cylindrical, spherical, etc.
By change in capacity: constant (capacity does not change), variable (changing physical properties, change the capacity), tuning. Capacitance can be changed by changing temperature, mechanical or electrical. The capacitance of tuning capacitors changes by changing the area of the plates.
By dielectric type: gas, liquid, solid dielectric.
By type of dielectric: glass, paper, mica, metal-paper, ceramic, thin-layer films of various compositions.

Depending on the type, other capacitors are also distinguished. The energy of a charged capacitor depends on the properties of the dielectric. The main quantity is called dielectric constant. Electrical capacity is directly proportional to it.

Flat capacitor

Let's consider the simplest device for collecting electric charge- flat capacitor. This is a physical system of two parallel plates, between which there is a dielectric layer.

The shape of the plates can be rectangular or round. If there is a need to obtain variable capacity, then the plates are usually taken in the form of half-disks. Rotation of one plate relative to the other leads to a change in the area of the plates.

С = εε 0 S/d.

Energy of a parallel plate capacitor

We see that the capacitance of the capacitor is directly proportional to the total area of one plate and inversely proportional to the distance between them. The proportionality coefficient is the electrical constant ε 0. An increase in the dielectric constant of the dielectric will increase the electrical capacity. Reducing the plate area makes it possible to obtain tuning capacitors. The energy of the electric field of a charged capacitor depends on its geometric parameters.

We use the calculation formula: W = CU 2 /2.

The energy of a charged flat capacitor is determined using the formula:

W = εε 0 S U 2 /(2d).

Using capacitors

The ability of capacitors to smoothly collect electrical charge and release it quickly enough is used in various fields of technology.

Connection with inductors allows you to create oscillatory circuits, current filters, and feedback circuits.

Photoflashes and stun guns, in which an almost instantaneous discharge occurs, use the ability of a capacitor to create a powerful current pulse. The capacitor is charged from a direct current source. The capacitor itself acts as an element that breaks the circuit. The discharge in the reverse direction occurs through a low ohmic resistance lamp almost instantly. In a stun gun, this element is the human body.

Capacitor or battery

The ability to retain the accumulated charge for a long time provides a wonderful opportunity to use it as an information storage device or energy storage device. This property is widely used in radio engineering.

Unfortunately, the capacitor is not able to replace the battery, since it has the ability to discharge. The energy accumulated by it does not exceed several hundred joules. The battery can store a large supply of electricity for a long time and with virtually no losses.

Like any system of charged bodies, a capacitor has energy. It is not difficult to calculate the energy of a charged flat capacitor with a uniform field inside it.

Energy of a charged capacitor.

In order to charge a capacitor, work must be done to separate positive and negative charges. According to the law of conservation of energy, this work is equal to the energy of the capacitor. You can verify that a charged capacitor has energy if you discharge it through a circuit containing an incandescent lamp designed for a voltage of several volts (Fig. 4). When the capacitor discharges, the lamp flashes. The energy of the capacitor is converted into other forms: heat, light.

Let us derive a formula for the energy of a flat capacitor.

The field strength created by the charge of one of the plates is equal to E/2, Where E is the field strength in the capacitor. There is a charge in a uniform field of one plate q, distributed over the surface of another plate (Fig. 5). According to the formula W p = qEd. for the potential energy of a charge in a uniform field, the energy of the capacitor is equal to:

It can be proven that these formulas are valid for the energy of any capacitor, and not just for a flat one.

Electric field energy.

According to the theory of short-range action, all the energy of interaction between charged bodies is concentrated in the electric field of these bodies. This means that energy can be expressed through the main characteristic of the field - intensity.

Since the electric field strength is directly proportional to the potential difference

(U = Ed), then according to the formula

the energy of the capacitor is directly proportional to the strength of the electric field inside it: W p ~ E 2 . A detailed calculation gives the following value for the field energy per unit volume, i.e. for energy density:

where ε 0 is the electrical constant

Application of capacitors.

