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1 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. Part I. Calculation of resistance Ohm's law. Resistance. Serial and parallel connection. symmetrical circuits. Bridges. Star-delta conversion. Jumper chains. Infinite chains and grids. Determine the equivalent resistance of the wire structures shown in the figure. The resistance of each link in the structure, i.e. the wires between the nodes, regardless of the length, are equal. a) b) c) d) e) f) g) 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 the Wheatstone bridge, the resistances are selected in such a way that the sensitive galvanometer shows zero. a) Considering the resistances, 2 and r known, determine the resistance value rx. b) if you swap the battery and the galvanometer, then again you 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 = ohm, 2 = 2 ohm, 3 = 3 ohm, 4 = 4 ohm, 5 = 5 ohm, 6 = 6 ohm, 7 = 7 ohm and four jumpers. The voltage 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. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 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 of U = 0 V. Assuming that = 3 Ohm is known, determine the current in the 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 with resistances. The two resistors are replaced with ideal jumpers, as shown in the figure. Find the total resistance of the resulting system between the pins 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 are carrying current I = 2, what is the total current going into the system at the node? What current flows through the ideal jumper `? ` K M C L B B` 9. Determine the resistance of the wire mesh between the indicated terminals. The frame marked with a thick line has a negligible resistance. The resistance of each of the other links of the grid is equal. 0. Determine the equivalent resistance of the semi-infinite chains of resistors 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 square mesh endless mesh is made of wire. The resistance of each mesh edge is equal. In figure C, the middle of edge B. It is known that when an ohmmeter is connected between points and B, it shows resistance /2. What resistance will show an ohmmeter connected between the points and C? 3. Determine the resistance of infinite flat grids with the resistance of one side of the cell, measured between nodes and B. a) b) c) B B C 4. Determine the resistance of an infinite volumetric cubic grid with the resistance of one side of the cell, measured between neighboring nodes and B.
3 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 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 the strip on the "equator" with a width a = 2 mm, which is made of aluminum. When voltage U = 0, mV was applied to the "poles" of the ball, a current I = 5.2 went through it. The experiment was repeated with another ball, which had an iron strip instead of an aluminum strip. What current will go through this ball? Resistivity aluminum 0.03 Ohm mm 2 /m, iron 0.0 Ohm mm 2 /m. 6. A ring of radius r = 0 cm is made of wire with a cross section of S = 5 mm 2. The material of the wire is inhomogeneous and its resistivity depends on the angle φ as shown in the graph. The resistance between all possible pairs of points of the ring is measured with an ohmmeter. What is the maximum resistance that can be obtained with such measurements?
4 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. Current and voltage measurements. mmeter, voltmeter and ohmmeter. Part II. Measuring instruments. Determine the unknown parameters of the electrical circuit. (Instruments are considered ideal). U 0 2 a) U 0 \u003d 24 B \u003d 2 I -? U v-? 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 \u003d.5 V, resistance of each resistor \u003d com. 3. In the circuit section, the diagram of which is shown in the figure, resistors with resistances \u003d 6 Ohm, 2 \u003d 3 Ohm, 3 \u003d 5 Ohm, 4 \u003d 8 Ohm are included. Readings of the first ammeter I = 0,. Find the second ammeter reading. 4. Based on the known readings of the instruments, determine the unknown ones. Consider the resistance of ammeters to be much smaller than the resistance of resistors. The devices are the same. 6 a) b) c) C 5 B 5 2 3C d) e) C How will the readings of ideal instruments change when the slider of the rheostat/potentiometer 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, ideal batteries, a voltmeter and an ammeter. The rheostat slider is shifted, slightly increasing its resistance. In what direction will the readings of the voltmeter and ammeter change?
