Goals: educational: systematize students’ knowledge and skills in solving problems and calculating equivalent resistances using models, frames, etc.

Developmental: development of logical thinking skills, abstract thinking, skills to replace equivalence schemes, simplify the calculation of schemes.

Educational: fostering a sense of responsibility, independence, and the need for skills acquired in class in the future

Equipment: wire frame of a cube, tetrahedron, mesh of an endless chain of resistance.

DURING THE CLASSES

Update:

1. Teacher: “Let’s remember the series connection of resistances.”

Students draw a diagram on the board.

and write down

U rev =U 1 +U 2

Y rev =Y 1 =Y 2

Teacher: remember the parallel connection of resistances.

A student sketches a basic diagram on the board:

Y rev =Y 1 =Y 2

; for for n equal

Teacher: Now we will solve problems on calculating the equivalent resistance of a section of the circuit presented in the form geometric figure, or metal mesh.

Task No. 1

A wire frame in the form of a cube, the edges of which represent equal resistances R. Calculate the equivalent resistance between points A and B. To calculate the equivalent resistance of a given frame, it is necessary to replace it with an equivalent circuit. Points 1, 2, 3 have the same potential, they can be connected into one node. And points (vertices) of the cube 4, 5, 6 can be connected into another node for the same reason. Students have such a model on each desk. After completing the described steps, draw an equivalent circuit.

In the AC section the equivalent resistance is ; on CD; on DB; and finally for the series connection of resistances we have:

By the same principle, the potentials of points A and 6 are equal, B and 3 are equal. Students combine these points on their model and get an equivalent diagram:

Calculating the equivalent resistance of such a circuit is simple

Task No. 3

The same model of a cube, with inclusion in the circuit between points 2 and B. Students connect points with equal potentials 1 and 3; 6 and 4. Then the diagram will look like this:

Points 1,3 and 6,4 have equal potentials, and no current will flow through the resistances between these points and the circuit is simplified to the form; the equivalent resistance of which is calculated as follows:

Problem No. 4

Equilateral triangular pyramid, the edge of which has a resistance R. Calculate the equivalent resistance when connected to the circuit.

Points 3 and 4 have equal potential, so no current will flow along edge 3.4. The students clean it up.

Then the diagram will look like this:

The equivalent resistance is calculated as follows:

Problem No. 5

Metal mesh with link resistance equal to R. Calculate the equivalent resistance between points 1 and 2.

At point 0 you can separate the links, then the diagram will look like:

- the resistance of one half is symmetrical at 1-2 points. There is a similar branch parallel to it, so

Problem No. 6

The star consists of 5 equilateral triangles, the resistance of each .

Are you so familiar with Ohm’s law (connections of conductors)? // Quantum. - 2012. - No. 1. - P. 32-33.

By special agreement with the editorial board and editors of the journal "Kvant"

The currents continue indefinitely at a constant rate, ... but they always stop the moment the circuit is broken.
Andre Ampere
The transfer of electricity between two nearby elements, otherwise equal conditions is proportional to the difference in electroscopic forces in these elements.
Georg Ohm
If given a system n conductors that are arbitrarily connected to each other, and an arbitrary electromotive force is applied to each conductor, then the required number linear equations to determine the currents flowing through conductors can be obtained using... two theorems.
Gustav Kirchhoff
...by translating the essential features of real circuit elements into the language of idealizations, it is possible to analyze an electrical circuit relatively simply.
Richard Feynman

Our first encounters with electrical circuits occur when we plug in household appliances at home or come across intricacies of wiring under the cover of some electronic device, or when we notice power lines on high supports and thick wires along which the current collectors of electric trains, trolleybuses and trams slide. Later we draw diagrams at school, perform simple experiments and learn about the laws of electrical, primarily direct, current, flowing - how could it be otherwise! - by wire.

But at the same time, we use mobile phones, wireless local networks, “stick ourselves in the air” to connect to the Internet, and increasingly hear that wireless transmission of not only information, but also electricity is just around the corner. How archaic then will all these bulky circuits, wires, terminals, rheostats and the laws describing them seem!