The energy of a capacitor is usually not very high - no more than hundreds of joules. In addition, it does not last long due to the inevitable charge leakage. Therefore, charged capacitors cannot replace, for example, batteries as sources of electrical energy.

But this does not mean at all that capacitors as energy storage devices have not received practical use. They have one important property: capacitors can accumulate energy for a more or less long time, and when discharged through a low-resistance circuit, they release energy almost instantly. This property is widely used in practice.

A flash lamp used in photography is powered by the electric current of a capacitor discharge, which is pre-charged by a special battery. Excitation of quantum light sources - lasers is carried out using a gas-discharge tube, the flash of which occurs when a battery of high-capacity capacitors is discharged.

However, capacitors are mainly used in radio engineering. You will become acquainted with this in the 11th grade.

The energy of a capacitor is proportional to its electrical capacity and the square of the voltage between the plates. All this energy is concentrated in the electric field. The field energy density is proportional to the square of the field strength.

Rice. 1 Fig. 2

LAWS OF DC CURRENT.

Stationary electric charges are rarely used in practice. In order to make electric charges serve us, they need to be set in motion - to create an electric current. Electric current illuminates apartments, sets machines in motion, creates radio waves, and circulates in all electronic computers.

We will start with the simplest case of the movement of charged particles - consider a direct electric current.

ELECTRICITY. CURRENT STRENGTH

Let us give a strict definition of what is called electric current.

Let us recall what value the current is quantitatively characterized by.

Let's find how fast electrons move through the wires in your apartment.

When charged particles move in a conductor, electric charge is transferred from one place to another. However, if charged particles undergo random thermal motion, such as free electrons in metal, then charge transfer does not occur (Fig. 1). An electric charge moves through the cross section of a conductor only if, along with random movement, electrons participate in ordered movement (Fig. 2 ). In this case, they say that the explorer is installed electricity.

From the VIII grade physics course you know that electric current is the ordered (directed) movement of charged particles.

Electric current arises from the ordered movement of free electrons or ions.

If you move a generally neutral body, then, despite the ordered movement of a huge number of electrons and atomic nuclei, no electric current arises. The total charge transferred through any section of the conductor will be equal to zero, since charges of different signs have the same average speed.

Electric current has a certain direction. The direction of current is taken to be the direction of movement of positively charged particles. If the current is formed by the movement of negatively charged particles, then the direction of the current is considered opposite to the direction of movement of the particles.

Actions of current. We do not directly see the movement of particles in a conductor. The presence of electric current must be judged by the actions or phenomena that accompany it.

Firstly, the conductor through which the current flows heats up.

Secondly, electric current can change chemical composition conductor, for example, to isolate its chemical components (copper from a solution of copper sulfate, etc.).

Third, the current exerts a force on neighboring currents and magnetized bodies. This action is called magnetic. Thus, a magnetic needle near a current-carrying conductor rotates. The magnetic effect of current, in contrast to the chemical and thermal effect, is fundamental, since it manifests itself in all conductors without exception. The chemical effect of current is observed only in solutions and melts of electrolytes, and heating is absent in superconductors.

Current strength.

If an electric current is established in a circuit, this means that an electric charge is constantly transferred through the cross-section of the conductor. The charge transferred per unit time serves as the main quantitative characteristic of the current, called the current strength.

Thus, the current strength is equal to the charge ratio q, transferred through the cross section of the conductor over a time interval t, to this time interval. If the current strength does not change over time, then the current is called constant.

The strength of the current, like a charge,— the quantity is scalar. She might be like positive, so and negative. The sign of the current depends on which direction along the conductor is taken as positive. Current strength / > 0, if the direction of the current coincides with the conventionally selected positive direction along the conductor. Otherwise /< 0.

The strength of the current depends on the charge carried by each particle, the concentration of the particles, the speed of their directional movement and the cross-sectional area of the conductor. Let's show this.