5 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 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. I 2 I 3 I the resistance of one resistor. 9. An electric circuit is a grid consisting of identical links having the same resistance. One of the links has been replaced with an ideal voltmeter. 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. 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 voltage U \u003d 6 V is applied to contacts B. The resistance of the voltmeters is much greater than the resistance of the resistors. 2. The section of the circuit consists of unknown resistances. How, having a source, an ideal ammeter and voltmeter, connecting wires with zero resistance, how to measure the resistance connected to points and B without breaking a single contact in the circuit? W C K B N D C L E D F G and U2 = 0.9 V, respectively. Then the physics expert connected the 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 U \u003d 9.6 V, the first ammeter I \u003d 9.5 m, the second ammeter I2 \u003d 9.2 m. Based on 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? 2B
6 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 6. The electrical circuit consists of two identical voltmeters, and two ammeters. Their indications are U = 0 V, U2 = 20 V, I = 50 m, I2 = 70 m, respectively. Determine the resistance of the resistor by getting the answer in general view. 7. The electric circuit consists of a battery, six resistors, whose resistance values \u003d Ohm, 2 \u003d 2 Ohm, 3 \u003d 3 Ohm, 4 \u003d 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 identical voltmeters. The resistances of voltmeters are much greater than the resistances of resistors = 0 ohms. 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 resistance of the devices is 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 identical ohmmeters in the circuits shown 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 are replaced with identical ohmmeters. Look for 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 instruments shows resistance = 2000 ohms. Determine the sum of the readings of the two remaining ohmmeters. 24. Construct 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. Ohmmeters are considered the same. 0-2
7 Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. Part III. Voltage sources. Nonlinear elements Joule-Lenz law. Voltage sources. Electromotive force of the current source. Ohm's law for a complete circuit. 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 bulbs in ascending order of brightness. 2. A circuit of four resistors is connected to the regulated voltage source, as shown in the figure. The meter shows a current of 2.5. Two resistors are allocated power 50 watts, the other two 200 watts. Key K is closed, and the source voltage is changed so that the ammeter again shows 2.5. What power will be released on the resistors after that? 3. A chain of two series-connected resistors is connected to a constant voltage source U = 2 V. The resistance of one of them is 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 that have the shape of a regular cylinder. Side surface each resistor is well 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 \u003d 38 C. Then three such resistors were connected in series to this battery, tightly aligning their ends and ensuring good electrical contact. How hot are the resistors? Room temperature t0 = 20 C. The heat transfer power is proportional to the temperature difference of 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 specific resistances ρ and ρ2 and an inhomogeneous section of length L connecting them. What heat power is released in an inhomogeneous section if the voltage per unit length of the conductor with specific resistance ρ is u and L resistivity of the inhomogeneous section varies linearly from ρ to ρ2? 6. A resistance 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 the 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 capacity c. What is the thermal power given off by the resistor to the environment? 7. The resistance of a 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 the 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. To the resistor, the resistance of which depends on temperature according to the law (t) \u003d 0 (+ αt), where t is the temperature in C, α and 0 are unknown coefficients, a current source is connected. After some time, the source is disconnected from the resistor. A plot of resistor temperature versus time is shown in the figure. The heat transfer power of the resistor to 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 picture cannot be displayed now. Fund "Talent and Success". Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 9. Find the EMF and internal resistance of the equivalent source (Ɛe \u003d 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 the points and b (on the arc SW) there are m current sources. Find the potential difference between the points and B. What will be this potential difference if the elements face each other with the same poles?. The experimenter assembled an electric circuit consisting of different batteries with negligible internal resistances and identical fuses, the resistance of which is also very low, and drew a diagram of it (the fuses in the diagram are indicated by black rectangles). The experimenter remembers that on that day during the experiment all the fuses remained intact. The voltages of some batteries are known. Restore unknown voltage values. 