# Kirchhoff's Rules and Applying Them

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2 Part C Apply the loop rule to loop 2 (the smaller loop on the right). Sum the voltage changes across each circuit element around this loop going in the direction of the arrow. Remember that the current meter is ideal. Hint C.1 Elements in series have same current Hint C.2 Sign of voltage across resistors Hint C.3 Voltage drop across ammeter Express the voltage drops in terms of,,, the given resistances, and any other given quantities. Answer not displayed Part D Now apply the loop rule to loop 1 (the larger loop spanning the entire circuit). Sum the voltage changes across each circuit element around this loop going in the direction of the arrow. Express the voltage drops in terms of,,, the given resistances, and any other given quantities. Answer not displayed Measurements in Electric Circuits Learning Goal: To understand the role of the internal resistance of various devices and the use of the ammeter and the voltmeter. Consider the circuit shown. All wires are considered ideal; that is, they have zero resistance. We will assume for now that all other elements of the circuit are ideal, too: The value of resistance is a constant, the internal resistances of the battery ( ) and the ammeter ( ) are zero, and the internal resistance of the voltmeter ( ) is infinitely large. What is the reading of the voltmeter? Express your answer in terms of the EMF. = Answer not displayed Part B 2 of 23 17/4/07 15:39

4 What is the new reading of the ammeter? Express your answer in terms of,,, and. = Answer not displayed Now assume that the ammeter again has zero resistance, but the resistance of the voltmeter is less than infinity. The battery still has nonzero internal resistance. Part H Compared to their values when is some large but finite value? Hint H.1 Consider the voltmeter first Hint H.2 The change in the battery current Hint H.3 The reading of the ammeter Answer not displayed, how would the readings of the ammeter and the voltmeter change when Suppose now that the piece of ideal wire between points 1 and 2 is removed and replaced by a nonideal wire with a nonzero resistance. Part I How would this change affect the readings of the ammeter and the voltmeter? Hint I.1 How to approach this part Answer not displayed Resistor and Battery Circuits Two Resistors in Series Ohmic Resistors in Series Consider the setup shown of two ideal ohmic resistors. Resistor 1 has resistance, and resistor 2 has resistance. They are connected in series with a constant voltage of magnitude. When the two resistors are connected in this way, they form a system equivalent to a single resistor of resistance, as shown in the next diagram. 4 of 23 17/4/07 15:39

5 What is the effective resistance of the two-resistor system?.1 Voltage drop across.2 Voltage drop across Hint A.3 How to put it all together Express the effective resistance in terms of and. = Answer not displayed Three Resistors in Series Now consider three resistors set up in series, as shown. The resistances are,, and, and the applied constant voltage is again. Part B Find the effective resistance,, of the three-resistor system. Hint B.1 Different strategies to use Express the effective resistance in terms of,, and. = Answer not displayed Two Resistors Two resistors of resistances and, with, are connected to a voltage source with voltage. When the resistors are connected in series, the current is. When the resistors are connected in parallel, the current from the source is equal to. Let be the ratio. Find..1 Calculate the source voltage When resistors are connected in series, the total current must travel through both resistors. Therefore, in a series circuit, the current is equal everywhere. The voltage drops by a certain amount through each resistor, so the voltage drop for the series is the sum of the voltage drops through each resistor. Use this information to find the 5 of 23 17/4/07 15:39

6 source voltage. Express your answer in terms of and the resistances and. =.2 Find another expresssion for the source voltage When resistors are connected in parallel, the current splits, with a portion of it going through each resistor. Therefore, the net current is the sum of the currents through each resistor. The voltage drop across each resistor is the same, and this drop is equal to the source voltage. Use this information to find another expression for the source voltage. Hint A.2.a The equivalent resistance of resistors in parallel Express your answer in terms of the resistances and and the observed parallel current. = Hint A.3 Equate the two expressions for voltage Equate the two expressions for the voltage that you have obtained, making the substitutions and. Then solve for. Keep in mind that. Hint A.4 Formula for the roots of a quadratic equation The roots of are. Hint A.5 General answer More generally, if the ratio of currents is, then is the smaller of the two roots of the equation. That is,. This would be unity for, which would result if the two resistors were equal. Note that the other root of the quadratic equation is greater than 1 and is equal to. Round your answer to the nearest thousandth Resistance and Wire Length You have been given a long length of wire. You measure the resistance of the wire, and find it to be cut the wire into identical pieces, and connect them in parallel, as shown in the figure.. You then 6 of 23 17/4/07 15:39