Take your time. Firstly, no matter what we transmit - signals or energy, there are emitters and receivers that will not operate without currents flowing through the conductors stuffed into them. Secondly, not everything can be miniaturized, for example, transport or power plants. Therefore, we will have to deal with electrical networks, and therefore with connections of conductors of various types, for a long time. We will continue this topic in the next issue of Kaleidoscope, at the end of which we will place a general list of “Quantum” publications on the topic “Ohm’s Law”.

Questions and tasks

1. Why can birds perch safely on high voltage wires?

2. A garland is assembled from series-connected light bulbs for a flashlight, designed to be connected to a 220 V network. Each light bulb has a voltage of only about 3 V, but if you unscrew one of the light bulbs from the socket and put your finger in it, it will “jerk” strongly. . Why?

3. The battery is closed by three conductors of equal length connected in series. Figure 1 shows a graph showing the voltage drop across them. Which conductor has the highest and which has the least resistance?

4. Calculate the total resistance of the circuit shown in Figure 2 if R= 1 Ohm.

5. Five conductors of equal resistance were connected so that under the influence total voltage 5 V, the current in the circuit turned out to be 1 A. Determine the resistance of one conductor. Does the problem have a single solution?

6. From identical resistors with a resistance of 10 Ohms, you need to create a circuit with a resistance of 6 Ohms. Which least amount Will you need resistors for this? Draw a diagram of the circuit.

7. Give an example of a circuit that is not a combination of series and parallel connections.

8. How will the resistance of a circuit consisting of five identical conductors change? r each, if we add two more of the same conductors, as shown by the dashed lines in Figure 3?

9. What is the resistance R of each of two identical resistors (Fig. 4), if the voltmeter has a resistance R V= 3 kOhm when switched on according to schemes a) and b) shows the same voltage? The voltage in the circuit is the same in both cases.

10. An electrical circuit consisting of resistors with resistances R 1, R 2 and R 3 is connected to two sources DC voltage U 1 and U 2 , as shown in Figure 5. Under what conditions will the current through a resistor with resistance R 1 be zero?

11. Find the resistance of the “star” (Fig. 6) between points A and B, if the resistance of each link is equal r.

12. A hollow cube was soldered from thin homogeneous sheets of tin, and conductors were soldered to the two opposite vertices of the large diagonal, as shown in Figure 7. The resistance of the cube between these conductors turned out to be 7 Ohms. Find the strength of the electric current crossing edge AB of the cube if the cube is connected to a 42 V source.

13. Determine the currents in each side of the cell shown in Figure 8, the total current from node A to node B, and the total resistance between these nodes. Each side of the cell has a resistance r, and the current flowing along the indicated side is equal to i.

14. Two jumpers CE and DF were soldered into an electrical circuit consisting of six identical resistors with resistance R, as shown in Figure 9. What was the resistance between terminals A and B?

15. The galvanic element is closed into two parallel conductors with resistances R 1 and R 2. Will the currents in these conductors decrease if their resistance is increased?

Microexperience

How can you determine the length of insulated copper wire rolled into a large coil without unwinding it?

It's interesting that...

Ohm's experiments, which seem trivial today, are remarkable in that they marked the beginning of clarifying the root causes electrical phenomena, which remained for a little less than two hundred years very vague and devoid of any experimental justification.

Not being familiar with Ohm's law, the French physicist Pouille, through experimentation, came to similar conclusions in 1837. Having learned that the law had been discovered a decade ago, Pouille set about thoroughly checking it. The law was confirmed with high accuracy, and a “by-product” was the study of Ohm’s law by French schoolchildren until the 20th century under the name of Pouillet’s law.

... when deducing his law, Ohm introduced the concepts of “resistance”, “current strength”, “voltage drop” and “conductivity”. Along with Ampere, who introduced the terms “electric circuit” and “electric current” and determined the direction of current in a closed circuit, Ohm laid the foundation for further electrodynamic research on the path to the practical use of electricity.