2. The figure shows the idealized current-voltage characteristics of the diode and resistor. Build the current-voltage characteristic of a circuit section 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 the circuit section containing the 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 non-linear 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 Construct the current-voltage characteristic of the non-linear element. -,0 0.5 5. Determine through which non-linear element the greater 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. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year. 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 non-linear element x is included in one of the arms of the bridge, for which the dependence of the current Ix on the applied voltage Ux is given by the formula: ix \u003d Ux 3, where \u003d 0.25 / V 3. Find the power Nx released on the non-linear element under 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 was inserted into a table lamp, on which power W = 60 W was dissipated, it turned out that power W2 = 0 mW was dissipated on the connecting wires of the lamp. What power will be dissipated on the connecting wires if you put a light bulb with a power of W3 = 00 W? The voltage in the network in both cases is considered equal to U = 220 V. 9. The resistance of the X element varies depending on the voltage on it. If the voltage U< Uкр, то сопротивление равно, а при U >Ukr 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 a resistance = Ohm and a non-linear element can be changed. The dependence of the ammeter readings on the source voltage is given on the graph. The positive direction of the current is set on 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 \u003d 2 \u003d 3 \u003d and three identical diodes D, D2, D3. The current-voltage characteristic of the diode is shown in the graph. Determine the current strength through the ammeter I depending on the voltage UB between the points and V. The ideal meter. Plot I vs. UB by plotting current and voltage values at characteristic points. 22. You have an unlimited number of resistors of arbitrary resistance and diodes at your disposal. Diodes pass current in only one direction, while the voltage drop across them is V (see Fig. a). What 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. to 0 ohm (on one of the resistors connected in parallel) 2. 3/30
10 3.0/m 5.4m Talent and Success Foundation. Sirius Educational Center. Direction "Science". Prel's physical change. 207 year.
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Goals: educational: to systematize the knowledge and skills of students to solve problems and calculate equivalent resistances using models, frames, etc.
Developing: development of logical thinking skills of abstract thinking, the ability to replace equivalence schemes, simplify the calculation of schemes.
Educational: fostering a sense of responsibility, independence, the need for skills acquired in the lesson in the future
Equipment: a wire frame of a cube, a tetrahedron, an infinite chain of resistance grids.
DURING THE CLASSES
Update:
1. Teacher: "Remember the series connection of resistances."
Students draw a diagram on the board.
and write down
U about \u003d U 1 + U 2
Y about \u003d Y 1 \u003d Y 2
Teacher: remember the parallel connection of resistances.
The student draws an elementary diagram on the board:
Y about \u003d Y 1 \u003d Y 2
; for for n equal
Teacher: And now we will solve problems for calculating the equivalent resistance of the circuit section is presented in the form geometric figure or metal mesh.
Task #1
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 this frame, it is necessary to replace the equivalent circuit. Points 1, 2, 3 have the same potential, they can be connected into one node. And the points (vertices) of the cube 4, 5, 6 can be connected to another node for the same reason. Students have a model on each desk. After performing the described steps, an equivalent circuit is drawn.
On 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 the equivalent circuit:
The calculation of the equivalent resistance of such a circuit is simple. 
Task #3
The same cube model, with inclusion in the circuit between points 2 and B. Students connect points with equal potentials 1 and 3; 6 and 4. Then the circuit will look like this:
Points 1.3 and 6.4 have equal potentials, and the current through the resistances between these points will not flow, and the circuit is simplified to the form; the equivalent resistance of which is calculated as follows:
Task #4
An equilateral triangular pyramid whose edge has resistance R. Calculate the equivalent resistance when included in the circuit.
Points 3 and 4 have an equal potential, so no current will flow along edge 3.4. Students remove it.
Then the diagram will look like this:
The equivalent resistance is calculated as follows:
Task number 5
Metal mesh with link resistance R. Calculate the equivalent resistance between points 1 and 2.
At point 0, you can separate the links, then the circuit will look like:
- resistance of one half symmetrical in 1-2 points. Parallel to it is the same branch, therefore 
Task number 6
The star consists of 5 equilateral triangles, the resistance of each 
.
Are you familiar with Ohm's law (connection of conductors)? // Quantum. - 2012. - No. 1. - C. 32-33.
By special agreement with the editorial board and the editors of the journal "Quantum"
The currents continue indefinitely at a constant speed, ... but they always stop the moment the circuit breaks.
André Ampère
The transfer of electricity between two nearest elements, other things being equal, 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 of linear equations for determining the currents flowing through the 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 in a relatively simple way.