Let the conductor (Fig. 3) have a cross section with area S. Let us take the direction from left to right as the positive direction in the conductor. The charge of each particle is equal q 0 . In the volume of the conductor, limited by cross sections 1 and 2 , contained nSl particles, where P — particle concentration. Their total charge q = q Q nSl. If particles move from left to right with average speed υ, then in time

All particles contained in the volume under consideration will pass through cross section 2 . Therefore, the current strength is:

formula (2) where e— electron charge modulus.

Let, for example, the current strength I = 1 A, and the cross-sectional area of the conductor S = 10 -6 m 2. Electron charge modulus e = 1.6 - 10 -19 C. The number of electrons in 1 m 3 of copper is equal to the number of atoms in this volume, since one of the valence electrons of each copper atom is collectivized and is free. This number is P= 8.5 10 28 m -3 Therefore,

Fig No. 1. Fig No. 2 Fig No. 3

CONDITIONS REQUIRED FOR THE EXISTENCE OF ELECTRIC CURRENT

What is needed to create an electric current? Think about it yourself and only then read this paragraph.

For the emergence and existence of a constant electric current in a substance, it is necessary, firstly, the presence of free charged particles. If positive and negative charges are bonded to each other in atoms or molecules, then their movement will not lead to the appearance of electric current.

The presence of free charges is not yet sufficient for the occurrence of current. To create and maintain the ordered movement of charged particles, secondly, a force acting on them in a certain direction is necessary. If this force ceases to act, then the ordered movement of charged particles will cease due to the resistance provided to their movement by ions of the crystal lattice of metals or neutral molecules of electrolytes.

Charged particles, as we know, are acted upon by an electric field with a force . Usually it is the electric field inside the conductor that serves as the cause that causes and maintains the ordered movement of charged particles. Only in the static case, when the charges are at rest, the electric field inside the conductor is zero.

If there is an electric field inside the conductor, then there is a potential difference between the ends of the conductor in accordance with the formula. When the potential difference does not change over time, a constant electric current is established in the conductor. Along the conductor, the potential decreases from the maximum value at one end of the conductor to the minimum at the other. This decrease in potential can be detected by simple experiment.

Let's take a not very dry wooden stick as a conductor and hang it horizontally. (Such a stick, although poorly, still conducts current.) Let the voltage source be an electrostatic machine. To record the potential of different sections of the conductor relative to the ground, you can use pieces of metal foil attached to the stick. We connect one pole of the machine to the ground, and the second to one end of the conductor (stick). The chain will be open. When we rotate the handle of the machine, we will find that all the leaf points deviate at the same angle (Fig. 1 ).

This means the potential everyone the points of the conductor relative to the ground are the same. This is how it should be if the charges on the conductor are in balance. If now the other end of the stick is grounded, then when the machine handle is rotated, the picture will change. (Since the earth is a conductor, grounding the conductor makes the circuit closed.) At the grounded end, the leaves will not diverge at all: the potential of this end of the conductor is almost equal to the potential of the ground (the potential drop in a metal wire is small). The maximum angle of divergence of the leaves will be at the end of the conductor connected to the machine (Fig. 2). A decrease in the angle of divergence of the leaves as they move away from the machine indicates a drop in potential along the conductor.

Electricity can only be obtained in a substance that contains free charged particles. For them to start moving, you need to create in the explorer electric field.

Fig No. 1 Fig No. 2

OHM'S LAW FOR A CIRCUIT SECTION. RESISTANCE

Ohm's law was studied in VIII grade. This law is simple, but so important that it needs to be repeated.

Volt-ampere characteristics.

In the previous paragraph, it was established that for the existence of current in a conductor, it is necessary to create a potential difference at its ends. The current strength in the conductor is determined by this potential difference. The greater the potential difference, the greater the electric field strength in the conductor and, consequently, the greater the speed of directional movement of charged particles. According to the formula, this means an increase in current strength.

