Speech therapy portal / Parents / Communication between values. Geometrical tasks resulting in differential equations. Mathematically, the Ohm law was described as

Communication between values. Geometrical tasks resulting in differential equations. Mathematically, the Ohm law was described as

26.07.2020

The fundamental position describing the dependence of the current, resistance and voltage from each other is the Ohm law for the AC circuit. The main difference from the same position for the regimen of the chain is to take into account the full resistance. This value depends on the active and reactive component of the line, that is, takes into account capacity and inductance. Therefore, the calculation of the parameters for the full chain compared to the site will be more complicated.

Basic concepts

All the science of electrical engineering is built on operating with such concepts as the charge and potential. In addition, important phenomena in the circuit are electrical and magnetic fields. In order to understand the essence of the Ohm law, it is necessary to understand what these quantities are, and from which certain electromagnetic processes depend on.

Electricity is called the phenomenon due to the interaction of charges between themselves and their movement. This word was put into use William Gilbert in 1600 after opening them the ability of some bodies to be electrified. Since he spent his experiments with a piece of amber, then the property attributing or pushing them with them, the other substances were called "amber", which in translating from Greek sounds like electricity.

In the future, various scientists, such as Ersted, Ampere, Joule, Faraday, Volt, Lenz and Ohm, a number of phenomena were opened. Thanks to their research, the use of concepts appeared: electromagnetic induction and field, galvanic element, current and potential. They discovered between electricity and magnetism, which led to the emergence of science examining the theory of electromagnetic phenomena.

In 1880, the Russian engineer Lachinov theoretically pointed out which conditions are necessary for the transmission of electricity at distances. And after 8 years, Henry Rudolf Hertz during experiments registered electromagnetic waves.

Thus it was found that electrical charges are able to create electrical radiation around themselves. Conditionally divided into particles with a positive and negative charge sign. It was established that the elements of the charges are attracted, and the varied - repel. To make their movement to the physical body, it is necessary to attach any energy. When they are moved, a magnetic field occurs.

The property of materials to ensure the movement of charges received the name of the conductivity, and the value inverse it is resistance. The ability to pass through itself charges depends on the structure of the crystal lattice of the substance, its connections, defects and the content of impurities.

Determination of voltage

Scientists have established that there are two types of movement of charges - chaotic and directed. The first type does not lead to any processes, since the energy is in a balanced state. But if the body is applied to the body, forcing the charges to follow one way, it will arise electricity. There are two types:

Permanent - the power and direction of which remain constant in time.
A variable - having a different value at a certain point of time and changing its movement, with its change repeating at equal intervals (cycle). This variability is described by the harmonic law of sine or cosine.

The charge is characterized by such a concept as potential, that is, the amount of energy it possesses. The necessary force to move the charge from one point of the body to another is called voltage.

It is determined relative to the change in the charge potential. The strength of the current is determined by the ratio of the amount of charge pasted through the body per unit of time, to the magnitude of this period. Mathematically described by the expression: IM \u003d ΔQ / ΔT, is measured in amperes (a).

Regarding the variable signal, an additional value is introduced - the frequency F, which determines the cyclicality of the signal flow F \u003d 1 / T, where T is the period. For its unit of measurement, hertz (Hz) was adopted. Based on this, the sinusoidal current is expressed by the formula:

I \u003d im * sin (w * t + ψ), where:

IM is the power of the current at a certain point in time;
Ψ is a phase determined by the displacement of the current wave relative to the voltage;
w is a circular frequency, this value depends on the period and equal to W \u003d 2 * p * f.

The voltage is characterized by the work that the electric field makes the charge from one point to another. It is determined as the potential difference: Um \u003d φ1 - φ2. The expending work is made up of two forces: electrical and third-party, called electromotive (EMF). It depends on magnetic induction. The potential is equal to the ratio of the energy of the interaction of the surrounding field to the value of its magnitude.

therefore for a harmonic signal change, the voltage value is expressed as:

U \u003d UM * SIN (W * T + ψ).