Richard Feynman
Our first encounters with electrical circuits occur when we plug in household appliances at home or stumble upon the intricacies of wiring under the cover of some electronic device, or when we notice power lines on high poles and thick wires along which current collectors of electric trains, trolleybuses and trams slide. Later we draw diagrams at school, set up the simplest experiments and learn about the laws of electric, primarily direct, current, current - how could it be otherwise! - by wire.
But at the same time, we use mobile phones, wireless local area networks, "stick in the air" to connect to the Internet, and increasingly we hear that not just around the corner - wireless transmission of not only information, but also electricity. How archaic, then, will all these cumbersome circuits, wires, terminals, rheostats and the laws that describe them seem!
Take your time. Firstly, no matter what we transmit - signals or energy, there are emitters and receivers, which will not work without currents flowing through the conductors stuffed into them. Secondly, not everything lends itself to miniaturization, such as transport or power plants. Therefore, we will have to deal with electrical networks, and hence with connections of conductors of various types, for a long time to come. 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 safely perch on high voltage wires?
2. From series-connected bulbs for a flashlight, a garland was assembled, designed to be connected to a 220 V network. Each of the bulbs has a voltage of only about 3 V, but if you unscrew one of the bulbs from the cartridge and put your finger in there, it will “twitch” strongly . Why?
3. The battery is closed by three conductors of the same length connected in series. Figure 1 shows a graph showing the voltage drop across them. Which conductor has the most 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 the same resistance were connected so that under the action of a total voltage of 5 V, the current in the circuit turned out to be 1 A. Determine the resistance of one conductor. Does the problem have the only solution?
6. From identical resistors with a resistance of 10 ohms, it is required to make a circuit with a resistance of 6 ohms. What is the smallest number of resistors needed for this? Draw a circuit diagram.
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 with resistance change r each, if we add two more of the same conductors, as shown by dashed lines in Figure 3?
9. What is the resistance R of each of the two identical resistors (Fig. 4), if the voltmeter resistance R V\u003d 3 kOhm when turned 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 of constant voltage U 1 and U 2, as shown in Figure 5. Under what conditions will the current through the resistor 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 r.
12. A hollow cube was soldered from thin homogeneous sheets of tin, conductors were soldered to two opposite vertices of the large diagonal of which, 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 the 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 impedance between these nodes. Each side of the cell has resistance r, and the current flowing on the indicated side is equal to i.
14. In an electrical circuit consisting of six identical resistors with a resistance R, two jumpers CE and DF were soldered, as shown in Figure 9. What was the resistance between terminals A and B?
15. The galvanic cell is closed to two parallel conductors with resistances R 1 and R 2. Will the currents in these conductors decrease if their resistances are increased?
Microexperience
How can you determine the length of an insulated copper wire rolled into a large coil without unwinding it?
It's curious that...
Ohm's experiments, which seem trivial today, are remarkable in that they initiated the clarification of the root causes of electrical phenomena, which remained very vague for a little less than two hundred years and devoid of any experimental justification.
Unfamiliar with Ohm's law, the French physicist Pouille, experimenting, came to similar conclusions in 1837. Upon learning that the law had been discovered a decade ago, Pouyet began to scrutinize it. The law was confirmed with high accuracy, and the "side result" was the study of Ohm's law by French schoolchildren until the 20th century under the name of Pouillet's law.
... when deriving his law, Ohm introduced the concepts of "resistance", "current", "voltage drop" and "conductivity". Along with Ampere, who introduced the terms "electric circuit" and "electric current" and determined the direction of the current in a closed circuit, Ohm laid the foundations for further electrodynamic research on the way to the practical use of electricity.
... in 1843, the English physicist Charles Wheatstone, applying 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 proved by Kirchhoff. Somewhat earlier, he also established the laws of current distribution in branched circuits, and later built general theory current flow in conductors, assuming the existence of two equal counter flows of positive and negative electricity in them.