For each conductor - solid, liquid and gaseous - there is a certain dependence of the current strength on the applied potential difference at the ends of the conductor. This dependence is expressed by the so-called volt - ampere characteristic of the conductor. It is found by measuring the current strength in the conductor at various voltage values. Knowledge of the current-voltage characteristic plays a big role in the study of electric current.

Ohm's law.

The simplest form is the volt-ampere characteristic of metal conductors and electrolyte solutions. It was first established (for metals) by the German scientist Georg Ohm, therefore the dependence of current on voltage is called Ohm's law. In the section of the circuit shown in Figure 109, the current is directed from point 1 to point 2 . The potential difference (voltage) at the ends of the conductor is equal to: U = φ 1 - φ 2. Since the current is directed from left to right, the electric field strength is directed in the same direction and φ 1 > φ 2

According to Ohm's law, for a section of a circuit, the current strength is directly proportional to the applied voltage U and inversely proportional to the conductor resistance R:

Ohm's law has a very simple form, but it is quite difficult to prove its validity experimentally. The fact is that the potential difference in a section of a metal conductor, even with a high current strength, is small, since the resistance of the conductor is low.

The electrometer in question is unsuitable for measuring such low voltages: its sensitivity is too low. An incomparably more sensitive device is needed. Then, by measuring the current with an ammeter and the voltage with a sensitive electrometer, you can make sure that the current is directly proportional to the voltage. The use of conventional instruments for measuring voltage - volt meters - is based on the use of Ohm's law.

The principle of the device, a voltmeter, is the same as an ampere meter. The angle of rotation of the device arrow is proportional to the current strength. The strength of the current passing through the voltmeter is determined by the voltage between the points of the circuit to which it is connected. Therefore, knowing the resistance of the voltmeter, you can determine the voltage by the current strength. In practice, the device is calibrated so that it immediately shows the voltage in volts.

Resistance. The main electrical characteristic of a conductor is resistance. The current strength in the conductor at a given voltage depends on this value. The resistance of a conductor is a measure of the conductor’s resistance to the establishment of an electric current in it. Using Ohm's law, you can determine the resistance of a conductor:

To do this, you need to measure the voltage and current.

Resistance depends on the material of the conductor and its geometric dimensions. The resistance of a conductor of length l with a constant cross-sectional area S is equal to:

where p is a value that depends on the type of substance and its state (primarily on temperature). The value p is called specific resistance of the conductor. Resistivity numerically equal to the resistance of a conductor shaped like a cube with an edge 1m, if the current is directed along the normal to two opposite faces of the cube.

The unit of conductor resistance is established based on Ohm's law and is called ohm. The nick wire has resistance 1 Ohm, if at potential difference 1 V current strength in it 1 A.

The unit of resistivity is 1 Ohm?m. The resistivity of metals is low. Dielectrics have very high resistivity. The table on the flyleaf gives examples of resistivity values for some substances.

The meaning of Ohm's law.

Ohm's law determines the current strength in an electrical circuit at a given voltage and known resistance. It allows you to calculate the thermal, chemical and magnetic effects of current, as they depend on the strength of the current. It follows from Ohm's law that it is dangerous to close a conventional lighting network with a conductor of low resistance. The current will be so strong that it can have serious consequences.

Ohm's law is the basis of all direct current electrical engineering. The formula must be well understood and firmly remembered.

ELECTRICAL CIRCUITS. SERIES AND PARALLEL CONDUCTOR CONNECTIONS

From a current source, energy can be transferred through wires to devices that consume energy: an electric lamp, a radio receiver, etc. For this, they make up electrical circuits of varying complexity. An electrical circuit consists of an energy source, devices that consume electrical energy, connecting wires and switches to complete the circuit. Often And the electrical circuit includes devices that control current strength And voltage at various parts of the circuit, - ammeters and volt meters.