7 Brightness of Light Bulbs Consider a circuit containing five identical lightbulbs and an ideal battery. Rank the brightness of the five bulbs (A through E) from brightest to dimmest. (The more current flowing through a bulb, the brighter it will be.) Hint A.1 Comparing bulb A to bulb B Hint A.2 Comparing bulb D to bulb E Hint A.3 Comparing bulb C to bulb D or E Hint A.4 Comparing bulb C to bulb A or B List the bulbs in order from brightest to dimmest. Between each pair of bulbs, use the symbol > to indicate that the left-hand bulb is brighter than the right-hand bulb, or = to indicate that the bulbs have the same brightness. For example, "B=C=E>A>D" means that bulbs B, C, and E all have the same brightness, and that they are brighter than bulb A, which in turn is brighter than bulb D. Answer not displayed Now consider what happens when a switch in the circuit is opened. Part B What happens to bulb A? Hint B.1 How to approach this part Answer not displayed 7 of 23 17/4/07 15:39

8 Part C What is the current now flowing in bulb C? Express your answer in terms of the applied voltage bulb. = Answer not displayed and, the resistance of a single Part D What happens to bulb C? Part D.1 Current in bulb C earlier Answer not displayed Comparing brightness of light bulbs Consider five identical light bulbs (A - E) connected to a battery as shown in the circuit below. Rank the brightness of all five bulbs from brightess to dimmest. Hint A.1 Brightness The brightness of a light bulb is related to the power dissipated by that bulb. Recall that power depends upon the square of the current that passes through the bulb and the resistance of the bulb. Since each light bulb in this circuit has the same resistance, the lightbulb with the largest current passing through it will be the brightest..2 Compare the brightness of bulbs A and B The brightness of bulb A is equal to the brightness of bulb B..3 Compare the brightness of bulbs D and E The brightness of bulb D is equal to the brightness of bulb E..4 Compare the brightness of bulbs C and D.4.a Compare the current through bulbs C and D The brightness of bulb C is greater than the brightness of bulb D..5 Compare the brightness of bulbs C and A.5.a Compare the current through bulbs C and A The brightness of bulb C is greater than the brightness of bulb A. 8 of 23 17/4/07 15:39

9 .6 Compare the brightness of bulbs D and A.6.a Compare the current through bulbs C and A The brightness of bulb D is less than the brightness of bulb A. Rank the bulbs from brightest to dimmest. View How does this change effect the circuit? Suppose bulb E is unscrewed and removed from its socket. (The empty socket remains in the circuit.) Part B Does bulb A get brighter, dimmer, or stay the same brightness? Hint B.1 How to approach the problem When light bulb E is unscrewed and removed from the circuit, the current that leaves the battery changes. This will change the amount of current that flows through light bulb A. Keep in mind that the brightness of a light bulb is related to the square of the current that passes through the bulb and the resistance of the bulb. Part B.2 Determine how the current changes When light bulb E is unscrewed and removed from the circuit, will the current that leaves the battery increase or decrease? Hint B.2.a Relationship between resistance and current Part B.2.b Determine how the total resistance changes It will increase. It will decrease Light bulb A gets brighter. Light bulb A gets dimmer. Light bulb A stays the same brightness. Resistors in Parallel A circuit contains a source of constant voltage and two resistors connected in parallel, as shown in the figure. Resistor 1 has resistance, and resistor 2 has resistance. 9 of 23 17/4/07 15:39

10 Which two of the following statements are true? 1. The voltage source provides a constant voltage, a part of which,, drops off across resistor 1, and the remainder,, across resistor 2. Hence. 2. The voltage drops across resistor 1 and across resistor 2 are the same, and they are independent of the values of and. Both and are equal to the voltage provided by the source. Hence. 3. The current provided by the voltage source splits: A part of it,, flows through resistor 1, while the remainder,, flows through resistor 2. Hence. 4. The currents through resistor 1 and through resistor 2 are the same, and they are equal to the current provided by the source. Hence. Answer not displayed Part B Part C Series And Parallel Connections Learning Goal: To learn to calculate the equivalent resistance of the circuits combining series and parallel connections. Resistors are often connected to each other in electric circuits. Finding the equivalent resistance of combinations of resistors is a common and important task. Equivalent resistance is defined as the single resistance that can replace the given combination of resistors in such a manner that the currents in the rest of the circuit do not change. Finding the equivalent resistance is relatively straighforward if the circuit contains only series and parallel connections of resistors. An example of a series connection is shown in the diagram: For such a connection, the current is the same for all individual resistors and the total voltage is the sum of the voltages across the individual resistors. Using Ohm's law ( ), one can show that, for a series connection, the equivalent resistance is the sum of the individual resistances. Mathematically, these relationships can be written as: An example of a parallel connection is shown in the diagram: For resistors connected in parallel the voltage is the same for all individual resistors because they are all connected to the same two points (A and B on the diagram). The total current is the sum of the currents through the individual 10 of 23 17/4/07 15:39

11 resistors. This should makes sense as the total current "splits" at points A and B. Using Ohm's law, one can show that, for a parallel connection, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances. Mathematically, these relationships can be written as: NOTE: If you have already studied capacitors and the rules for finding the equivalent capacitance, you should notice that the rules for the capacitors are similar - but not quite the same as the ones discussed here. In this problem, you will use the the equivalent resistance formulas to determine resistors. for various combinations of For the combination of resistors shown, find the equivalent resistance between points A and B. Express your answer in Ohms. = 9 These resistors are connected in series; the current through each is the same. Part B For the set-up shown, find the equivalent resistance between points A and B. Express your answer in Ohms. = 2 This is a parallel connection since the voltage across each resistor is the same. Part C 11 of 23 17/4/07 15:39

12 For the combination of resistors shown, find the equivalent resistance between points A and B. Hint C.1 How to approach the question Part C.2 What kind of connection is this? Express your answer in Ohms. = 5 In this case, you cannot say that all three resistors are connected either in series or in parallel. You have a combination of a series and a parallel connection. Some circuits may contain a large number of resistors connected in various ways. To determine the equivalent resistance of such circuits, you have to take several steps, carefully selecting the "sub-combinations" of resistors connected in relatively obvious ways. Good record-keeping is essential here. The next question helps you practice this skill. Part D For the combination of resistors shown, find the equivalent resistance between points A and B. Hint D.1 How to approach the question Part D.2 Find for the "4-6-12" combination Part D.3 Find for the top branch Part D.4 Find for the bottom branch Express your answer in Ohms. 12 of 23 17/4/07 15:39

13 = 3 The next level of analyzing a circuit is to determine the voltages across and the currents through the various branches of the circuit. You will practice that skill in the future. Of course, there are circuits that cannot possibly be represented as combinations of series and parallel connections. However, there are ways to analyze those, too. Finding Current by Changing Resistors A battery provides a voltage and has unknown internal resistance. When the battery is connected across a resistor of resistance, the current in the circuit is. If the external resistance is then changed to (i.e., by replacing the resistor), what will be the value of the new current in the circuit? Hint A.1 How to approach the problem Find two expessions for the internal resistance, one involving the resistance and one involving. Eliminate the internal resistance from the resulting system of equations, and solve for. Hint A.2 Internal resistance explained All sources of EMF have an attribute called internal resistance. For example, the chemical reactions that provide the EMF in a DC battery occur in a medium that has its own resistance to the flow of electrons as current. We can model this as an an extra resistor with resistance connected in series with the EMF source. (In reality, this internal resistance is an inherent part of the EMF source and is not something we can remove.).3 Find the internal resistance in terms of Find the internal resistance Hint A.3.a Kirchhoff's rules of the EMF source. Remember that according to Kirchhoff's rules, the sum of the voltages around a closed loop must equal zero. Express your answer in terms of,, and. =.4 Find the internal resistance in terms of Find the internal resistance of the EMF source. Hint A.4.a Kirchhoff's rules Express your answer in terms of,, and. 13 of 23 17/4/07 15:39

14 More Challenging Resistor and Battery Circuits Series Resistors with Different Areas Four wires are made of the same highly resistive material, cut to the same length, and connected in series. 1. Wire 1 has resistance and cross-sectional area. 2. Wire 2 has resistance and cross-sectional area. 3. Wire 3 has resistance and cross-sectional area. 4. Wire 4 has resistance and cross-sectional area. A voltage is applied across the series, as shown in the figure. Find the voltage across wire 2..1 Resistance of wires.2 Find the current through wire 2 Hint A.3 The voltage across wire 2 Give your answer in terms of, the voltage of the battery. = Answer not displayed Batteries in Series or Parallel You are given two circuits with two batteries of emf and internal resistance each. Circuit A has the batteries connected in series with a resistor of resistance, and circuit B has the batteries connected in parallel to an equivalent resistor. Note that the symbol should be entered in your answers as EMF. 14 of 23 17/4/07 15:39

15 In which direction does the current in circuit A flow? Hint A.1 Conventions Remember that the conventional current flows from a positive to a negative terminal. clockwise counterclockwise Part B What is the current through the resistor of resistance in circuit A? Hint B.1 Which formula to use You are given the voltage and the resistance; hence Ohm's law is the relevant one here. Hint B.2 Total resistance of the circuit One can do this by inspection or by applying the Kirchhoff's loop rule. Kirchhoff's loop rule states that. For circuit A, is the change in total potential difference due to the presence of electrical components in the circuit. Remember that each resistor contributes voltage equal to. Express the current in terms of,, and. = Part C Calculate the current through the resistor of resistance for circuit B. Hint C.1 Which rule to use Apply Kirchhoff's loop rule to the circuit. You should divide the circuit into two loops, then apply Kirchhoff's loop rule to each loop to obtain two equations. To obtain the equation for, you will also need to apply Kirchhoff's junction rule (current rule). Part C.2 What is the emf for loop 1? The diagram shows the circuit divided into two loops: is the current in the topmost branch, is the current in the branch below it, while is the current in the lowest branch, which contains. Find an expression for the emf using the voltage drops across the two resistors in loop 1. Express your answer in terms of,,, and. = Part C.3 What is the emf for loop 2? Now look at loop 2. Find an expression for the emf using the voltage drops across the two resistors in the loop. Express your answer in terms of,,, and. 15 of 23 17/4/07 15:39

16 = Part C.4 Application of Kirchhoff's junction rule (current rule) You should now have two equations involving all the variables in the circuit diagram. To solve for, you need a relationship between and. Choose the correct relation by applying Kirchhoff's junction rule to one of the junctions. Recall that Kirchhoff's junction rule states that the algebraic sum of all the currents into a junction is zero:. Now solve the three equations you have obtained for the currents in each branch to obtain an expression for ( ). To do this, you could either add the two equations other than the one above, or substitute for and from the other equations into this one. Express your answer in terms of,, and. = Part D What is the power dissipated by the resistor of resistance for circuit A, given that,, and? Hint D.1 What formula to use There are two formulas for working out the power dissipated by a resistor: and where is the resistance of a resistor and is the voltage across the resistor. You have worked out from a previous part of the question, so the latter equation looks more useful. Calculate the power to two significant figures. = W, Part E For what ratio of and would power dissipated by the resistor of resistance be the same for circuit A and circuit B? Hint E.1 Getting started Hint E.2 Finding 16 of 23 17/4/07 15:39

17 = 1 Part F Under which of the following conditions would power dissipated by the resistance that of circuit B? Hint F.1 How to think about the problem Some answer choices overlap; choose the most restrictive answer. in circuit A be bigger than Heating a Water Bath In the circuit in the figure, a 20-ohm resistor sits inside 101 of pure water that is surrounded by insulating Styrofoam. If the water is initially at temperature 10.4, how long will it take for its temperature to rise to 57.0? Hint A.1 How to approach the problem First reduce the system of resistors to a single equivalent resistor; then use this simplified circuit to calculate the current flowing through the battery. Determine the current flowing through the resistor in the water and calculate its power output. Finally, use the calculated power output to calculate the time needed to heat the water bath..2 Calculate the resistance of the circuit Calculate the total resistance of the network of resistors shown in the figure. Hint A.2.a Reducing a network of resistors to an equivalent resistor.2.b Combining the resistors in the middle section Hint A.2.c Combining the rest of the resistors Express your answer in ohms using three significant figures. = Calculate the current in the equivalent resistor 17 of 23 17/4/07 15:39

18 Circuits with Capacitors and Varying Current An R-C Circuit Learning Goal: To understand the behavior of the current and voltage in a simple R-C circuit A capacitor with capacitance is initially charged with charge. At time a resistor with resistance is connected across the capacitor. Use the Kirchhoff loop rule and Ohm's law to express the voltage across the capacitor flowing through the circuit. in terms of the current Express your answer in terms of and. = Answer not displayed Part B Part C Part D Charging and Discharging a Capacitor in an R-C Circuit Learning Goal: To understand the dynamics of a series R-C circuit. Consider a series circuit containing a resistor of resistance and a capacitor of capacitance connected to a source of EMF with negligible internal resistance. The wires are also assumed to have zero resistance. Initially, the switch is open and the capacitor discharged. Let us try to understand the processes that take place after the switch is closed. The charge of the capacitor, the current in the circuit, and, correspondingly, the voltages across the resistor and the capacitor, will be changing. Note that at any moment in time during the life of our circuit, Kirchhoff's loop rule holds and indeed, it is helpful:, where is the voltage across the resistor, and is the voltage across the capacitor. 18 of 23 17/4/07 15:39

21 . Graphs of these functions are shown in the figure. Part J Find the time that it would take the charge of the capacitor to reach 99.99% of its maximum value given that and. Part J.1 Find an expression for the time Find the time that it takes the charge of the capacitor to reach 99.99% of its maximum value. Hint J.1.a How to approach this part What is the maximum charge on the capacitor? Set corresponding time. to be 99.99% of this value and solve for the Express your answer in terms of. Use three significant figures for any numerical terms. = It would take the same amount of time for the current to drop to 0.01% of its initial (maximum) value; compare the expressions for and to see this for yourself. Express your answer numerically in seconds. Use three significant figures in your answer. = Notice how quickly the circuit approaches steady state for these typical values of resistance and capacitance! Let us now consider a different R-C circuit. This time, the capacitor is initially charged ( ), and there is no source of EMF in the circuit. We will assume that the top plate of the capacitor initially holds positive charge. For this circuit, Kirchhoff's loop rule gives, or equivalently,. Part K Find the current as a function of time for this circuit. Part K.1 Find the charge on the capacitor 21 of 23 17/4/07 15:39

22 Changing Capacitance Yields a Current Each plate of a parallel-plate capacator is a square with side length, and the plates are separated by a distance. The capacitor is connected to a source of voltage. A plastic slab of thickness and dielectric constant is inserted slowly between the plates over the time period until the slab is squarely between the plates. While the slab is being inserted, a current runs through the battery/capacitor circuit. Assuming that the dielectric is inserted at a constant rate, find the current as the slab is inserted..1 What is the effect of the dielectric on capacitance?.2 What is the current in the circuit?.3 What is the initial capacitance?.4 What is the change in capacitance? Express your answer in terms of any or all of the given variables,,,,, and, the permittivity of free space. = Answer not displayed Charged Capacitor and Resistor Learning Goal: To study the behavior of a circuit containing a resistor and a charged capacitor when the capcaitor begins to discharge. A capacitor with capacitance is initially charged with charge. At time, a switch is thrown to close the circuit connecting the capacitor in series with a resistor of resistance. 22 of 23 17/4/07 15:39

23 Part B Summary 7 of 17 problems complete (36.15% avg. score) of 35 points 23 of 23 17/4/07 15:39

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