Calculation of resistance along length and section. Resistivity formula. Influence of neighboring conductors

We know what's the reason electrical resistance conductor is the interaction of electrons with ions of the metal crystal lattice (§ 43). Therefore, it can be assumed that the resistance of a conductor depends on its length and cross-sectional area, as well as on the substance from which it is made.

Figure 74 shows the setup for conducting such an experiment. Various conductors are included in the current source circuit in turn, for example:

1. nickel wires of the same thickness, but different lengths;
2. nickel wires of the same length, but different thicknesses (different cross-sectional areas);
3. nickel and nichrome wires of the same length and thickness.

The current in the circuit is measured with an ammeter, and the voltage with a voltmeter.

Knowing the voltage at the ends of the conductor and the current in it, using Ohm's law, you can determine the resistance of each of the conductors.

Fig. 74. Dependence of conductor resistance on its size and type of substance

After performing these experiments, we will establish that:

1. of two nickel wires of the same thickness, the longer wire has greater resistance;
2. of two nickelin wires of the same length, the wire with a smaller cross-section has the greater resistance;
3. Nickel and nichrome wires of the same size have different resistances.

Ohm was the first to study experimentally the dependence of the resistance of a conductor on its size and the substance from which the conductor is made. He found that resistance is directly proportional to the length of the conductor, inversely proportional to its cross-sectional area and depends on the substance of the conductor.

How to take into account the dependence of resistance on the material from which the conductor is made? To do this, calculate the so-called resistivity of a substance.

Specific resistance is a physical quantity that determines the resistance of a conductor made of a given substance with a length of 1 m and a cross-sectional area of ​​1 m 2.

Let's introduce letter designations: ρ is the resistivity of the conductor, I is the length of the conductor, S is its cross-sectional area. Then the conductor resistance R will be expressed by the formula

From it we get that:

From the last formula you can determine the unit of resistivity. Since the unit of resistance is 1 ohm, the unit of cross-sectional area is 1 m2, and the unit of length is 1 m, then the unit of resistivity is:

It is more convenient to express the cross-sectional area of ​​the conductor in square millimeters, since it is most often small. Then the unit of resistivity will be:

Table 8 shows the resistivity values ​​of some substances at 20 °C. Specific resistance changes with temperature. It has been experimentally established that for metals, for example, the resistivity increases with increasing temperature.

Table 8. Electrical resistivity of some substances (at t = 20 °C)

Of all the metals, silver and copper have the lowest resistivity. Therefore, silver and copper are the best conductors of electricity.

When wiring electrical circuits, aluminum, copper and iron wires are used.

In many cases, devices with high resistance are needed. They are made from specially created alloys - substances with high resistivity. For example, as can be seen from Table 8, the nichrome alloy has a resistivity almost 40 times greater than aluminum.

Porcelain and ebonite have such a high resistivity that they almost do not conduct electric current at all; they are used as insulators.

Questions

1. How does the resistance of a conductor depend on its length and cross-sectional area?
2. How to experimentally show the dependence of the resistance of a conductor on its length, cross-sectional area and the substance from which it is made?
3. What is the resistivity of a conductor?
4. What formula can be used to calculate the resistance of conductors?
5. What units is the resistivity of a conductor expressed in?
6. What substances are conductors used in practice made from?

The lesson reveals in detail the previously announced parameters of the conductor, on which its resistance depends. It turns out that to calculate the resistance of a conductor, its length, cross-sectional area and the material from which it is made are important. The concept of resistivity of a conductor is introduced, which characterizes the substance of the conductor.

Subject:Electromagnetic phenomena

Lesson: Calculation of conductor resistance. Resistivity

In previous lessons, we already raised the question of how electrical resistance affects the current strength in a circuit, but we did not discuss what specific factors the resistance of a conductor depends on. In today's lesson we will learn about the parameters of a conductor that determine its resistance, and we will learn how Georg Ohm investigated the resistance of conductors in his experiments.

To obtain the dependence of the current in the circuit on the resistance, Ohm had to conduct a huge number of experiments in which it was necessary to change the resistance of the conductor. In this regard, he was faced with the problem of studying the resistance of a conductor depending on its individual parameters. First of all, Georg Ohm drew attention to the dependence of the resistance of a conductor on its length, which was already discussed in passing in previous lessons. He concluded that as the length of the conductor increases, its resistance also increases in direct proportion. In addition, it was found that the resistance is also influenced by the cross-section of the conductor, i.e., the area of ​​​​the figure that is obtained from a transverse section. Moreover, the larger the cross-sectional area, the lower the resistance. From this we can conclude that the thicker the wire, the lower its resistance. All these facts were obtained experimentally.