...in 1843, the English physicist Charles Wheatstone, using Ohm's law, invented a method for measuring resistance, now known as the Wheatstone bridge.

...the identity of the “electroscopic forces” included in the formulation of Ohm’s law with electric potentials was proven by Kirchhoff. Somewhat earlier, he had established the laws of current distribution in branched circuits, and later constructed general theory movement of current in conductors, assuming the existence in them of two equal counter flows of positive and negative electricity.

...the intensive development of electrical measurement methods in the 19th century was facilitated by technical demands: the creation of overhead telegraph lines, the laying of underground cables, the transmission of electric current through uninsulated overhead wires and, finally, the construction of an underwater transatlantic telegraph. The theorist of the last project was the outstanding English physicist William Thomson (Lord Kelvin).

…some practical problems economics and logistics - such as, for example, the search for the minimum cost distribution of goods, found their solution when modeling transport flows using electrical networks.

Questions and tasks

1. The resistance of the bird’s body is much greater than the resistance of the section of wire parallel to it between its legs, therefore the current strength in the bird’s body is small and harmless.

2. The finger has a very high resistance compared to the resistance of the light bulb. When it is “turned on” in series with the light bulbs, the same current flows through the finger and the light bulbs, so the voltage drop across the finger will be significantly greater than the voltage drop across the light bulbs, i.e. Almost all the mains voltage will be applied to the finger.

3. Conductor 3 has the highest resistance, conductor 2 has the least.

4. Rtot = R = 1 Ohm.

5. When five conductors are connected in series, the resistance of each conductor is R = 1 Ohm. Another solution is possible: the conductors are connected in parallel to each other into 2 groups, one of which has 3 conductors, the other - 2, and these groups are connected to each other in series. Then R = 6 Ohm.

6. Four resistors; see fig. 10.

7. Figure 11 shows the so-called bridge circuit, when currents flow through all resistors.

Transcript

1 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 Part I. Calculation of resistance Ohm's Law. Resistance. Serial and parallel connection. symmetrical circuits. Bridges. Star-delta conversion. Jumper chains. Infinite chains and meshes.. Determine the equivalent resistance of the wire structures shown in the figure. The resistance of each link of the structure, i.e. wires between nodes, regardless of length, are equal. a) b) c) d) e) f) f) 2. N points are connected to each other by identical conductors with resistance each. Determine the equivalent circuit resistance between two adjacent points. 3. In a Wheatstone bridge, the resistances are selected in such a way that the sensitive galvanometer shows zero. a) Assuming resistances 2 and r are known, determine the value of resistance rx. b) if you swap the battery and galvanometer, you will again get a bridge circuit. Will the balance be maintained in the new scheme? 4. Find the equivalent resistance of the circuit section. a) 2 b) 2 c) Determine the equivalent resistance of the circuit section containing jumpers with negligible resistance. a) b) The electrical circuit is made up of seven series-connected resistors = com, 2 = 2 com, 3 = 3 com, 4 = 4 com, 5 = 5 com, 6 = 6 com, 7 = 7 com and four jumpers. A voltage of U = 53.2 V is applied to the input. Indicate the resistors through which the minimum and maximum currents flow, and determine the values ​​of these currents.

2 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 7. A circuit consisting of three resistors and four identical jumpers (the two lower ones are connected in parallel) is connected to a source with a voltage U = 0 V. Assuming known = 3 Ohms, determine the current strength in jumper B. The resistance of the jumpers is much less than the resistance of the resistors. U 2 V 8. The cube is assembled from identical resistors having resistances. The two resistors are replaced with ideal jumpers as shown in the picture. Find the total resistance of the resulting system between contacts and B. Which of the remaining resistors can be removed without changing the total resistance of the system? If it is known that most of the resistors in the circuit carry a current I = 2, what is the total current entering the system at the node? What current flows through an ideal jumper `? ` K M C L V V ` 9. Determine the resistance of the wire mesh between the indicated terminals. The frame marked with a thick line has negligible resistance. The resistance of each of the remaining links of the grid is equal. 0. Determine the equivalent resistance of the semi-infinite resistor chains shown in the figure. 2 2 a) b) c) Determine the equivalent resistance of an infinitely branching chain consisting of resistors with resistance. 2. The endless mesh with square cells is made of wire. The resistance of each grid edge is equal. Figure C shows the middle of edge B. It is known that when an ohmmeter is connected between points and B, it shows a resistance of /2. What resistance will the ohmmeter connected between points and C show? 3. Determine the resistance of infinite flat meshes with the resistance of one side of the cell, measured between nodes and B. a) b) c) B B 4. Determine the resistance of an infinite volumetric cubic mesh with the resistance of one side of the cell, measured between adjacent nodes and B.

3 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 5. A hollow metal ball has a radius r = 0 cm and a wall thickness d = mm. It is made of copper, with the exception of a strip at the “equator” with a width of a = 2 mm, which is made of aluminum. When a voltage U = 0, mV was applied to the “poles” of the ball, a current I = 5.2 passed through it. The experiment was repeated with another ball, which had an iron strip instead of an aluminum strip. What current will flow through this ball? The resistivity of aluminum is 0.03 Ohm mm 2 /m, iron is 0.0 Ohm mm 2 /m. 6. A ring of radius r = 0 cm is made of wire with a cross section S = 5 mm 2. The material of the wire is non-uniform and its resistivity depends on the angle φ as shown in the graph. The resistance between all possible pairs of points on the ring is measured with an ohmmeter. What is the maximum resistance that can be obtained from such measurements?

4 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 Current and voltage measurements. ammeter, voltmeter and ohmmeter. Part II.Measuring instruments. Determine the unknown parameters of the electrical circuit. (The devices are considered ideal). U 0 2 a) U 0 = 24 B = 2 I -? Uv-? b) 2 U I 4 = = 2 2 = 3 3 = 2 4 = 20 5 = 0 U-? I 6 =? 2. Determine the ammeter reading in the circuit shown in the figure. Source voltage U =.5 V, resistance of each resistor = com. 3. In the section of the circuit, the diagram of which is shown in the figure, resistors with resistances = 6 Ohms, 2 = 3 Ohms, 3 = 5 Ohms, 4 = 8 Ohms are included. The readings of the first ammeter I = 0. Find the reading of the second ammeter. 4. Using known instrument readings, determine the unknown ones. The resistance of ammeters is considered to be much less than the resistance of resistors. The devices are the same. 6 a) b) c) B 5 B 5 2 3B d) e) B How will the readings of ideal instruments change when the rheostat/potentiometer slider is moved in the direction indicated by the arrow or when the key is opened? 3 a) b) c) d) Ɛ,r Ɛ,r 6. The circuit is assembled from a number of different resistors, a rheostat, an ideal battery, a voltmeter and an ammeter. The rheostat slider is moved, slightly increasing its resistance. In what direction will the voltmeter and ammeter readings change?

