Resistance formula through force. Electrical resistance

One of the physical properties of a substance is the ability to conduct an electric current. Electrical conductivity (conductor resistance) depends on several factors: the length of the electrical circuit, structural features, the presence of free electrons, temperature, current, voltage, material and cross-sectional area.

The flow of electric current through the conductor leads to the directed movement of free electrons. The presence of free electrons depends on the substance itself and is taken from the table of D. I. Mendeleev, namely from the electronic configuration of the element. The electrons start to hit crystal lattice element and transfer energy to the latter. In this case, a thermal effect occurs when the current acts on the conductor.

During this interaction, they slow down, but then, under the influence of an electric field that accelerates them, they begin to move at the same speed. The electrons collide a huge number of times. This process is called conductor resistance.

Therefore, the electrical resistance of a conductor is considered to be a physical quantity that characterizes the ratio of voltage to current strength.

What is electrical resistance: a value that indicates the property of a physical body to convert electrical energy into thermal energy, due to the interaction of electron energy with the crystal lattice of a substance. By the nature of the conductivity are distinguished:

1. Conductors (capable of conducting electric current, as free electrons are present).
2. Semiconductors (can conduct electricity, but under certain conditions).
3. Dielectrics or insulators (have great resistance, no free electrons, making them unable to conduct current).

This characteristic is denoted by the letter R and measured in Ohms (Ohm). The use of these groups of substances is very significant for the development of electrical circuit diagrams of devices.

To fully understand the dependence of R on something, you need to pay special attention to the calculation of this value.

Calculation of electrical conductivity

To calculate the R of a conductor, Ohm's law is applied, which states that current (I) is directly proportional to voltage (U) and inversely proportional to resistance.

The formula for finding the conductivity characteristic of a material R (a consequence of Ohm's law for a circuit section): R = U / I.

For a complete section of the circuit, this formula takes the following form: R \u003d (U / I) - Rin, where Rin is the internal R of the power source.

The ability of a conductor to transmit electric current depends on many factors: voltage, current, length, cross-sectional area and material of the conductor, as well as on the ambient temperature.

In electrical engineering, for making calculations and manufacturing resistors, the geometric component of the conductor is also taken into account.

What resistance depends on: on the length of the conductor - l, specific resistance - p and on the cross-sectional area (with a radius r) - S \u003d Pi * r * r.

Formula R conductor: R = p * l / S.

From the formula it is clear what depends on conductor resistivity: R, l, S. There is no need to calculate it this way, because there is a much better way. Resistivity can be found in the relevant reference books for each type of conductor (p is a physical quantity equal to R of a material 1 meter long and with a cross-sectional area equal to 1 m².

However, this formula is not enough to accurately calculate the resistor, so temperature dependence is used.

Influence of ambient temperature

It has been proven that every substance has a resistivity that depends on temperature.

To demonstrate this, the following experiment can be performed. Take a spiral of nichrome or any conductor (indicated in the diagram as a resistor), a power source and a regular ammeter (it can be replaced with an incandescent lamp). Assemble the chain according to scheme 1.

Scheme 1 - Electrical circuit for the experiment

It is necessary to power the consumer and carefully monitor the readings of the ammeter. Next, heat R without turning it off, and the ammeter readings will begin to fall as the temperature rises. There is a dependence according to Ohm's law for the circuit section: I \u003d U / R. In this case, the internal resistance of the power source can be neglected: this will not affect the demonstration of the dependence of R on temperature. Hence it follows that the temperature dependence of R is present.

The physical meaning of the increase in the value of R is due to the influence of temperature on the amplitude of oscillations (increase) of ions in the crystal lattice. As a result, electrons collide more often and this causes R to increase.

According to the formula: R = p * l / S, we find an indicator that temperature dependent(S and l - do not depend on temperature). It remains p conductor. Based on this, the temperature dependence formula is obtained: (R - Ro) / R \u003d a * t, where Ro at a temperature of 0 degrees Celsius, t is the ambient temperature and a is the proportionality factor (temperature coefficient).

For metals, "a" is always greater than zero, and for electrolyte solutions, the temperature coefficient is less than 0.