The simplest and most common connections of conductors include serial and parallel connections.

Series connection of conductors.

When connected in series, the electrical circuit has no branches. All conductors are connected to the circuit one after another. Figure 1 shows a series connection of two conductors 1 and 2 , having resistance R 1, and R2. These can be two lamps, two windings of an electric motor, etc.

The current strength in both conductors is the same, i.e. (1)

since in conductors the electric charge in the case of direct current does not accumulate and the same charge passes through any cross-section of the conductor over a certain time.

The voltage at the ends of the section of the circuit under consideration is the sum of the voltages on the first and second conductors:

We hope that you can handle the proof of this simple relationship yourself.

Applying Ohm's law for the entire section as a whole and for sections with resistance R 1 And R2, it can be proven that the total resistance of the entire section of the circuit when connected in series is equal to:

This rule can be applied to any number of conductors connected in series.

The voltages on the conductors and their resistances in a series connection are related by the relationship:

Prove this equality.

Parallel connection of conductors.

Figure 2 shows a parallel connection of two conductors 1 and 2 with resistances R 1 And R2. In this case, electric current 1 branches into two parts. We denote the current strength in the first and second conductors by I 1 and I 2. Since at the point A- branching of conductors (this point is called node) - electric charge does not accumulate, then the charge entering the node per unit time is equal to the charge leaving the node during the same time. Therefore, I = I 1 + I 2

The voltage U at the ends of conductors connected in parallel is the same.

The lighting network maintains a voltage of 220 or 127 V. Devices that consume electrical energy are designed for this voltage. Therefore, parallel connection is the most common way to connect different consumers. In this case, the failure of one device does not affect the operation of the others, whereas with a series connection, the failure of one device opens the circuit.

Applying Ohm's law for the entire section as a whole and for sections with resistances R 1 and R 2 , it can be proven that the reciprocal of the impedance of the section ab, equal to the sum of the reciprocal values of the resistances of individual conductors:

The current strength in each of the conductors and the resistance of the conductors in a parallel connection are related by the relation

Various conductors in a circuit are connected to each other in series or in parallel. In the first case, the current strength is the same in all conductors, and in the second case, the voltages on the conductors are the same. Most often, various current consumers are connected in parallel to the lighting network.

CURRENT AND VOLTAGE MEASUREMENT

Everyone should know how to measure current with an ampere meter and voltage with a voltmeter.

Current measurement.

To measure the current strength in a conductor, an ammeter is connected in series with this conductor(Fig. 1). But you need to keep in mind that the ampere meter itself has some resistance R a. Therefore, the resistance of the circuit section with the ampere meter turned on increases, and at a constant voltage, the current decreases in accordance with Ohm’s law. In order for the ammeter to have as little influence as possible on the current it measures, its resistance is made very small. This must be remembered and never try to measure the current in the lighting network by connecting the ammeter to the outlet. will happen short circuit; The current strength with low resistance of the device will reach such a large value that the winding of the ammeter will burn out.

Voltage measurement.

In order to measure the voltage across a section of a circuit with resistance R, A voltmeter is connected to it in parallel. The voltage on the voltmeter coincides with the voltage on the circuit section (Fig. 2).

If the voltmeter resistance RB, then after connecting it to the circuit, the resistance of the section will no longer be R, A . Because of this, the measured voltage in the circuit section will decrease. In order for the voltmeter not to introduce noticeable distortions into the measured voltage, its resistance must be large compared to the resistance of the section of the circuit on which the voltage is measured. The voltmeter can be connected to the network without the risk that it will burn out, if only it is designed for a voltage exceeding the network voltage.

The ammeter is connected in series with the conductor in which the current is measured. The voltmeter is connected in parallel to the conductor on which the voltage is measured.

DC OPERATION AND POWER

Electric current is so widely used because it carries energy. This energy can be converted into any form.