5 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 7. Determine the readings of a voltmeter connected between two nodes of a fragment of an electrical circuit if the readings of ammeters and 3 are equal to I = 2, I3 = 9, respectively, and the resistance of the resistors = 0 Ohm. I2 I3 I 8. An electric circuit in the shape of a tetrahedron contains four identical resistors, an ideal constant voltage source and an ideal ammeter that shows the current I = 2. If you replace the ammeter with an ideal voltmeter, it will show a voltage U = 2 V. Determine the voltage U0 of the source and the resistance of one resistor. 9. An electrical circuit is a grid consisting of identical links having the same resistance. One of the links has been replaced with an ideal voltmeter. The voltage U0 = 9.7 V is applied to the circuit. Find the voltmeter reading. U0 0. An electrical circuit is a grid consisting of identical links having the same resistance. One of the links has been replaced with an ideal voltmeter. The voltage U0 = 73 V is applied to the circuit. Find the voltmeter reading. U0. The experimenter assembled the circuit shown in the figure from several identical resistors and identical voltmeters. What will be the sum of the readings of all voltmeters if a voltage of U = 6 V is applied to contacts B? The resistance of voltmeters is much greater than the resistance of resistors. 2. A section of the circuit consists of unknown resistances. How, given a source, an ideal ammeter and voltmeter, connecting wires with zero resistance, measure the resistance connected to points A and B without breaking a single contact in the circuit? W C K V N D C L E D F G 3. A physics expert assembled a circuit of three identical resistors, connected it to a constant voltage source (which can be considered ideal) and measured the voltage with a voltmeter, first between points and D, and then between points and B. He got U = 3 V and U2 = 0.9 V respectively. Then the physics expert connected points and C with a wire (the resistance of which can be neglected) and measured the voltage between points B and D. What did he get? 4. The circuit shown in the figure contains 50 different ammeters and 50 identical voltmeters. The readings of the first voltmeter are U = 9.6 V, the first ammeter I = 9.5 m, the second ammeter I2 = 9.2 m. Using these data, determine the sum of the readings of all voltmeters. 5. If only the first voltmeter is connected to the battery, then it shows 4 V. If only the second one is connected, then it shows 4.5 V. If both of these voltmeters are connected in series to the battery, then together they show 5 V. What will be the readings of these two voltmeters if they are connected to the same battery in parallel? 2 B

6 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 6. An electrical circuit consists of two identical voltmeters and two ammeters. Their readings are U = 0 V, U2 = 20 V, I = 50 m, I2 = 70 m, respectively. Determine the resistance of the resistor, obtaining the answer in general form. 7. The electrical circuit consists of a battery, six resistors, the resistance values ​​of which = Ohm, 2 = 2 Ohm, 3 = 3 Ohm, 4 = 4 Ohm, and three identical ammeters, the internal resistance r of which is small. Calculate the readings of the ammeters if the battery voltage is U = 99 V. 8. Find the readings of the same voltmeters. The resistance of voltmeters is much greater than the resistance of resistors = 0 Ohm. Input voltage U = 4.5 V. 9. An ammeter and a voltmeter are connected in series to a battery with an emf Ɛ = 9 V and an unknown internal resistance. The resistances of the devices are unknown. If a resistance is connected in parallel to the voltmeter (its value is also unknown), then the ammeter reading is doubled, and the voltmeter reading is halved. What was the voltmeter reading after connecting the resistance? 20. Determine the readings of the same ohmmeters in the circuits presented in the figure. The resistance of each of the resistors in the circuits is equal. a) b) c) 2. An electrical circuit is a grid consisting of identical links having the same resistance. Two of the links were replaced with identical ohmmeters. Find the ohmmeter readings. 22. Determine the sum of the ohmmeter readings in the circuit shown in the figure. Ɛ,r Ɛ 2,r The circuit shown in the figure was assembled from identical ohmmeters. One of the devices shows resistance = 2000 Ohms. Determine the sum of the readings of the two remaining ohmmeters. 24. Draw a graph of the readings of the right ohmmeter depending on the resistance of the rheostat, which can vary from 0 to 2. Own resistance of the ohmmeter. The ohmmeters are considered the same. 0-2