The formula for finding p used in the calculations: p \u003d (1 + a * t) * po, where ro is the specific resistance value taken from the reference book for a particular conductor. In this case, the temperature coefficient can be considered constant. The dependence of power (P) on R follows from the power formula: P \u003d U * I \u003d U * U / R \u003d I * I * R. The specific resistance value also depends on the deformation of the material, at which the crystal lattice is broken.

When metal is processed in a cold environment at a certain pressure, plastic deformation occurs. In this case, the crystal lattice is distorted and R of the electron flow increases. In this case, the resistivity also increases. This process is reversible and is called recrystallization annealing, due to which some of the defects are reduced.

Under the action of tension and compression forces on the metal, the latter undergoes deformations, which are called elastic. Resistivity decreases during compression, as there is a decrease in the amplitude of thermal vibrations. directed charged particles it becomes easier to move. When stretched, the specific resistance increases due to an increase in the amplitude of thermal vibrations.

Another factor affecting conductivity is the type of current flowing through the conductor.

Resistance in AC networks behaves a little differently, because Ohm's law only applies to circuits with DC voltage. Therefore, calculations should be made differently.

Impedance is denoted by the letter Z and consists of the algebraic sum of active, capacitive and inductive resistances.

When active R is connected to an alternating current circuit, under the influence of a potential difference, a sinusoidal current begins to flow. In this case, the formula looks like: Im \u003d Um / R, where Im and Um are the amplitude values ​​of the current and voltage. The resistance formula takes the following form: Im = Um / ((1 + a * t) * po * l / 2 * Pi * r * r).

Capacitance (Xc) is due to the presence of capacitors in circuits. It should be noted that an alternating current passes through the capacitors and, therefore, it acts as a conductor with a capacitance.

Xc is calculated as follows: Xc = 1 / (w * C), where w is the angular frequency and C is the capacitance of a capacitor or group of capacitors. The angular frequency is defined as follows:

1. The AC frequency is measured (typically 50 Hz).
2. Multiplied by 6.283.

Inductive reactance (Xl) - implies the presence of inductance in the circuit (choke, relay, circuit, transformer, and so on). Calculated as follows: Xl = wL, where L is the inductance and w is the corner frequency. To calculate the inductance you need to use specialized online calculators or a reference book on physics. So, all quantities are calculated according to the formulas and it remains only to write down Z: Z * Z = R * R + (Xc - Xl) * (Xc - Xl).

To determine the final value, it is necessary to extract the square root of the expression: R * R + (Xc - Xl) * (Xc - Xl). It follows from the formulas that the frequency of alternating current plays a big role, for example, in a circuit of the same design, with increasing frequency, its Z also increases. It must be added that in circuits with alternating voltage Z depends on such indicators:

1. Conductor lengths.
2. Sectional areas - S.
3. Temperatures.
4. material type.
5. Capacities.
6. inductance.
7. Frequencies.

Therefore, Ohm's law for a section of the circuit has a completely different form: I=U/Z. The law for the complete chain also changes.

Resistance calculations require a certain amount of time, so special electrical measuring instruments called ohmmeters are used to measure their values. The measuring device consists of a pointer indicator, to which a power source is connected in series.

Measure R all combined appliances such as testers and multimeters. Separate instruments for measuring only this characteristic are used extremely rarely (megaohmmeter for checking the insulation of a power cable).

The device is used to test electrical circuits for damage and serviceability of radio components, as well as to test cable insulation.

When measuring R, it is necessary to completely de-energize the circuit section in order to avoid damage to the device. To do this, the following precautions must be taken:

In expensive multimeters, there is a continuity function, duplicated by an audible signal, so there is no need to look at the instrument panel.

Thus, electrical resistance plays an important role in electrical engineering. It depends in permanent circuits on temperature, current strength, length, type of material and area transverse conductor section. In AC circuits, this dependence is supplemented by such quantities as frequency, capacitance and inductance. Thanks to this dependence, it is possible to change the characteristics of electricity: voltage and current strength. Ohmmeters are used to measure the resistance value, which are also used to detect wiring problems, continuity of various circuits and radio components.