In addition to geometric parameters, the resistance of a conductor is also influenced by a quantity that describes the type of substance of which the conductor is composed. In his experiments, Om used conductors made of various materials. When using copper wires, the resistance was one way, silver - another, iron - another, etc. The value that characterizes the type of substance in this case is called resistivity.

Thus, we can obtain the following dependences for the conductor resistance (Fig. 1):

1. Resistance is directly proportional to the length of the conductor, which is measured in m in SI;

2. Resistance is inversely proportional to the cross-sectional area of ​​the conductor, which we will measure in mm 2 due to its smallness;

3. Resistance depends on the specific resistance of the substance (read “rho”), which is a tabular value and is usually measured in .

Rice. 1. Explorer

As an example, here is a table of the resistivity values ​​of some metals, which were obtained experimentally:

Resistivity,

It is worth noting that among good guides, which are metals, the best are precious metals, while silver is considered the most the best guide, because it has the smallest low resistivity. This explains the use of precious metals when soldering particularly important elements in electrical engineering. From the resistivity values ​​of substances, one can draw conclusions about their practical application- substances with high resistivity are suitable for the manufacture of insulating materials, and those with low resistivity are suitable for conductors.

Comment. In many tables, resistivity is measured in , which is related to the SI measurement of area in m2.

Physical meaning of resistivity- resistance of a conductor with a length of 1 m and a cross-sectional area of ​​1 mm 2.

The formula for calculating the electrical resistance of a conductor, based on the above considerations, is as follows:

If you pay attention to this formula, you can conclude that it expresses the resistivity of the conductor, i.e., by determining the current and voltage on the conductor and measuring its length with cross-sectional area, you can use Ohm’s law and the specified formula to calculate resistivity. Then, its value can be compared with the data in the table and determine what substance the conductor is made of.

All parameters that affect the resistance of conductors must be taken into account when designing complex electrical circuits, such as power lines, for example. In such projects, it is important to balance the ratios of lengths, cross-sections and materials of conductors to effectively compensate for the thermal effect of current.

The next lesson will look at the design and operating principle of a device called a rheostat, the main characteristic of which is resistance.

Bibliography

1. Gendenshtein L.E., Kaidalov A.B., Kozhevnikov V.B. Physics 8 / Ed. Orlova V.A., Roizena I.I. - M.: Mnemosyne.
2. Peryshkin A.V. Physics 8. - M.: Bustard, 2010.
3. Fadeeva A.A., Zasov A.V., Kiselev D.F. Physics 8. - M.: Enlightenment.

1. Internet portal Exir.ru ().
2. Cool physics ().

Homework

1. Page 103-106: questions No. 1-6. Peryshkin A.V. Physics 8. - M.: Bustard, 2010.
2. The length and cross-sectional area of ​​aluminum and iron wires are the same. Which conductor has the greater resistance?
3. What is the resistance of a copper wire 10 m long and with a cross-sectional area of ​​0.17 mm 2?
4. Which of the solid iron rods of different diameters has the greater electrical resistance? The masses of the rods are the same.

Knowing the cause of electrical resistance, we can conclude that resistance depends on the dimensions of the conductor (length and thickness) and on the material from which it is made. Experience confirms this conclusion.

Figure 262 shows a setup for conducting such an experiment. The current source circuit is turned on in turn various conductors, for example:

• nickel wires of the same thickness, but different lengths;
• nickel wires of the same length, but different thicknesses (different cross-sectional areas);
• nickel and nichrome wires of the same length and thickness.

The current in the circuit is measured with an ammeter, and the voltage with a voltmeter.

Knowing the voltage at the ends of the conductor and the current in it, using Ohm's law, you can determine the resistance of each of the conductors.

Ohm was the first to study the dependence of the resistance of a conductor on its size and material through experiments. He found that resistance directly proportional to the length of the conductor, is inversely proportional to its cross-sectional area and depends on the material of the conductor.