7 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 Part III. Voltage sources. Nonlinear elements Joule-Lenz law. Voltage sources. Electromotive force of a current source. Ohm's law for complete chain. Connections of current sources. Nonlinear elements.. The circuit shown in the figure is assembled from identical light bulbs and connected to a voltage source. Arrange the light bulbs in ascending order of brightness. 2. A circuit of four resistors is connected to an adjustable voltage source, as shown in the figure. The meter shows a current of 2.5. Two resistors produce 50 W of power, and the other two 200 W. Switch K is closed, and the source voltage is changed so that the ammeter again shows 2.5. What power will be released in the resistors after this? 3. A chain of two series-connected resistors is connected to a constant voltage source U = 2 V. Resistance of one of them = 4 Ohms. At what value of resistance 2 of the second resistor will the thermal power released on it be maximum? Find this maximum power. 4. There are identical resistors, shaped like a regular cylinder. Side surface Each resistor is well thermally insulated, and when it is heated, heat transfer occurs only through the ends. One of the resistors was connected to an ideal battery. At the same time, it heated up to a temperature of t = 38 C. Then three such resistors were connected in series to this battery, tightly aligning their ends and ensuring good electrical contact. To what temperature will the resistors heat up? Room temperature t0 = 20 C. Heat transfer power is proportional to the temperature difference between the resistor and environment. The resistance of resistors does not change when heated. 5. A cylindrical conductor of radius r consists of two homogeneous sections with resistivity ρ and ρ2 and a non-uniform section of length L connecting them. What thermal power is released in the non-uniform section if the voltage per unit length of the conductor with resistivity ρ is equal to u and L the resistivity of the inhomogeneous section varies linearly from ρ to ρ2? 6. A resistor is connected to an ideal current source. The source voltage is equal to U. It turned out that the temperature of the resistor T depends on time t as T = T0 + αt (T0 and α are known constants). The resistor has mass m and is made of a substance with specific heat c. What is the thermal power emitted by the resistor to the environment? 7. The resistance of the resistor increases linearly with temperature, and the heat transfer power from its surface is directly proportional to the temperature difference between the resistor and the environment. If a very small current is passed through a resistor, its resistance is 0. When the amount of current flowing through the resistor approaches I0, the resistor quickly heats up and melts. What voltage will be across the resistor if current I0/2 is passed through it? 8. A current source is connected to a resistor whose resistance depends on temperature according to the law (t) = 0 (+ αt), where t is the temperature in C, α and 0 are unknown coefficients. After some time, the source is disconnected from the resistor. A graph of resistor temperature versus time is shown in the figure. The heat transfer power of the resistor into the environment is proportional to the temperature difference between the resistor and the environment: P = βt, where β is an unknown coefficient. Assuming that the temperature of the resistor is the same at all its points, find α.

8 The drawing cannot be displayed now. Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 9. Find the EMF and internal resistance of the equivalent source (Ɛе = φ φb) Ɛ Ɛ 2 a) b) c) 2r r B 2 r B Ɛ Ɛ r d) e) r f) 2Ɛ B B Ɛ Ɛ r Ɛ 2 Ɛ r B Ɛ r 2Ɛ Ɛ B r 0. There is a circuit containing N = 000 identical current sources with emf Ɛ and internal resistance r each. Between points ub (on arc NE) there are m current sources. Find the potential difference between the points and B. What will this potential difference be if the elements face each other with like poles? The experimenter assembled an electrical circuit consisting of different batteries with negligible internal resistance and identical fuses, the resistance of which is also very small, and drew its diagram (the fuses in the diagram are indicated by black rectangles). The experimenter remembers that during the experiment that day all the fuses remained intact. The voltages of some batteries are known. Restore unknown values stress. 2. The figure shows the idealized current-voltage characteristics of the diode and resistor. Plot the current-voltage characteristic of a section of a circuit containing a diode and a resistor connected: a) in parallel; b) sequentially. I0 0 I U0 2U0 D U 3. The figure shows the idealized current-voltage characteristics of the diode and resistor. Plot the current-voltage characteristic of a circuit section containing a diode and two resistors. -0.4-0.2 3.0 2.0.0 0 -.0 I, D 0.2 0.4 0.6 U, V a) b) 4. The figure shows the current-voltage characteristics of the resistor and the circuit section , consisting of a resistor and a nonlinear element connected: a) in series; b) in parallel. -0.4-0.2 3.0 2.0.0 0 I, Σ 0.2 0.4 0.6 U, V Plot the current-voltage characteristic of the nonlinear element. -.0 0.5 5. Determine through which nonlinear element the larger current will flow, 2 0.4 if it is connected to a source with U0 = 0.5 V and r = Ohm. 3 0.3 0.2 I, 0, 0 0, 0.2 0.3 0.4 0.5 0.6 U,V