One of the main characteristics of an electrical circuit is the current strength. It is measured in amps and determines the load on the conductive wires, busbars or tracks of the boards. This value reflects the amount of electricity that has flowed in the conductor per unit of time. You can determine it in several ways, depending on the data you know. Accordingly, students and novice electricians because of this often encounter problems in solving educational tasks or practical situations. In this article, we will tell you how to find the current strength through power and voltage or resistance.

If power and voltage are known

Let's say you need to find the current in the circuit, while you only know the voltage and power consumption. Then, to determine it without resistance, use the formula:

After simple ones, we get a formula for calculations

It should be noted that this expression is valid for DC circuits. But in calculations, for example, for an electric motor, its full power or cosine Phi is taken into account. Then for a three-phase motor, it can be calculated as follows:

We find P taking into account the efficiency, usually it lies in the range of 0.75-0.88:

Р1 = Р2/η

Here P2 is the active useful power on the shaft, η - efficiency, both of these parameters are usually indicated on the nameplate.

We find the total power, taking into account cosФ (it is also indicated on the nameplate):

S = P1/cosφ

We determine the consumed current by the formula:

Inom = S/(1.73 U)

Here 1.73 is the root of 3 (used to calculate a three-phase circuit), U is the voltage, depends on the engine being turned on (triangle or star) and the number of volts in the network (220, 380, 660, etc.). Although in our country 380V is most common.

If voltage or power and resistance are known

But there are tasks when you know the voltage in the circuit section and the magnitude of the load, then to find the current strength without power, use it, with its help we calculate the current strength through resistance and voltage.

But sometimes it happens that you need to determine the current without voltage, that is, when you know only the power of the circuit and its resistance. In this case:

In this case, according to the same Ohm's law:

P=I 2 *R

So the calculation is carried out according to the formula:

I 2 =P/R

Or take the expression on the right side of the expression under the root:

I=(P/R) 1/2

If EMF, internal resistance and load are known

Student problems with a catch include cases when you are given the value of the EMF and the internal resistance of the power source. In this case, you can determine the current in the circuit using Ohm's law for the complete circuit:

I=E/(R+r)

Here E is the EMF, r is the internal resistance of the power source, R is the load.

Joule-Lenz law

Another task that can put even a more or less experienced student into a stupor is to determine the current strength if time, resistance and the amount of heat generated by the conductor are known. For this, let's remember.

Its formula looks like this:

Q=I 2 Rt

Then do the calculation like this:

I 2 \u003d QRt

Or enter the right side of the equation under the root:

I=(Q/Rt) 1/2

A few examples

As a conclusion, we propose to consolidate the information obtained on several examples of tasks in which it is necessary to find the current strength.

It is clear from the condition that two answers must be given for each of the connection options. Then, to find the current in series connection, first add the resistances of the circuit to get the total.

I=U/R=12/3=4 Amps

When two elements are connected in parallel, Rtotal can be calculated as follows:

Rtot=(R1*R2)/(R1+R2)=1*2/3=2/3=0.67

Then further calculations can be carried out as follows:

First of all, you need to find the R total of R2 and R3 connected in parallel, using the same formula that we used above.

- an electrical quantity that characterizes the property of a material to prevent the flow of electric current. Depending on the type of material, resistance can tend to zero - be minimal (mi/micro ohms - conductors, metals), or be very large (giga ohms - insulation, dielectrics). The reciprocal of electrical resistance is .

unit of measurement electrical resistance - Ohm. It is denoted by the letter R. The dependence of resistance on current and in a closed circuit is determined.

Ohmmeter- a device for direct measurement of circuit resistance. Depending on the range of the measured value, they are divided into gigaohmmeters (for large resistance - when measuring insulation), and into micro / milliohmmeters (for small resistances - when measuring transient resistance of contacts, motor windings, etc.).

There is a wide variety of ohmmeters by design from different manufacturers, from electromechanical to microelectronic. It is worth noting that a classic ohmmeter measures the active part of the resistance (the so-called ohms).

Any resistance (metal or semiconductor) in an AC circuit has an active and a reactive component. The sum of active and reactance is AC circuit impedance and is calculated by the formula:

where, Z is the total resistance of the AC circuit;

R is the active resistance of the AC circuit;

Xc is the capacitive reactance of the AC circuit;

(C is the capacitance, w is the angular velocity of the alternating current)

Xl is the inductive reactance of the AC circuit;

(L is the inductance, w is the angular velocity of the alternating current).