Where Um is an amplitude voltage value. A variable voltage voltage (B) is measured.

Impedance chain

Each physical body has its own resistance. It is due internal structure Substances. This value is characterized by the property of the conductor to prevent the passage of the current and depends on the specific electrical parameter. Determined by the formula: R \u003d ρ * L / S, where ρ is a specific resistance, which is a scalar value, OM * M; L - Explorer length; m; S is the cross-sectional area, m 2. In this expression, constant resistance is determined inherent in passive elements.

At the same time, impedance, impedance, is as the sum of the passive and reactive component. The first is determined only by the active resistance consisting of the resistive load of the power supply and resistors: R \u003d R0 + R. The second is located as a difference between the capacitive and inductive resistance: x \u003d xl-xc.

If you put the perfect capacitor (without loss) to the electrical circuit, then after the variable signal goes on it, it will charge. The current will begin to come further, in accordance with the periods of its charge and discharge. The amount of electricity flowing into the chain is: Q \u003d C * U, where C is the capacity of the element, F; U - power supply voltage or capacitor plays, V.

Since the rate of change of current and voltage is directly proportional to the frequency W, the following expression will be fair: i \u003d 2 * p * f * C * U. It turns out that it turns out that capacitive impedance is calculated by the formula:

Xc \u003d 1/2 * p * f * c \u003d 1 / W * C, Ohm.

The inductive resistance occurs due to the appearance of its own field in the conductor, called EL self-induction. It depends on the inductance and speed of change. In turn, the inductance depends on the forms and sizes of the conductor, the magnetic permeability of the medium: L \u003d F / I, is measured in Teslas (TL). Since the voltage applied to inductance is equally equal to self-induction EMF, it is true EL \u003d 2 * P * F * L * I. The rate of current change is proportional to the frequency w. Based on this, inductive resistance is:

XL \u003d W * L, Ohm.

Thus, the impedance of the chain is calculated as: z \u003d (R 2 + (x C-x L) 2) ½, Ohm.

AC law

The classic law was opened by a physicist from Germany Simon Omom in 1862. Conducting experiments, it discovered the connection between the current and voltage. As a result, the scientist formulated the assertion that the current strength is proportional to the potential difference and inversely proportional to the resistance. If the current circuit decreases several times, then the voltage in it will become less than the same.

Mathematically, the Ohm law was described as:

therefore ohma law for alternating current is described by the formula:

I \u003d u / z, where:

I is the strength of alternating current, and;
U - the difference of potentials, in;
Z - Full chain resistance, Ohm.

The total resistance depends on the frequency of the harmonious signal and is calculated according to the following formula:

Z \u003d ((R + R) 2 + (W * L - 1 / W * C) 2) ½ \u003d ((R + R) 2 + x 2) ½.

When the current of the variable value is passed, the electromagnetic field does work, while due to the resistance rendered in the chain, heat is highlighted. That is, electrical energy goes to thermal. The power is proportional to the current and voltage. The formula describing the instantaneous value looks like: p \u003d i * u.

At the same time, for an alternating signal, it is necessary to take into account the amplitude and frequency component. Therefore:

P \u003d I * U * Cosw * T * COS (W * T + ψ), where i, u are amplitude values, and ψ is a phase shift.

For analyzing processes in electrical circuits of alternating current, the concept of a complex number is introduced. This is due to the displacement of the phases that appear between the current and the difference of potentials. It is indicated by the number of Latin letter J and consists of imaginary im and real Re parts.

Since there is a power transformation into heat on the active resistance, and on the reactive it is converted to the energy of the electromagnetic field, it is possible to transitions from any form to any. It can be written: z \u003d u / i \u003d z * e j * ψ.

From here the complete resistance of the chain: z \u003d r + j * x, where R and X are respectively active and reactive resistance. If the phase shift is taken equal to 90 0, then the complex number can not be taken into account.