... intensive development of electrical measurement methods in the 19th century was facilitated by the demands of technology: the creation of overhead telegraph lines, the laying of underground cables, the transmission of electric current through overhead uninsulated wires, and, finally, the construction of an underwater transatlantic telegraph. The theorist of the latter project was the outstanding English physicist William Thomson (Lord Kelvin).
… some practical problems of economics and logistics, such as, for example, the search for the minimum cost distribution of goods, have found their solution when modeling traffic flows using electrical networks.
Questions and tasks
1. The resistance of the bird's body is much greater than the resistance of the parallel section of the wire between its legs, so the current in the bird's body is small and harmless.
2. The finger has a very high resistance compared to the resistance of a light bulb. When it is “turned on” in series with the bulbs, the same current flows through the finger and the bulbs, so the voltage drop on the finger will be much greater than the voltage drop on the bulbs, i.e. almost all the mains voltage will be applied to the finger.
3. Conductor 3 has the most resistance, conductor 2 has the least.
4. R total \u003d R \u003d 1 Ohm.
5. When five conductors are connected in series, the resistance of each conductor is R = 1 Ohm. Another solution is also possible: the conductors are connected in parallel to each other in 2 groups, in one of which there are 3 conductors, in the other - 2, and these groups are connected to each other in series. Then R = 6 ohms.
6. Four resistors; see fig. 10.
7. Figure 11 shows the so-called bridge circuit, when currents flow through all resistors.
Size: px
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1 9 class 1. Minimal path A car traveling at a speed υ at some moment starts moving with such a constant acceleration that during the time τ the path s traveled by it turns out to be minimal. Define this path s. 2. Reflection in flight In the ballistics laboratory, during an experiment on the study of elastic reflection from moving obstacles u, a small ball was fired from a small υ catapult mounted on a horizontal surface. At the same time, from the point where, according to calculations, the ball was supposed to fall S, a massive vertical wall began to move towards it at a constant speed (see figure). After elastic reflection from the wall, the ball fell at some distance from the catapult. Then the experiment was repeated, changing only the velocity of the wall. It turned out that in two experiments the ball hit the wall at the same height h. Determine this height if it is known that the flight time of the ball before reflection in the first case was t1 = 1 s, and in the second t2 = 2 s. What is the maximum height H reached by the balloon during the entire flight? What is the initial speed of the ball υ if the distance between the places of its fall on a horizontal surface in the first and second experiments was L = 9 m? Determine the velocities of uniform motion of the wall u1 and u2 in these experiments and the initial distance S between the wall and the catapult. Consider g = 1 m/s 2. Note. In the reference frame associated with the wall, the modules of the ball velocity before and after the collision are the same, and the angle of reflection of the ball equal to the angle fall. 3. Three-cylinder A body glued together from three coaxial cylinders of different cross sections and different heights is immersed in some liquid and the dependence of the Archimedes force F acting on the body on the depth h of its immersion is removed. It is known that the cross-sectional area of the narrowest (not the fact that the lowest) cylinder S \u003d 1 cm 2. Plot the dependence F (h) and use it to determine the height of each of the cylinders, the cross-sectional areas of the other two cylinders and the density of the liquid. During the experiment, the axis of rotation of the cylinders remained vertical, g = 1 m/s 2. h, cm F a, H.3.9 1.8 2.4 3.6 4.2 4.8 6, 7.2 7, 3 7.5 7.6 7.7 7.8 7.9 7.9
2 4. Two in a cube The cube is assembled from identical resistors having resistances R. Two resistors are replaced by ideal jumpers, as shown in the figure. Find the total resistance of the resulting system between pins A and B. Which of the remaining resistors can be removed so that this does not change the total resistance of the system? If you know that most of the resistors in the circuit carry a current I = 2 A, what is the current in the wire connected to node A (or B)? Calculate the current flowing through the ideal jumper AA? 5. Ice spot Determine what is the maximum mass mp of water vapor taken at a temperature of 1 C that may be required to heat the ice in the calorimeter to the melting point (without melting). The exact mass of ice and its initial temperature are not known, but these values may lie in the region marked on the -3 m/m diagram. Specific heat -4 of vaporization L = 2.3 MJ/kg, specific heat of ice melting λ = 34 kJ/kg, specific heat of water c = 4 2 J/(kg C), specific heat of ice c1 = 2 1 J/(kg WITH). The mass of ice m on the diagram is given in arbitrary units, showing how many times the mass of ice is less than m = 1 kg. Ignore the heat capacity of the calorimeter and heat losses t, C