Active resistance- this is part of the impedance of the electrical circuit, the energy of which is completely converted into other types of energy (mechanical, chemical, thermal). A distinctive feature of the active component is the complete consumption of all electricity (energy is not returned to the network back to the network), and reactance returns part of the energy back to the network (a negative property of the reactive component).

The physical meaning of active resistance

Each medium where electric charges pass creates obstacles in their path (it is believed that these are the nodes of the crystal lattice), into which they seem to hit and lose their energy, which is released in the form of heat.

Thus, there is a drop (loss of electrical energy), part of which is lost due to the internal resistance of the conductive medium.

The numerical value characterizing the ability of a material to prevent the passage of charges is called resistance. It is measured in Ohms (Ohm) and is inversely proportional to the electrical conductivity.

Different elements of the periodic system of Mendeleev have different specific electrical resistances (p), for example, the smallest sp. silver (0.016 Ohm * mm2 / m), copper (0.0175 Ohm * mm2 / m), gold (0.023) and aluminum (0.029) have resistance. They are used in industry as the main materials on which all electrical engineering and energy are built. Dielectrics, on the other hand, have a high sp. resistance and used for insulation.

The resistance of a conducting medium can vary significantly depending on the cross section, temperature, magnitude and frequency of the current. In addition, different media have different charge carriers (free electrons in metals, ions in electrolytes, "holes" in semiconductors), which are the determining factors of resistance.

The physical meaning of reactance

In coils and capacitors, when applied, energy is accumulated in the form of magnetic and electric fields, which requires some time.

Magnetic fields in alternating current networks change following the changing direction of movement of charges, while providing additional resistance.

In addition, there is a stable phase shift and current strength, and this leads to additional losses of electricity.

Resistivity

How to find out the resistance of a material if it does not flow through it and we do not have an ohmmeter? There is a special value for this - electrical resistivity of the material in

(these are tabular values ​​that are determined empirically for most metals). With this value and the physical quantities of the material, we can calculate the resistance using the formula:

where, p- resistivity (units of measurement ohm * m / mm 2);

l is the length of the conductor (m);

S - cross section (mm 2).

Electricity itself is invisible, although that does not make it any less dangerous. Quite the opposite: that's why it's more dangerous. After all, if we saw him, as we see, for example, water pouring from a tap, then we would certainly have avoided a lot of trouble.

Water. Here it is, the water pipe, and here is the closed faucet. Nothing flows, nothing drips. But we know for sure: inside the water. And if the system is working properly, then this water is under pressure there. 2, 3 atmospheres, or how much is there? No matter. But the pressure is there, otherwise the system would not work. Somewhere pumps are buzzing, driving water into the system, creating this very pressure.

And here is our electrical wire. Somewhere far away, at the other end, generators are also buzzing, generating electricity. And there is also pressure in the wire from this ... No, no, not pressure, of course, here in this wire voltage. It is also measured, but in its own units: in volts.

Water presses against the walls in pipes, without moving anywhere, waiting for a way out to rush there in a powerful stream. And the voltage silently waits in the wire when the switch closes, so that the electron flows move to fulfill their purpose.

And then the faucet opened, a stream of water flowed. It flows throughout the pipe, moving from the pump to the outlet valve. And as soon as the switch contacts closed, electrons flowed in the wires. What is this movement? it current. Electrons flow. And this movement, this current also has its own unit of measurement: ampere.

And there's more resistance. For water, this is, figuratively speaking, the size of the hole in the outlet valve. The larger the hole, the less resistance to water movement. In wires, almost the same: the greater the resistance of the wire, the less current.

Here, something like this, if you figuratively imagine the main characteristics of electricity. And from the point of view of science, everything is strict: there is the so-called Ohm's law. It reads as follows: I = U/R.
I- current strength. Measured in amperes.
U- voltage. Measured in volts.
R- resistance. Measured in ohms.

There is one more concept - power, W. It is also simple with it: W=U*I. Measured in watts.