With the ordered movement of charged particles in a conductor the electric field does work; it is usually called current work. Now we will recall information about work and current power from the physics course VIII class.

Current work.

Let's consider an arbitrary section of the chain. This may be a homogeneous conductor, for example, the filament of an incandescent lamp, the winding of an electric motor, etc. Let a charge q pass through the cross section of the conductor during time t. Then the electric field will do the work A=qU.

Since the current strength , then this work is equal to:

The work done by the current on a section of the circuit is equal to the product of the current, voltage and time during which the work was done.

According to the law of conservation of energy, this work must be equal to the change in the energy of the section of the circuit under consideration. Therefore, the energy released in a given section of the circuit over time At, equal to the work of the current (see formula (1)).

If no mechanical work is performed on a section of the circuit and the current does not produce chemical effects, only heating of the conductor occurs. A heated conductor gives off heat to surrounding bodies.

Heating of the conductor occurs as follows. The electric field accelerates electrons. After colliding with ions of the crystal lattice, they transfer their energy to the ions. As a result, the energy of random motion of ions around equilibrium positions increases. This means an increase in internal energy. At the same time, the temperature of the conductor rises, and it begins to transfer heat to the surrounding bodies. A short time after the circuit is closed, the process is established, and the temperature stops changing over time. Due to the work of the electric field, energy is continuously supplied to the conductor. But its internal energy remains unchanged, since the conductor transfers to the surrounding bodies an amount of heat equal to the work of the current. Thus, formula (1) for the work of current determines the amount of heat transferred by the conductor to other bodies.

If in formula (1) we express either voltage in terms of current, or current in terms of voltage using Ohm’s law for a section of the circuit, we obtain three equivalent formulas:

(2)

The formula A = I 2 R t is convenient to use for connecting conductors in series, since the current strength in this case is the same in all conductors. For a parallel connection, the following formula is convenient: , since the voltage on all conductors is the same.

Joule-Lenz law.

The law that determines the amount of heat that a conductor with current releases into the environment was first established experimentally by the English scientist D. Joule (1818-1889) and the Russian scientist E. H. Lenz (1804-1865). The Joule-Lenz law was formulated as follows: the amount of heat generated by a conductor carrying current is equal to the product of the square of the current, the resistance of the conductor and the time it takes for the current to pass through the conductor:

(3)

We obtained this law using reasoning based on the law of conservation of energy. Formula (3) allows you to calculate the amount of heat generated in any section of the circuit containing any conductors.

Current power.

Any electrical device (lamp, electric motor) is designed to consume a certain energy per unit of time. Therefore, along with work, the concept of current power. Current power is equal to the ratio of current work over timet to this time interval.

According to this definition

(4)

This expression for power can be rewritten in several equivalent forms if we use Ohm’s law for a section of the circuit:

Most devices indicate their power consumption.

The passage of electric current through a conductor is accompanied by the release of energy in it. This energy is determined by the work of the current: the product of the transferred charge and voltage at the ends of the conductor.

ELECTROMOTIVE FORCE.

Any current source is characterized by electromotive force, or EMF. So, on a round flashlight battery it says: 1.5 V. What does this mean?

Connect two metal balls carrying charges of opposite signs with a conductor. Under the influence of the electric field of these charges, an electric current arises in the conductor (Fig. 1). But this current will be very short-term. The charges are quickly neutralized, the potentials of the balls will become the same, and the electric field will disappear.

Outside forces.

In order for the current to be constant, it is necessary to maintain a constant voltage between the balls. This requires a device (current source), which would move charges from one ball to another in the direction opposite to the direction of the forces acting on these charges from the electric field of the balls. In such a device, in addition to electrical forces, charges must be acted upon by forces of non-electrostatic origin (Fig. 2). The electric field of charged particles (Coulomb field) alone is not capable of maintaining a constant current in the circuit.

Any forces acting on electrically charged particles, with the exception of forces of electrostatic origin (i.e. Coulomb), are called extraneous forces.