3 1 class 1. Power time As a result of the experiment, the dependence of the power N of a constant horizontal force on the time t of its action on a bar of mass m = 2 kg initially resting on a smooth horizontal table was obtained. Some measurements may not be very accurate. determine the power of the force at the time τ = 6 s; find the value of the force F. N, W 1.4 2.8 4.5 5, 6, 1.4 14.7 16.6 18.3 t, s 1, 1.5 2, 2.5 3.2 5 , 7,2 8,4 9, 2. In the hole Rod AB touches ledge K of a hemispherical hole of radius R. Point A moves uniformly with a speed υ over the surface of the hole, starting from the bottom point N, to point M. Find the dependence of the modulus of velocity u of the end rod B from the angle α, which the rod makes with the horizon. The length of the rod AB is 2R. 3. Water with ice Some water and ice are mixed in the calorimeter. Their exact masses and initial temperatures are unknown, but these values lie in the shaded areas highlighted in the diagram. Find the maximum amount of heat that could be transferred by water to ice if, after the establishment of thermal equilibrium, the mass of ice did not change. Determine the possible mass of the contents of the calorimeter in this case. The specific heat of ice melting is λ = 34 kJ/kg, the specific heat of water is c = 42 J/(kg C), the specific heat of ice is c1 = 21 J/(kg C). The masses of water and ice on the diagram are given in arbitrary units, showing how many times their masses are less than m = 1 kg. Ignore the heat capacity of the calorimeter and heat losses t, С 1 m /m
4 4. Three cubed The cube is assembled from identical resistors with resistance R. Three resistors were replaced with ideal jumpers, as shown in the figure. Find the total resistance of the resulting system between pins A and B. Which of the remaining resistors can be removed so that this does not change the total resistance of the system? If you know that the current flowing through most of the resistors in an electrical circuit is equal, what is the current in the wire connected to node A (or B)? I 2A Calculate the amount of current flowing through the ideal jumper AA? 5. Conveyor on its side A belt conveyor lying on its side moves along a rough horizontal floor so that the plane of the belt is vertical. The speed of the conveyor belt is υ. The conveyor moves along the floor at a constant speed u perpendicular to the main sections of its belt. For some time, the conveyor has moved a distance s. Its new position is shown in the figure. The conveyor pushes a block having the shape of a rectangular parallelepiped along the floor. The figure shows a top view of this system. Neglecting the deflection of the belt and assuming the motion of the bar to be steady, find the displacement of the bar in time s/u. Determine the work done by the conveyor during this time to move the block. The coefficient of friction between the bar and the floor is μ1, and between the bar and the tape μ2.
5 11 class 1. Power in space A bar of mass m = 2 kg initially resting on a smooth horizontal table was acted upon by a constant horizontal force F. As a result, the dependence of power N on the displacement s of the bar was obtained. Some measurements may not be very accurate. In what coordinate axes is the experimental dependence of power on displacement linear? Determine the power of the force at a point with coordinate s \u003d 1 cm. Find the value of the force F. N, W, 28.4.57.75 1.2 1.1 1.23 1.26 1.5 s, cm 1, 2, 4, 7, "Dark Matter" Clusters of stars form collisionless systems of the galaxy, in which the stars move uniformly in circular orbits around the axis of symmetry of the system. The galaxy NGC 2885 consists of a cluster of stars in the form of a ball (a core with a radius of rb = 4 kpc) and a thin ring, the inner radius of which coincides with the radius of the core, and the outer one is equal to 15 rb. The ring consists of stars with a negligible mass compared to the core. The stars are evenly distributed in the core. It was found that line speed the motion of stars in the ring does not depend on the distance to the center of the galaxy: from the outer edge of the ring up to the edge of the core, the speed of the stars is υ = 24 km/s. Such a phenomenon can be explained by the presence of a non-luminous mass (“dark matter”) distributed spherically symmetrically about the center of the galaxy outside its core. 1) Determine the mass M of the nucleus of the galaxy. 2) Determine the average density ρth of the substance of the galactic nucleus. 3) Find the dependence of the density of "dark matter" ρт(r) on the distance to the center of the galaxy. 4) Calculate the ratio of the mass of "dark matter", which affects the movement of stars in the disk, to the mass of the nucleus. Note: 1 kpc = 1 kiloparsec = 3, m, gravitational constant γ = 6, N m 2 kg 2.