Use of formula

The use of the Ohm law allows you to build the time characteristics of various elements. With it, it is easy to calculate loads for electrical circuits, select the desired cross-section of the wires, correctly select protective machines and fuses. Understanding the law makes it possible to apply the correct power source.

The use of the Ohm law can be applied in practice to solve the problem. For example, let there is an electrical line consisting of sequentially connected elements, such as: capacitance, inductance and resistor. In this case, the capacity C \u003d 2 * F, the inductance l \u003d 10 mg, and the resistance R \u003d 10 com. It is required to calculate the impedance of the full chain and calculate the current strength. In this case, the power supply operates at a frequency equal to f \u003d 200 Hz and gives a signal with an amplitude U \u003d 12 0 V. The internal resistance of the power supply is R \u003d 1 com.

Inductive resistance is located from the expression: xl \u003d 2 * p * f * L. on f \u003d 200 Hz and it leaves: x * l \u003d 1.25 ohms. Full resistance of the RLC chain will be: z \u003d ((10 * 10 3 + 1 * 10 3) 2 + (588-1.25) 2) ½ \u003d 11 com.

The potential difference that varies by the harmonic law of sinus will be determined by: u (t) \u003d u * sin (2 * p * f * t) \u003d 120 * sin (3.14 * T). The current will be equal to: I (t) \u003d 10 * 10 -3 + sin (3.14 * T + P / 2).

According to the calculated data, you can construct a current schedule corresponding to a frequency of 100 Hz. To do this, in the Cartesian coordinate system, the current dependence on time is displayed.

It should be noted, the OMA's law for an alternating signal differs from only the impedance and frequency of the signal only for the classical calculation. And it is important to consider them, since any radio component has both active and reactive resistance, which ultimately affects the work of the whole scheme, especially at high frequencies. Therefore, when designing electronic structures, in particular impulse devices, it is used for calculations that is the full law of Oma.

§nine. Communication between physical quantities. Physical theories

✓ What do they call the physical size?

✓ Give examples of the relationship of physical quantities.

1. As you know, physical values \u200b\u200bare used to describe physical phenomena and properties of bodies and substances.

Conducting experiments, scientists noticed that the values \u200b\u200bthat characterize the same phenomenon are mutually related.

For example, when the temperature changes, their volume and length change. They increase due to temperature rise and decrease with its decrease. The water temperature in the kettle during heating depends on the heating time.

2. To conclude that the relationship between values \u200b\u200bis not accidental, justice is checked for many similar phenomena.

If the links between the values \u200b\u200bcharacterizing the phenomenon are constantly manifested, they are called physical laws.

There are physical laws relating to the relationship of only certain physical phenomena. For example, there are laws that describe mechanical phenomena, or laws that are subject to thermal phenomena. In addition, there are more general laws fair for all physical phenomena. A combination of phenomena, which are described by law, is determined by the limits of their applicability.

Sure, physical Law Write in the formula.

3. The knowledge of the surrounding world would be incomplete if people only observed and described the phenomena, established laws. You need to be able to explain the phenomena of nature. A man studying in nature, always seeks to answer not only the question "What is happening?" But the question "Why is this happening?".

The answer to the question "Why is this or that phenomenon?" can be obtained using theoretical knowledge that are the basis of physical theory. So, mechanical phenomena, for example, the nature of the movement of vehicles or earth satellites, explain the theory called mechanics. Explained why the bodies are expanding when heated, why the spoon is heated, lowered into a glass with hot tea, gives the molecular-kinetic theory of the structure of the substance. There are theories explaining electrical, optical and magnetic phenomena.

Thus, physical phenomena - mechanical, thermal, electrical and others are explained by the corresponding physical theories. The theory contains general, systematic knowledge of physical phenomena.

The theory allows not only to explain why the phenomenon occurs, but also to predict its move.

Questions for self-test

1. What does physical law express?

3. What is the role of physical theory?

4. What phenomena explains the mechanics?

In this lesson, new concepts are considered in detail: "Mass of one subject", "number of objects", "the mass of all objects". It is concluded about the relationship of these concepts among themselves. Students are given the opportunity to practice in solving simple and composite tasks based on the knowledge gained.