The conclusion about the need for external forces to maintain a constant current in the circuit will become even more obvious if we turn to the law of conservation of energy. The electrostatic field is potential. The work of this field when moving charged particles along a closed electrical circuit is zero. The passage of current through the conductors is accompanied by the release of energy - the conductor heats up. Consequently, in any circuit there must be some source of energy supplying it to the circuit. In it, in addition to the Coulomb forces, third-party non-potential forces must act. The work of these forces along a closed loop must be different from zero. It is in the process of doing work by these forces that charged particles acquire energy inside the current source and then give it to the conductors of the electrical circuit.

Third-party forces set in motion charged particles inside all current sources: in generators at power plants, in galvanic cells, batteries, etc.

When a circuit is closed, an electric field is created in all conductors of the circuit. Inside the current source, charges move under the influence of external forces against Coulomb forces (electrons from a positively charged electrode to a negative one), and throughout the rest of the circuit they are driven by an electric field (see Fig. 2).

Analogy between electric current and fluid flow.

To better understand the mechanism of current generation, let us turn to the similarity between electric current in a conductor and the flow of liquid through pipes.

In any section of a horizontal pipe, liquid flows due to the pressure difference at the ends of the section. The liquid moves in the direction of decreasing pressure. But the pressure force in a liquid is a type of elasticity force, which is potential, like Coulomb forces. Therefore, the work of these forces on a closed path is zero and these forces alone are not capable of causing long-term circulation of liquid through the pipes. The flow of liquid is accompanied by energy losses due to the action of friction forces. A pump is needed to circulate water.

The piston of this pump acts on liquid particles and creates a constant pressure difference at the inlet and outlet of the pump (Fig. 3). This allows the liquid to flow through the pipe. The pump is similar to a current source, and the role of external forces is played by the force acting on the water from the moving piston. Inside the pump, fluid flows from areas with lower pressure to areas with higher pressure. The pressure difference is similar to voltage.

The nature of external forces.

The nature of external forces can be varied. In power plant generators, an external force is a force acting from a magnetic field on electrons in a moving conductor. This was briefly discussed in the VIII class physics course.

In a galvanic cell, for example a Volta cell, chemical forces act. The Volta cell consists of zinc and copper electrodes placed in a sulfuric acid solution. Chemical forces cause zinc to dissolve in acid. Positively charged zinc ions pass into the solution, and the zinc electrode itself becomes negatively charged. (Copper dissolves very little in sulfuric acid.) A potential difference appears between the zinc and copper electrodes, which determines the current in a closed electrical circuit.

Electromotive force.

The action of external forces is characterized by an important physical quantity called electromotive force (abbreviated EMF).

The electromotive force in a closed circuit is the ratio of the work done by external forces when moving a charge along the circuit to the charge:

Electromotive force is expressed in volts.

We can talk about electromotive force at any part of the circuit. This is the specific work of external forces (work to move a unit charge) not throughout the entire circuit, but only in a given area. Electromotive force of a galvanic cell there is the work of external forces when moving a single positive charge inside an element from one pole to another. The work of external forces cannot be expressed through a potential difference, since external forces are not potential and their work depends on the shape of the trajectory. So, for example, the work of external forces when moving a charge between the terminals of a current source outside the source itself is zero.

Now you know what EMF is. If the battery says 1.5 V, this means that external forces (chemical in this case) perform 1.5 J of work when moving a charge of 1 C from one pole of the battery to the other. Direct current cannot exist in a closed circuit if there are no external forces acting in it, i.e. there is no EMF

Fig No. 1 Fig. No. 2 Fig. No. 3

OHM'S LAW FOR A COMPLETE CIRCUIT

Electromotive force determines the current strength in a closed electrical circuit with a known resistance.

Using the law of conservation of energy, we will find the dependence of current strength on EMF and resistance.