The resistance of a conductor 1 m long with a cross-sectional area of ​​1 m2 is called resistivity. Let us introduce the letter designations: p - resistivity, I - length and S - cross-sectional area of ​​the conductor. Then resistance conductor R will be expressed by the formula:

From this formula you can determine the unit of resistivity:

units p = units R * units S/ units l

Since units R = 1 Ohm, units. S = 1 m2, units. l = 1 m, then by unit

1 Ohm * 1 m2/1 m, or 1 Ohm * m

It is more convenient to express the cross-sectional area of ​​the conductor in square millimeters, since it is most often small. Then by one resistivity will be:

1 Ohm *mm2/m

We will use this unit in the future.

Table 13 shows the resistivity values ​​of some substances at 20° C. (The temperature is indicated because conductor resistance with change temperature changes.)

Of all the metals, silver and copper have the lowest resistivity. Therefore, silver and copper are the best conductors of electricity.

When wiring electrical circuits, aluminum, copper and iron wires are used.

Questions. 1. How does the resistance of a conductor depend on its length and cross-sectional area? 2. How to show experimentally the dependence of the resistance of a conductor on its length, cross-sectional area and material? 3. What is the specific resistance of a conductor? 4. What formula can be used to calculate the resistance of conductors? 5. In what units is the resistivity of a conductor measured? 6. Which of the metals given in Table 13 have the lowest resistivity? 7. What material are conductors used in practice made of?

Content:

When designing electrical networks In apartments or private houses, it is mandatory to calculate the cross-section of wires and cables. To carry out calculations, indicators such as the value of power consumption and the current strength that will flow through the network are used. Resistance is not taken into account due to the short length of cable lines. However, this indicator is necessary for long power lines and voltage drops in different areas. The resistance of the copper wire is of particular importance. Such wires are increasingly used in modern networks, so their physical properties must be taken into account when designing.

Concepts and meaning of resistance

The electrical resistance of materials is widely used and taken into account in electrical engineering. This value allows you to set the basic parameters of wires and cables, especially with a hidden method of laying them. First of all, the exact length of the laid line and the material used to produce the wire are established. Having calculated the initial data, it is quite possible to measure the cable.

Compared to conventional electrical wiring, resistance parameters are of critical importance in electronics. It is considered and compared in conjunction with other indicators present in the electronic circuits. In these cases, incorrectly selected wire resistance can cause a malfunction of all elements of the system. This can happen if you use a wire that is too thin to connect to the computer's power supply. There will be a slight decrease in voltage in the conductor, which will cause the computer to operate incorrectly.

The resistance in a copper wire depends on many factors, and primarily on the physical properties of the material itself. In addition, the diameter or cross-section of the conductor is taken into account, determined by a formula or a special table.

Table

The resistance of a copper conductor is influenced by several additional physical quantities. First of all, the ambient temperature must be taken into account. Everyone knows that as the temperature of a conductor increases, its resistance increases. At the same time, the current decreases due to the inversely proportional dependence of both quantities. This primarily applies to metals with a positive temperature coefficient. An example of a negative coefficient is tungsten alloy used in incandescent lamps. In this alloy, the current strength does not decrease even at very high temperatures.

How to calculate resistance

There are several methods for calculating the resistance of a copper wire. The simplest is the tabular version, which shows interrelated parameters. Therefore, in addition to resistance, the current strength, diameter or cross-section of the wire is determined.

In the second case, various ones are used. A set of physical quantities of copper wire is inserted into each of them, with the help of which accurate results are obtained. Most of these calculators use 0.0172 Ohm*mm 2 /m. In some cases, such an average may affect the accuracy of the calculations.

The most difficult option is considered to be manual calculations using the formula: R = p x L/S, in which p is the resistivity of copper, L is the length of the conductor and S is the cross-section of this conductor. It should be noted that the table defines the resistance of copper wire as one of the lowest. Only silver has a lower value.

Any body through which electric current flows exhibits a certain resistance to it. The property of a conductor material to prevent passage through it electric current called electrical resistance.

The greater the resistance of a conductor, the worse it conducts electric current, and, conversely, the lower the resistance of the conductor, the easier it is for electric current to pass through this conductor.

The resistance of various conductors depends on the material from which they are made. To characterize the electrical resistance of various materials, the concept of so-called resistivity has been introduced.