We solve the tasks and learn how the concepts of "mass of one subject", "number of objects", "the mass of all subjects are connected with each other.

Read the first task.

Package mass with flour - 2 kg. Find out a lot of 4 such packages (Fig. 1).

Fig. 1. Illustration for the task

When solving the problem, we argue: 2 kg is the mass of one package, such packages - 4 pieces. We learn how much all packages weigh, the action of multiplication.

We write down the decision.

Answer: 8 kg weigh four packages.

We conclude: To find a lot of all objects, you need a mass of one subject to multiply by the number of items.

Let me read the second task.

Mass 4 identical packages with flour - 8 kg. Find out a lot of one package (Fig. 2).

Fig. 2. Illustration to the task

We submit data from the task to the table.

When solving the problem, we argue: 8 kg is the mass of all packages, such packages - 4 pieces. We learn how much one package weighs, the action of division.

We write down the decision.

Answer: 2 kg weighs one package.

We conclude: To find a lot of one subject, you need to share the mass of all items to the number of items.

Read the third task.

Mass of one package with flour - 2 kg. How many packages will need to decompose 8 kg equal in them (Fig. 3)?

Fig. 3. Illustration for the task

We submit data from the task to the table.

When solving the problem, we argue as follows: 8 kg is the mass of all packages, each package weighs 2 kg. Since all flour, 8 kg, laid out equally, two kilograms, we will find out how many packages will be required, the action of division.

We write down the decision.

Answer: 4 packages will be required.

We conclude: To find the number of items, you need to share the mass of all items on a mass of one subject.

Practice to correlate the text of the task with a brief record.

We will select a brief record to each task (Fig. 4).

Fig. 4. Illustration for the task

Consider the first task.

In 3 identical boxes of 6 kg of cookies. Sk kg weighs one biscuit box?

We will argue like that. This task suits a brief entry in Table 2. It contains the mass of all boxes - 6 kg, the number of boxes - 3. You need to know how much one biscuit box weighs. Recall the rule and learn the action of divisions.

Answer: 2 kg weighs one biscuit box.

Consider the second task.

Mass of one biscuit box - 2 kg. How many kg weigh 3 of the same biscuits?

We will argue like that. A brief entry is suitable for this task in Table 3. It shows the mass of one biscuit box - 2 kg, the number of boxes - 3. You need to know how much all biscuits are weighing. To find out, you need a mass of one box to multiply by the number of boxes.

Answer: 6 kg weigh three biscuits.

Consider the third task.

Mass of one biscuit box - 2 kg. How many boxes need to decompose 6 kg of biscuits equally?

We argue like that. This task suits a brief entry in Table 1. It shows the mass of one box - 2 kg, the mass of all boxes is 6 kg. You need to know the number of boxes to decompose the cookies. Recall that in order to find the number of boxes, we need a mass of all items to divide the mass of one subject.

Answer: 3 boxes will need.

Note that all three tasks that we decided were simple, since we could answer the question of the task by performing one action.

Knowing the relationship between the values \u200b\u200bof the "mass of one subject", the "number of objects", "the mass of all items" you can solve the composite tasks, that is, in 2, 3 actions.

Practice and solve the composite problem.

In 7 identical boxes of 21 kg of grapes. How many kg grapes in 4 of the same boxes?

We write these tasks in the table.

We will argue. To answer the question of the task, you need to multiply the mass of one box by the number of boxes. We will find a mass of one box: since 7 boxes weigh 21 kg, then in order to find a mass of one box, 21: 7 \u003d 3 (kg). Now we know how much one box weighs, we can find out how much 4 boxes weigh. For this we are 3 * 4 \u003d 12 (kg).

We write down the decision.