9 Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207 6. Find the amount of current flowing through the diode in the circuit shown in the figure. The ideal source voltage U and resistance are known. 4 U 2 7. A nonlinear element x is included in one of the arms of the bridge, for which the dependence of the current strength Ix on the applied voltage Ux is given by the formula: ix = Ux 3, where = 0.25 / V 3. Find the power Nx released on the nonlinear element under the conditions when there is no current through the galvanometer G. The resistances of the remaining arms of the bridge = 2 Ohms, 2 = 4 Ohms and 3 = Ohms. 8. When a light bulb with a power dissipation of W = 60 W was inserted into a table lamp, it turned out that the power of W2 = 0 mW was dissipated on the connecting wires of the lamp. What power will be dissipated on the connecting wires if you install a light bulb with power W3 = 00 W? The voltage in the network in both cases is considered equal to U = 220 V. 9. The resistance of element X varies depending on the voltage on it. If voltage U< Uкр, то сопротивление равно, а при U >Ucr resistance is equal to 2. The circuit shown in the figure is assembled from three elements X. Find the dependence of the current through the circuit on the voltage across it. 20. The voltage of a source connected to a circuit consisting of identical resistors with resistance = Ohm and a nonlinear element can be changed. The dependence of the ammeter readings on the source voltage is shown on the graph. The positive direction of the current is specified in the electrical circuit diagram. Restore the current-voltage characteristic of the nonlinear element from these data. 2. The electrical circuit, the diagram of which is shown in the figure, contains three identical resistors = 2 = 3 = and three identical diodes D, D2, D3. The current-voltage characteristic of the diode is presented in the graph. Determine the current through the ammeter I depending on the voltage UВ between points and V. The ammeter is ideal. Construct a graph of I versus UB, indicating the values ​​of current and voltage at characteristic points. 22. You have an unlimited number of resistors of arbitrary resistance and diodes at your disposal. Diodes pass current only in one direction, and the voltage drop across them is equal to V (see Fig. a). What kind of circuit needs to be assembled so that it has such a dependence of current on voltage, as shown in Fig. b? Try to use as few elements as possible. Test. D.C. at 0 Ohm (on one of the parallel connected resistors) 2. 3/30

10 3. 0/ m 5. 4 m Talent and Success Foundation. Educational center "Sirius". Direction "Science". Prelsky physical change. 207

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For development creativity Students are interested in solving DC resistor circuits using the equipotential node method. The solution to these problems is accompanied by a sequential transformation of the original circuit. Moreover, it undergoes the greatest change after the first step, when it is used this method. Further transformations involve equivalent replacement of series or parallel resistors.

To transform a circuit, they use the property that in any circuit points with the same potentials can be connected into nodes. And vice versa: the nodes of the circuit can be divided if after this the potentials of the points included in the node do not change.

IN methodological literature often written like this: if a circuit contains conductors with equal resistances located symmetrically relative to any axis or plane of symmetry, then the points of these conductors, symmetrical relative to this axis or plane, have the same potential. But the whole difficulty is that no one indicates such an axis or plane on the diagram and it is not easy to find it.

I propose another, simplified way to solve such problems.

Problem 1. A wire cube (Fig. 1) is included in the circuit between the points A to B.

Find its total resistance if the resistance of each edge is equal R.

Place the cube on its edge AB(Fig. 2) and “cut” it into twoparallel halves plane AA 1 B 1 B, passing through the lower and upper edge.

Let's look at the right half of the cube. Let's take into account that the lower and upper ribs split in half and became 2 times thinner, and their resistance increased 2 times and became 2 times R(Fig. 3).

1) Find resistanceR 1three upper conductors connected in series:

4) Find the total resistance of this half of the cube (Fig. 6):

Find the total resistance of the cube:

It turned out to be relatively simple, understandable and accessible to everyone.

Problem 2. The wire cube is connected to the circuit not by an edge, but by a diagonal AC any edge. Find its total resistance if the resistance of each edge is equal R (Fig. 7).