Specific resistance is the resistance of a conductor with a length of 1 m and a cross-sectional area of ​​1 mm2. Resistivity is denoted by the letter p (rho) of the Greek alphabet. Each material from which a conductor is made has its own resistivity.

For example, the resistivity of copper is 0.0175, i.e. a copper conductor with a length of 1 m and a cross-section of 1 mm2 has a resistance of 0.0175 ohms. The resistivity of aluminum is 0.029, the resistivity of iron is 0.135, the resistivity of constantan is 0.48, and the resistivity of nichrome is 1-1.1.

The resistance of a conductor is directly proportional to its length, i.e. the longer the conductor, the greater its electrical resistance.

The resistance of a conductor is inversely proportional to its cross-sectional area, i.e. the thicker the conductor, the lower its resistance, and, conversely, the thinner the conductor, the greater its resistance.

The conductor resistance can be determined by the formula:

where r is the conductor resistance in (Ohm); ρ—conductor resistivity (Ohm*m); l is the length of the conductor in (m); S - conductor cross-section in (mm2).

Example: Determine the resistance of 200 m of copper wire with a cross section of 1.5 mm2.

Example: Determine the resistance of 200 m of copper wire with a cross section of 2.5 mm2.

Insulation

Insulation in electrical engineering is a design element of equipment that prevents the passage of electric current through it, for example, to protect people.

Materials with dielectric properties are used for insulation: glass, ceramics, numerous polymers, mica. There is also air insulation, in which air plays the role of an insulator, and structural elements fix the spatial configuration of the insulated conductors so as to provide the necessary air gaps.

Insulating covers can be produced:

• made of electrical insulating rubber;
• made of polyethylene;
• made of cross-linked and foamed polyethylene;
• from silicone rubber;
• made of polyvinyl chloride plastic compound (PVC);
• made of impregnated cable paper;
• made of polytetrafluoroethylene.

Rubber insulation

Rubber insulation can only be used with a rubber hose sheath (if available). Since rubber made from natural rubber is quite expensive, almost all rubber used in the cable industry is artificial. Add to rubber:

• vulcanizing agents (elements that allow the transformation of linear bonds in rubber into spatial bonds in insulation, for example, sulfur);
• vulcanization accelerators (reduce time consumption);
• fillers (reduce the price of the material without significantly reducing technical characteristics);
• softeners (increase plastic properties);
• antioxidants (added to shells for resistance to solar radiation);
• dyes (to give the desired color).

Rubber allows you to assign large bending radii to cable products, therefore, together with a stranded core, it is used in conductors for movable connections (cables of the KG, KGESH brand, RPSh wire).
Specialization: used in general industrial cables for mobile connection of consumers.

Positive properties:

• low cost of artificial rubber;
• good flexibility;
• high electrical insulation characteristics (6 times higher than the value for PVC plastic);
• practically does not absorb water vapor from the air.

Negative qualities:

• decrease in electrical resistance when the temperature rises to +80°C;
• exposure to solar radiation (light oxidation) followed by characteristic cracking of the surface layer (in the absence of a shell);
• it is necessary to introduce special substances into the composition to obtain a certain chemical resistance;
• spreads the fire.

Read also:

Calculation of wire resistance. Online calculator.
Dependence of resistance on conductor material, length, diameter or cross-section. Calculation of cross-sectional area of ​​wires depending on load power.

At first glance, it may seem that this article is from the “Notes for Electricians” section.
On the one hand, why not, on the other hand, we, inquisitive electronics engineers, sometimes need to calculate the resistance of the winding of an inductor, or a homemade nichrome resistor, and, let’s be honest, an acoustic cable for high-quality sound-reproducing equipment.

The formula here is quite simple R = p*l/S, where l and S are the length and cross-sectional area of ​​the conductor, respectively, and p is the resistivity of the material, so these calculations can be carried out independently, armed with a calculator and the A minor thought that all collected data must be lead to the SI system.

Well, for normal guys who decided to save their time and not get nervous over trifles, we’ll draw a simple table.

TABLE FOR CALCULATING CONDUCTOR RESISTANCE

The page turned out to be lonely, so I’ll put a table here for those who want to connect their time with laying electrical wiring, connect a powerful source of energy consumption, or just look into the eyes of the electrician Vasily and, “sipping from the pot,” ask a fair question: “Why, exactly? Maybe decided to ruin me? Why do I need four squares of oxygen-free copper for two light bulbs and a refrigerator? Why?”