1. 21: 7 \u003d 3 (kg) - Mass of one box

2. 3 * 4 \u003d 12 (kg)

Answer: 12 kg grapes in 4 boxes

Today, at the lesson, we solved the tasks and learned how the magnitude of the "mass of one subject", the "number of objects", "the mass of all items" are connected with each other, learned to solve problems, applying these knowledge.

Bibliography

M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: In 2 parts, part 1. - M.: Enlightenment, 2012.
M.I. Moro, MA Bantova and others. Mathematics: Tutorial. Grade 3: in 2 parts, part 2. - M.: "Education", 2012.
M.I. Moro. Mathematics lessons: Guidelines for teacher. Grade 3. - M.: Enlightenment, 2012.
Regulatory document. Control and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
"School of Russia": programs for elementary school. - M.: "Enlightenment", 2011.
S.I. Volkov. Mathematics: test work. Grade 3. - M.: Enlightenment, 2012.
V.N. Rudnitskaya. Tests. - M.: Exam, 2012.

Nsportal.ru ().
Prosv.ru ().
Do.gendocs.ru ().

Homework

1. Finished phrases:

to find a lot of all items, you need ...;

to find a lot of one subject, you need ...;

to find the number of items, you need ....

2. Select a brief entry to the task and solve it.

In the three identical boxes of 18 kg of cherry. How many kg cherry in one box?

3. Solve the task.

In 4 identical boxes 28 kg of apples. How many kg of apples in 6 of the same boxes?

We transfer data to Gable. 6.2 on the schedule (Fig. 6.1). Having settled on the horizontal axis of labor, and on the vertical - the volume of release, you can build curves of cumulative, medium and limiting products. Graphically magnitude MR It is determined by the tangent angle of inclination tangent to the curve of the common product at a point corresponding to its specific volume, the value AR - Tangent angle of inclination of the beam, walking from the beginning of the coordinates to the same point.

When constructing a limit product curve, the corresponding values MR need to postpone in the middle of the segment L. (if a MR \u003d 39, then on the chart this value is postponed when L. = 2,5).

As follows from Table. 6.2 and graphs in Fig. 6.1, I and B, entering additional units of variable resource (in our case - labor) at a fixed capital value leads to a permanent increase in total product Tr. However, a more thorough analysis shows that this growth occurs unevenly: on the site (o - l t) of the increment of the DTR with the same increments of the DL increase (curve Tr. has a concave view), and with a further increase in the number of increment units applied ATR Reduced (curve GR becomes convex).

Fig. 6.1. Curves total (but), Middle and limit (b)

product

A similar change in the volume of production of goods and services, depending on the increase in the introduced units of the variable factor, reflects the effect of one of the fundamental laws of the economy - the law of the reduced return of the resource. According to this law the introduction of additional units of an alternating resource with a constant value of a permanent factor will certainly lead to a situation where each subsequent unit of the variable factor will begin to add less than its previous unit.

This is equivalent to the statement that with the above conditions above, the moment will come when a further increase in the units of the variable factor used will cause a decrease in the limiting product, so sometimes this law is called the law of an indispensable reduction in the limit product.

The general meaning of the law of the declining return is that the use of a permanent factor in the production of goods limits the increments of the volume of this product with a consistent increase in the number of units of the variable resource used.

How can I explain the action of the law of the decreasing return of the resource? With one fixed factor (capital), the introduction of additional units of the variable factor (labor) at the first pores (section OL () Allows you to effectively use the division of labor. This leads to the fact that each additional worker makes an increasing amount of goods and services, i.e. Growing a limit product. However, at some point, the next worker will be superfluous - all the possibilities of separation of labor are exhausted, and he will have to wait when the machine is free to apply his work. From this point on, the service of each subsequent worker will be increasingly useless, which will cause further reducing the limit product. Theoretically, the situation may occur when an additional worker will begin to interfere with production, and this will cause a decrease in production volumes. In this case, the values \u200b\u200bof the limit product will become negative and curve MR Absolute the axis of the abscissa, and the curve Tr. will decrease (hypothetically similar situation occurs at the point L 3. In fig. 6.1, but and b).