Actually, this is all the necessary and sufficient theory for us. From these four units of measurement, in accordance with the above two formulas, a number of others can be derived:

You say: - Why do I need all this? Formulas, numbers... I'm not going to do calculations.

And I will answer this way: - Re-read the previous article. How can you be sure without knowing the simplest truths and calculations? Although, in fact, in everyday practical terms, only formula 7 is most interesting, where the current strength is determined at known voltage and power. As a rule, these 2 values ​​are known, and the result (current strength) is certainly necessary to determine the allowable wire cross-section and to select protection.

There is another circumstance that should be mentioned in the context of this article. In the electric power industry, the so-called "alternating" current is used. That is, those same electrons do not always move in the same direction in the wires, they constantly change it: forward-backward-forward-backward... And this change of direction is 100 times per second.

Wait, but it says everywhere that the frequency is 50 hertz! Yes, that's exactly what it is. Frequency is measured in cycles per second, but in each cycle, the current reverses twice. In other words, in one period there are two vertices that characterize the maximum value of the current (positive and negative), and it is in these vertices that the direction changes.

We will not go into details more deeply, but still: why exactly alternating current, and not direct current?

The whole problem is the transmission of electricity over long distances. This is where Ohm's inexorable law comes into play. At heavy loads, if the voltage is 220 volts, the current strength can be very large. To transmit electricity with such a current, wires of a very large cross section will be required.

There is only one way out: to raise the tension. The seventh formula says: I=W/U. It is quite obvious that if we apply a voltage of not 220 volts, but 220 thousand volts, then the current strength will decrease by a thousand times. And this means that the cross section of the wires can be taken much less.

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Among other indicators characterizing the electrical circuit, the conductor, it is worth highlighting the electrical resistance. It determines the ability of the atoms of a material to prevent the directed passage of electrons. Assistance in determining this value can be provided both by a specialized device - an ohmmeter, and mathematical calculations based on knowledge of the relationship between quantities and the physical properties of the material. The indicator is measured in Ohms (Ohm), the symbol is R.

Ohm's law - a mathematical approach to determining resistance

The ratio established by Georg Ohm defines the relationship between voltage, current, resistance, based on the mathematical relationship of concepts. The validity of the linear relationship - R \u003d U / I (ratio of voltage to current strength) - is not observed in all cases.
Unit [R] = B/A = Ohm. 1 ohm is the resistance of a material carrying a current of 1 ampere at a voltage of 1 volt.

Empirical formula for calculating resistance

Objective data on the conductivity of a material follow from its physical characteristics, which determine both its own properties and reactions to external influences. Based on this, the conductivity depends on:

• size.
• Geometry.
• Temperatures.

Atoms of a conducting material collide with directed electrons, preventing their further advancement. At a high concentration of the latter, the atoms are not able to resist them and the conductivity is high. Large resistance values ​​are typical for dielectrics, which are characterized by almost zero conductivity.

One of the defining characteristics of each conductor is its resistivity - ρ. It determines the dependence of resistance on the conductor material and external influences. This is a fixed (within the same material) value, which represents the data of the conductor of the following dimensions - length 1 m (ℓ), cross-sectional area 1 sq.m. Therefore, the relationship between these quantities is expressed by the relation: R = ρ* ℓ/S:

• The conductivity of a material decreases as its length increases.
• An increase in the cross-sectional area of ​​the conductor entails a decrease in its resistance. This pattern is due to a decrease in the density of electrons, and, consequently, the contact of material particles with them becomes more rare.
• An increase in the temperature of the material stimulates an increase in resistance, while a decrease in temperature causes it to decrease.

It is advisable to calculate the cross-sectional area according to the formula S \u003d πd 2 / 4. A tape measure will help in determining the length.

Relationship with power (P)

Based on the formula of Ohm's law, U = I*R and P = I*U. Therefore, P = I 2 *R and P = U 2 /R.
Knowing the magnitude of the current strength and power, the resistance can be determined as: R \u003d P / I 2.
Knowing the magnitude of voltage and power, the resistance is easy to calculate by the formula: R \u003d U 2 /P.

The resistance of the material and the values ​​of other related characteristics can be obtained using special measuring instruments or based on established mathematical patterns.