Let's consider the simplest complete (closed) circuit, consisting of a current source (galvanic cell, battery or generator) and a resistor with a resistance R(Fig. 1). The current source has an emf ε and a resistance r. The source resistance is often called internal resistance in contrast to the external resistance R of the circuit. In a generator, r is the resistance of the windings, and in a galvanic cell, it is the resistance of the electrolyte solution and electrodes.

Ohm's law for a closed circuit relates the current in the circuit, the emf and total resistance R + r of the circuit. This connection can be established theoretically if we use the law of conservation of energy and the Joule-Lenz law.

Let it take time t an electric charge will pass through the cross section of the conductor q. Then the work of external forces when moving a charge?q can be written as follows: A st = ε · q. According to the definition of current strength q = It . That's why

(1)

When performing this work on the internal and external sections of the circuit, the resistance of which r and R, some heat is released. According to the Joule-Lenz law, it is equal to:

Q = I 2 Rt + I 2 rt.(2)

According to the law of conservation of energy, A = Q. Equating (1) and (2), we obtain:

ε = IR + Ir(3)

The product of current and resistance of a circuit section is often called voltage drop in this area. Thus, the EMF is equal to the sum of the voltage drops on the internal and external sections of the closed circuit.

Usually Ohm's law for a closed circuit is written in the form

(4)

Electric capacitance of the conductor.

Electrical capacity- characteristic of a conductor, a measure of its ability to accumulate electrical charge. In electrical circuit theory, capacitance is the mutual capacitance between two conductors; parameter of a capacitive element of an electrical circuit, presented in the form of a two-terminal network. Such capacitance is defined as the ratio of the magnitude of the electric charge to the potential difference between these conductors.

Capacitor. Capacitance of a parallel plate capacitor.

Connection of capacitors.

Parallel connection of capacitors
The capacitor plates are connected in pairs, i.e. two insulated conductors remain in the system, which represent the plates of the new capacitor

Conclusion: When connecting capacitors in parallel a) the charges add up, b) the voltages are the same, c) the containers are folded. That., *the total capacitance is greater than the capacitance of any of the parallel-connected capacitors*
Series connection of capacitors
Only one connection is made, and the two remaining plates - one from capacitor C 1 and the other from capacitor C 2 - play the role of plates of the new capacitor.

Conclusion: When connecting capacitors in series a) the stresses add up, b) the charges are the same, c) the reciprocals of the capacitances are added. That., *the total capacitance is less than the capacitance of any of the capacitors connected in series.*

Energy stored in a capacitor.

As is known from mechanics F=mg, similar in electrical F=qE, the role of mass is played by charge, and the role of attractive force is played by field strength.

The work of moving a charge in an electric field looks like this :A=qEd1-qEd2=qEd

On the other hand, work is also equal to the difference in potential energies A=W1-W2=W.

Thus, using these two expressions, we can conclude that the potential energy accumulated in the capacitor is equal to:

Formula 1 - Energy of a charged capacitor

It is not difficult to notice that the formula is very similar to potential energy from mechanics W=mgh.

In electrical engineering, the formula for energy in this form is not used. It is convenient to express it in terms of the capacitance of the capacitor and the voltage to which it is charged.

Therefore, the energy of the capacitor will be: W=q(E/2)d

Since tension can be expressed in terms of tension and distance (U=Ed) Substitute it into our formula and we get: W=qU/2

Now using the expression for capacity, C=q/U we get the final result.

The energy of a charged capacitor has the form:

Electric field energy.

An electric field has energy. The density of this energy is determined by the field strength and can be found using the formula

Electric field energy.The energy of a charged capacitor can be expressed in terms of quantities characterizing the electric field in the gap between the plates. Let's do this using the example of a flat capacitor. Substituting the expression for capacitance into the formula for capacitor energy gives
Private U / dequal to the field strength in the gap; workS· drepresents the volumeVoccupied by the field. Hence,