Of course, this law can be interpreted as a law of an indispensable reduction in the average product, since under similar conditions the moment necessarily comes when the further growth of the units of the variable factor will lead to a decrease in the average product.

Example 2. Suppose that 2 workers participate in the production of 42 units of goods, which produce on average, 21 units of goods per month, T.S. LR \u003d TP / L \u003d 42/2 \u003d 21. Let the firm hire another one, the third worker. If the recovery of an additional hired worker (i.e., an extreme product) is higher than each of the available workers, for example, 39 units, then the value of the average product, taking into account the hiring of three workers there will be more than 21 units:

This means that as long as MR\u003e Ar.. The magnitude of the limit product exceeds the value of the average product, the latter increases; At the same time on the graph (see Fig. 6.1) the curve of the limiting product is located above the average product curve. If MR and the limiting product curve passes below the medium product curve, then the value AR decreases. Consequently, curve MR Crossing curve AR At point where the curve AR Has a maximum.

Topic: Communication between values: speed, time and distance.

Objectives: 1. Description the knowledge of the connection between values \u200b\u200b(speed, time and distance) with the tasks.

2. Improve computational skills.

3. Wear student performance.

During the classes

1. Organizing time. Slide 1.

Today in the lesson we will repeat the relationship between speed, time and distance, consolidate the ability to solve problems on this topic. To be completely ready, let's make a massage for hands. (Spec. Ex.)

What are you ready? Slide 2.

To arouse the right thing

To fail to know failure in life,

We go to the hike without boldly -

In the world of mysteries and complex tasks!

2. Oral account.

Well, towards pencils.

Neither knuckles nor pens nor chalk

Verbal counting! We will create this business

Only the strength of mind and soul.

And now 5 challenges

Only a few minutes!

1) Ryabina chicken demolished the testicle, and the mouse took Irazbil. Then Ryaba demolished 3 more eggs. Mouse and these broke too. Ryaba drove up and demolished another 5 eggs, but the shameless mouse was chosen and these. From how many eggs could you cook my scrambled eggs and baba, if you did not break your mouse?

2) in the kitchen 39 flies. 6 flies drink tea from puddles on the table, 12 fly around the bulbs, the rest walk on the ceiling. How many flies go on foot on the ceiling?

3) The hunter shot at once in two hares and missed. The first hare weighed5 kilograms, and the second 2 times more. It is asked how many kilograms of the encounter ran away from the hunter?

4) 70 students learn at school. The remaining 430 students are lying fool. How many disciples in this school?

5) Firefighters teach to wear pants for 3 seconds. Consider how many pants will have time to put on a well-trained firefighter in 1 minute.

3. Fastening the studied material.

In which units is measured:

distance?

speed?

Finish offers. Slide 4.

How to find the distance at famous speed and time of movement?

How to find speed, knowing the distance and time of movement?

How to find traffic time, knowing distance and speed?

Guys, do you like to travel? What can travel on?

And where would you like to go?

Before people go to the road, they determine how much time they have, what path they are waiting for and at what speed they will move (type of transport, place).

Assek, you too will go on a little journey to the country "Tasks", but the type of transport you must choose (on the table card with tasks: elephant, horse, deer, camel, etc.) children solve the tasks obtained.

Fizminutka Game "Buratino". Slide 5.

And now we will go with you to Kyshtovka.

Who knows the distance from p. Bolshoi before s. Kyshtovka? (28)

We will go skiing and solve such a task. Slide 6.

Skiers (list the names of children from this class) went to Kyshtovka at a speed of 4 km / h. How long will they need to return? (7h)

What can you tell about number 7?

Slide 7. 7 minutes of life takes one repurchased cigarette.

Deciding all these tasks, we once again made sure that this topic Allowed by you well.

Independent work on options. Slide 8. (if the work is done correctly, then the guys make up proposal: Mathematics Mind in order leads!)

Game "Umbrella" (Questions for the time)

What is the result of subtraction?

How many ends of 3.5 sticks?