And we will make these calculations not freely and not even in accordance with folk wisdom, which says that “the required cross-sectional area of ​​the wire is equal to the maximum current divided by 10,” but in strict accordance regulatory documents Ministry of Energy of Russia on the rules for the installation of electrical installations.
These rules ignore wires with a cross-section smaller than 1.5 mm2. I will also ignore them, and the aluminum ones too, due to their blatant archaic nature.
So.

Electrical resistance and conductivity

CALCULATION OF THE SECTIONAL AREA OF WIRE DEPENDING ON THE LOAD POWER

Losses in conductors arise due to the non-zero value of their resistance, which depends on the length of the wire.
The power values ​​of these losses released in the form of heat into the surrounding space are given in the table.
As a result, the voltage reaches the energy consumer at the other end of the wire in a slightly reduced form - less than it was at the source. The table shows that, for example, with a network voltage of 220 V and a 100-meter wire length with a cross-section of 1.5 mm2, the voltage at a load consuming 4 kW will be not 220, but 199 V.
Is it good or bad?
For some devices it doesn’t matter, some will work, but at reduced power, and some will kick up and send you to a hairdryer along with your long wires and smart tables.
Therefore, the Ministry of Energy is the Ministry of Energy, and one’s own head will not hurt under any circumstances. If the situation develops in a similar way, there is a direct path to choosing wires with a larger cross-section.

The current strength in a conductor is directly proportional to the voltage across it.

Wire resistance.

This means that as the voltage increases, the current also increases. However, with the same voltage, but using different conductors, the current strength is different. You can say it differently. If you increase the voltage, then although the current strength will increase, it will be different everywhere, depending on the properties of the conductor.

The current versus voltage relationship for that particular conductor represents the resistance of that conductor. It is denoted by R and is found by the formula R = U/I. That is, resistance is defined as the ratio of voltage to current. The greater the current in a conductor at a given voltage, the lower its resistance. The greater the voltage for a given current, the greater the resistance of the conductor.

The formula can be rewritten in relation to current strength: I = U/R (Ohm's law). In this case, it is clearer that the greater the resistance, the less the current.

We can say that resistance prevents the voltage from creating a large current.

Resistance itself is a characteristic of the conductor. It does not depend on the voltage applied to it. If a large voltage is applied, the current will change, but the U/I ratio will not change, i.e. the resistance will not change.

What does the resistance of a conductor depend on? It's the envy of

• conductor length,
• its cross-sectional area,
• the substance from which the conductor is made,
• temperature.

To connect a substance and its resistance, the concept of specific resistance of a substance is introduced. It shows what the resistance will be in a given substance if a conductor made from it has a length of 1 m and a cross-sectional area of ​​1 m2. Conductors of the same length and thickness, made from different substances, will have different resistances. This is due to the fact that each metal (most often they are conductors) has its own crystal lattice, its own number of free electrons.

The lower the resistivity of a substance, the better conductor of electric current it is. For example, silver, copper, aluminum have low resistivity; much more for iron, tungsten; very large for various alloys.

The longer the conductor, the greater the resistance it has. This becomes clear if we take into account that the movement of electrons in metals is hindered by the ions that make up the crystal lattice. The more of them, i.e., the longer the conductor, the greater the chance for the electron to slow down its path.

However, increasing the cross-sectional area makes the road wider. It is easier for electrons to flow and not collide with the nodes of the crystal lattice. Therefore, the thicker the conductor, the lower its resistance.

Thus, the resistance is directly proportional to the resistivity (ρ) and length (l) of the conductor and inversely proportional to the area (S) of its cross-section. We get the resistance formula:

At first glance, this formula does not reflect the dependence of the resistance of the conductor on its temperature. However, the resistivity of a substance is measured at a certain temperature (usually 20 °C). Therefore, temperature is taken into account. For calculations, resistivities are taken from special tables.

For metal conductors, the higher the temperature, the greater the resistance. This is due to the fact that as the temperature increases, the lattice ions begin to vibrate more strongly and interfere more with the movement of electrons. However, in electrolytes (solutions where the charge is carried by ions rather than electrons), the resistance decreases with increasing temperature. Here this is due to the fact that the higher the temperature, the more dissociation into ions occurs, and they move faster in the solution.