6 3. Four cubed The cube is made up of identical resistors with resistances R. The four resistors are replaced with ideal jumpers, as shown in the figure. Find the total resistance of the resulting system between pins A and B. Through which resistors is the current current maximum, and through which it is minimum? Find these current values if the current entering node A is I = 1.2 A? What is the current flowing through the ideal jumper AA`? 4. Rhombus. The cyclic process performed over an ideal gas on the (p, V) plane is a rhombus (see the qualitative figure). Vertices (1) and (3) lie on the same isobar, and vertices (2) and (4) lie on the same isochore. During the cycle, the gas did work A. How much does the amount of heat Q12 supplied to the gas in section 1-2 differ from the amount of heat Q 3.4 in section 3-4?, removed from the gas by 5. There are no fluctuations! In an electrical circuit (see Fig.), Consisting of a resistor with resistance R, a coil with inductance L, a charge Q is located on a capacitor with a capacitance C. At some point in time, the key K is closed and at the same time they begin to change the capacitance of the capacitor so that an ideal voltmeter shows a constant voltage. 1) How does the capacitance of the capacitor C(t) depend on time when t changes from to t 1 C L? 2) What work did the external forces do during the time t1? Assume that t 1 L / R C L. Hint. The amount of heat released on the resistor during the time t1 is equal to t1 2 2 Q WR I () t Rdt. 3C
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Consider a classical problem. A cube is given, the edges of which are conductors with some identical resistance. This cube is included in the electrical circuit between its various points. Question: what is cube resistance in each of these cases? In this article, a tutor in physics and mathematics 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 that confirms all calculations.
So, the cube can be included in the circuit in three different ways.
Cube resistance between opposite vertices
In this case, the current, reaching the point A, is distributed among the three edges of the cube. In this case, since all three edges are equivalent in terms of symmetry, none of the edges can be given more or less "significance". Therefore, the current between these ribs must be distributed equally. That is, the current strength in each rib is equal to:
As a result, it turns out that the voltage drop on each of these three ribs is the same and equal to , where is the resistance of each rib. 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 the same and equal. For reasons of symmetry, the potentials of the points 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 get that the edges AC, AD And AE T. Similarly, ribs Facebook, GB And KB 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 edges are equivalent AD And AC. They will carry the same current. In addition, the equivalent are also KE And KF. They will carry the same current. We repeat once again that the current between the 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. So 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 the vertical section (directly between the points T And M) current does not flow. Indeed, the situation is analogous to a balanced measuring bridge. This means that this link can be excluded from the chain. After that, it will not be difficult to calculate the total resistance:
The resistance of the upper link is , the lower one is . Then the total resistance is:
Cube resistance 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 AC And AD. And, accordingly, the same potentials will have points C And D, as well as points symmetrical to them E And F:
Again we connect in pairs the points with the same potentials. 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 merge into a dot 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 by standard methods. Each segment of two resistors connected in parallel is replaced by 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 a single resistor with resistance , in parallel. The resistance of a circuit consisting of two resistors connected in parallel 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:
Well, the total resistance of two resistors connected in parallel with resistance and 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, the edges of which are the same resistors. I also have a multimeter, which I turned on in resistance measurement mode. The resistance of a single resistor is 38.3 kOhm:
