Kirchhoff's Rules and Applying Them Part 1

Kirchhoff's Rules and Applying Them Learning Goal: To understand the origins of both of Kirchhoff's rules and how to use them to solve a circuit problem. This problem introduces Kirchhoff's two rules for circuits: Kirchhoff's loop rule: The sum of the voltage changes across the circuit elements forming any closed loop is zero. Kirchhoff's junction rule: The algebraic sum of the currents into (or out of) any junction in the circuit is zero. The figure shows a circuit that illustrates the concept of loops, which are colored red and labeled loop 1 and loop 2. Loop 1 is the loop around the entire circuit, whereas loop 2 is the smaller loop on the right. To apply the loop rule you would add the voltage changes of all circuit elements around the chosen loop. The figure contains two junctions (where three or more wires meet)--they are at the ends of the resistor labeled . The battery supplies a constant voltage , and the resistors are labeled with their resistances. The ammeters are ideal meters that read and respectively. The direction of each loop and the direction of each current arrow that you draw on your own circuits are arbitrary. Just assign voltage drops consistently and sum both voltage drops and currents algebraically and you will get correct equations. If the actual current is in the opposite direction from your current arrow, your answer for that current will be negative. The direction of any loop is even less imporant: The equation obtained from a counterclockwise loop is the same as that from a clockwise loop except for a negative sign in front of every term (i.e., an inconsequential change in overall sign of the equation because it equals zero). Part A The junction rule describes the conservation of which quantity? Note that this rule applies only to circuits that are in a steady state. Hint 1. At the junction Think of the analogy with water flow. If a certain amount of water comes to a split in the pipe, what can you say (mathematically) about the sum of the three masses of water at this junction? ANSWER: Typesetting math: 33% charge voltage resistance Correct Part B Apply the junction rule to the junction labeled with the number 1 (at the bottom of the resistor of resistance ). Answer in terms of given quantities, together with the meter readings and and the current . Hint 1. Elements in series The current through resistance is not labeled. You should recognize that the current passing through the ammeter also passes through resistance because there is no junction in between the resistor and the ammeter that could allow it to go elsewhere. Similarly, the current passing through the battery must be also. Circuit elements connected in a string like this are said to be in series and the same current must pass through each element. This fact greatly reduces the number of independent current values in any practical circuit. ANSWER: = Correct If you apply the juncion rule to the junction above , you should find that the expression you get is equivalent to what you just obtained for the junction labeled 1. Obviously the conservation of charge or current flow enforces the same relationship among the currents when they separate as when they recombine. 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. Express the voltage drops in terms of , , Hint 1. Elements in series have same current , the given resistances, and any other given quantities. The current through the ammeter is , and this current has to go through the resistor of resistance because there is no junction in between that could add or subtract current. Similarly, the current passing through the battery must be also. Circuit elements connected in a string like this are said to be in series and the same current must pass through each element. This fact greatly reduces the number of independent current values in any practical circuit. Hint 2. Sign of voltage across resistors In determining the signs, note that if your chosen loop traverses a particular resistor in the same direction as the current through that resistor, then the end it enters through will have a more positive potential than the end from which it exits by the amount . Thus the voltage change across that resistor will be negative. Conversely, if your chosen loop traverses the resistor in the opposite direction from its current arrow, the voltage changes across the resistor will be positive. Let these conventions govern your equations (i.e., don't try to figure out the direction of current flow when using the Kirchhoff loop--decide when you put the current arrows on the resistors and stick with that choice). Hint 3. Voltage drop across ammeter An ideal ammeter has zero resistance. Hence there is no voltage drop across it. ANSWER: = Correct 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: = Correct There is one more loop in this circuit, the inner loop through the battery, both ammeters, and resistors and . If you apply Kirchhoff's loop rule to this additional loop, you will generate an extra equation that is redundant with the other two. In general, you can get enough equations to solve a circuit by either 1. selecting all of the internal loops (loops with no circuit elements inside the loop) or 2. using a number of loops (not necessarily internal) equal to the number of internal loops, with the extra proviso that at least one loop pass through each circuit element. pes Learning Goal: To practice Problem-Solving Strategy 26.2 Kirchhoff’s Rules. The circuit diagram below shows two emf sources and a bulb connected in parallel. Also connected in the circuit is a resistor with resistance = 0.2 . The resistance of the bulb is = 0.5 , and each of the sources has internal resistance: = 0.025 and = 0.02 . If = 13.0 and = 5.0 flowing in emf source . , calculate the current Problem-Solving Strategy: Kirchhoff’s rules IDENTIFY the relevant concepts: Kirchhoff’s rules are important tools for analyzing any circuit more complicated than a single loop. SET UP the problem using the following steps: 1. Draw a circuit diagram, and label all quantities, known and unknown, including an assumed direction for each unknown current and emf. If you use Kirchhoff’s rules correctly, they will give you the directions as well as the magnitudes of unknown currents and emfs. 2. When you label currents, it is usually best to use the junction rule immediately to express the currents using as few quantities as possible. 3. Determine which quantities are the target variables. EXECUTE the solution as follows: 1. Choose any closed loop in the network, and designate a direction (clockwise or counterclockwise) to travel around it. 2. Travel around the loop, adding potential differences as you cross them. Recall that a positive potential difference corresponds to an increase in potential and a negative potential difference corresponds to a decrease in potential. An emf is counted as positive when you traverse it from ( ) to (+) and negative when you go from (+) to ( ). An term is negative if you travel through the resistor in the same direction as the assumed current and positive if you pass through it in the opposite direction. 3. Equate the sum in Step 2 to zero. 4. If necessary, choose another loop to get a different relationship among the unknowns, and continue until you have as many independent equations as unknowns or until every circuit element has been included in at least one of the chosen loops. 5. Solve the equations simultaneously to determine the unknowns. Typesetting math: 33% 6. You can use this same system to find the potential of any point with respect to any other point . Start at and add the potential changes you encounter in going from to , using the same sign rules as in Step 2. The algebraic sum of these changes is . EVALUATE your answer: Check all the steps in your algebra. A useful strategy is to consider a loop other than the ones you used to solve the problem; if the sum of potential drops around this loop isn’t zero, you made an error somewhere. IDENTIFY the relevant concepts The circuit has more than one loop, so you must use both the junction rule and the loop rule. The bulb in the circuit can be considered as a resistor. SET UP the problem using the following steps Part A Using Kirchhoff's junction rule, determine which one of the following four diagrams would be the appropriate one to use for solving this problem. ANSWER: Typesetting math: 33% Typesetting math: 33% Correct Often, you will not know in advance the actual direction of an unknown current or emf, but this doesn’t matter. If the actual direction of a particular quantity is opposite to your assumption, the result of your calculations will come out with a negative sign. EXECUTE the solution as follows Part B Calculate the current flowing in emf source . Express your answer in amperes to three significant figures. Hint 1. How to approach the problem There are three closed loops in the network, as shown in the accompanying diagram. You have two unknown variables; therefore, you will need to use two of the loops to obtain two equations that you will solve simultaneously. Travel around each loop in a designated direction, calculating the potential difference for each circuit element and equating the sum to zero (the problem-solving strategy explains how to calculate potential differences). This question involves algebra and can be fairly complex. Be careful with algebraic manipulations; one sign error will prove fatal to the entire solution. Hint 2. Identify the potential drops in loop 1 Select from the list below all the expressions that represent a potential difference through a circuit element in loop 1. Select one expression for each circuit element in the loop; do not include expressions related to elements in other loops of the circuit. Determine the sign of each potential drop by traveling around the loop in the direction shown in the accompanying diagram. Typesetting math: 33% Check all that apply. ANSWER: Hint 3. Identify an equation for loop 2 Identify the equation that satisfies Kirchhoff's rules for loop 2 shown in Hint 1 above. Work in the same counterclockwise direction used to find the equation for loop 1 in Hint 2. ANSWER: Typesetting math: 33% ANSWER: = 22.6 Correct EVALUATE your answer Part C A useful strategy to evaluate your answer is to consider a loop other than the ones you used to solve the problem; if the sum of potential drops around this loop isn’t zero, you made an error somewhere in your calculations. Begin by calculating the current flowing through the emf source . Express your answer in amperes to three significant figures. ANSWER: = 33.5 Typesetting math: 33% Correct Below is a list of the equations obtained by applying Kirchhoff's rules to each loop shown in the diagram above (equation numbers refer to the corresponding loop in the circuit): 1. 2. 3. You can easily verify that your calculated values for and do satisfy all three equations. In other words, a sum check using your values of and will all equate to zero to two significant figures. 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. Typesetting math: 33% Part A What is the reading of the voltmeter? Express your answer in terms of the EMF . ANSWER: = Correct Part B The voltmeter, as can be seen in the figure, is connected to points 1 and 3. What are the respective voltages between points 1 and 2 and between points 2 and 3? ANSWER: Correct Typesetting math: 33%