Place the cube on edge AB again. “Saw” the cube into twoparallel halvesthe same vertical plane (see Fig. 2).

Again we look at the right half of the wire cube. We take into account that the upper and lower ribs split in half and their resistances became 2 each R.

Taking into account the conditions of the problem, we have the following connection (Fig. 8).

Let's consider a classic problem. Given a cube, the edges of which represent conductors with some identical resistance. This cube is included in an electrical circuit between all its possible points. Question: what is it equal to? cube resistance in each of these cases? In this article, a physics and mathematics tutor talks about how this classic problem is solved. There is also a video tutorial in which you will find not only a detailed explanation of the solution to the problem, but also a real physical demonstration confirming all the calculations.

So, the cube can be connected to the circuit in three different ways.

Resistance of a cube between opposite vertices

In this case, the current, having reached the point A, is distributed between three edges of the cube. Moreover, since all three edges are equivalent in terms of symmetry, no edge can be given more or less “significance”. Therefore, the current between these edges must be distributed equally. That is, the current strength in each edge is equal to:

The result is that the voltage drop across each of these three edges is the same and is equal to , where is the resistance of each edge. But the voltage drop between two points is equal to the potential difference between these points. That is, the potentials of the points C, D And E are the same and equal. For symmetry reasons, the point potentials F, G And K are also the same.

Points with the same potential can be connected by conductors. This will not change anything, because no current will flow through these conductors anyway:

As a result, we find that the edges A.C., AD And A.E. T. Likewise the ribs FB, G.B. And K.B. connect at one point. Let's call it a point M. As for the remaining 6 edges, all their “beginnings” will be connected at the point T, and all ends are at the point M. As a result, we get the following equivalent circuit:

Resistance of a cube between opposite corners of one face

In this case, the equivalent edges are AD And A.C.. The same current will flow through them. Moreover, equivalent are also KE And KF. The same current will flow through them. Let us repeat once again that the current between equivalent edges must be distributed equally, otherwise the symmetry will be broken:

Thus, in this case the points have the same potential C And D, as well as points E And F. This means that these points can be combined. Let the points C And D unite at a point M, and the points E And F- at the point T. Then we get the following equivalent circuit:

On a vertical section (directly between the points T And M) no current flows. Indeed, the situation is similar to a balanced measuring bridge. This means that this link can be excluded from the chain. After this, calculating the total resistance is not difficult:

The resistance of the upper link is equal to , the resistance of the lower link is . Then the total resistance is:

Resistance of a cube between adjacent vertices of the same face

This is the last possible option for connecting the cube to an electrical circuit. In this case, the equivalent edges through which the same current will flow are the edges A.C. And AD. And, accordingly, points will have identical potentials C And D, as well as points symmetric to them E And F:

We again connect points with equal potentials in pairs. We can do this because no current will flow between these points, even if we connect them with a conductor. Let the points C And D unite into a point T, and the points E And F- exactly M. Then we can draw the following equivalent circuit:

The total resistance of the resulting circuit is calculated using standard methods. We replace each segment of two parallel-connected resistors with a resistor with resistance . Then the resistance of the “upper” segment, consisting of series-connected resistors , and , is equal to .

This segment is connected to the “middle” segment, consisting of one resistor with a resistance of , in parallel. The resistance of a circuit consisting of two parallel-connected resistors with resistance and is equal to:

That is, the scheme is simplified to an even simpler form:

As you can see, the resistance of the “upper” U-shaped segment is equal to:

Well, the total resistance of two parallel connected resistors is equal to:

Experiment to measure the resistance of a cube

To show that all this is not a mathematical trick and that there is real physics behind all these calculations, I decided to conduct a direct physical experiment to measure the resistance of a cube. You can watch this experiment in the video at the beginning of the article. Here I will post photos of the experimental setup.

Especially for this experiment, I soldered a cube whose edges were identical resistors. I also have a multimeter that I turned on in resistance mode. The resistance of a single resistor is 38.3 kOhm: