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Tutorial: Analyzing Circuits Name____________________________Section __________ Created by Alexandru Maries 1 (Adapted from BU) You should work together in table groups throughout this assignment. Be sure to check in with your instructor at each of the checkpoints. I. Kirchoff’s Rules: Junction Rule A. A junction is a point at which at least three circuit paths meet. Identify all of the junctions in the circuit below. Junction Rule: At a junction, the sum of the currents going into the junction is equal to the sum of the currents coming out of the junction. B. For each the following junctions below, write down Ki rchoff’s junction rule. II. Batteries, resistors and electric potential – Kirchoff’s Loop Rule Consider the circuit shown in the figure on the right. A. Suppose the electric potential at point A is – 12 V. What is the electric potential at point B? Explain your reasoning. B. What is the electric potential at point C? C. What is the electric potential at point D? D. What is the direction of the electric current going through the 8Ω resistor?

Tutorial: Analyzing Circuits Created by Alexandru Maries 2 (Adapted from BU) E. Consider the 8Ω resistor shown on the right. There is a current of 1.5 A going up through the resistor. If the electric potential at the lower end of the resistor is 20 V, what is the electric potential at the upper end? Loop Rule: The sum of all the electric potential changes around a complete loop is equal to zero. To use this rule, it helps if we first mark the ends of each resistor/battery in a circuit with +/- to indicate which end is at a higher potential. For example, consider the circuit below. For the R 1 , since the electric current goes towards the left, the electric potential at the left end of R 1 is lower than the electric potential at the right of R 1 and we use + and – to mark the ends to represent this. F. Mark the ends of each of the other five resistors and the battery with + or – to indicate the ends at higher/lower electric potential. Now that we have marked each circuit element, we can write Kirchoff’s loop rule for the various loops in the circuit. Consider the outer loop (which includes the battery, R 1 , R 2 , R 3 and R 4 ). If we start at point A, and go once around the loop adding all the changes in potential we get: 120 V – I 1 R 1 – I 1 R 2 – I 2 R 3 – I 2 R 4 According to Kirchoff’s loop rule, these changes must add up to zero, therefore: 120 V – I 1 R 1 – I 1 R 2 – I 2 R 3 – I 2 R 4 = 0. G. Write the loop rule for the inner loop (which includes the battery, R 1 , R 2 , R 5 and R 6 ). H. Write the loop rule for the top loop (which includes R 3 , R 4 , R 5 and R 6 ).  Consult your group partners before you proceed. If possible, consult an instructor.

Tutorial: Analyzing Circuits Created by Alexandru Maries 3 (Adapted from BU) III. Using Kirchoff’s rules to solve circuit problems A. Label the current and the current direction in each branch. Note that in many cases it's hard to tell which the correct direction for the current is in a particular loop. That does NOT matter . Just make an educated guess. If you guess wrong, you'll get a negative value for that current, which is fine. The value is correct and the negative sign indicates that the direction is opposite to the way you drew it. However, always remember to use the negative sign in your calculations of other values. B. Mark the ends of the battery with +/- and use the direction of the currents you selected to label the ends of each resistor with +/- to indicate which end is at a higher/lower electric potential. C. Use Kirchoff’s junction rule to write equations for each junction. Note that usually, writing down equations for half the junctions is sufficient since the other junctions will give you equivalent equations. D. Use Kirchoff’s loop rule to write down loop equations for all the loops in the circuit. i. Top loop ii. Outer loop iii. Bottom loop We now have to solve for the unknown currents. To do this, we will ALWAYS use at least one junction equation and one loop equation. E. Write down the three equations you will need to use to solve for the three unknown currents. Pick the easier two of the three loop equations. General strategy to solve for the three currents: • You now have three equations, two loop equations which include only two currents, and one junction equation which includes all three. • Identify the current that is common in the loop equations. • Use those equations to solve for the other currents in terms of the common one. For example, if the common current is I 3 , use the two loop equations to solve for I 1 and I 2 in terms of I 3 . • Plug in the junction equation.

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Tutorial: Analyzing Circuits Created by Alexandru Maries 4 (Adapted from BU) F. Use the general strategy outlined on the previous page to solve for the three currents. IV. Practice A. For the circuit on the right, pick a direction for the current in each branch. B. Label the ends of batteries and resistors with +/- to indicate which end is at a higher/lower electric potential. C. Write down Kirchoff’s junction rule for half of the junctions (here, there are two junctions and they give you the same equation). D. Write down Kirchoff’s loop rule for as many loops as you need to solve for all the unknowns. E. Use the general strategy outlined on the previous page to solve for the currents.

Tutorial: Analyzing Circuits Created by Alexandru Maries 5 (Adapted from BU) F. Follow the steps you learned today to determine the three currents in the circuit on the right. G. What is the power used by the entire circuit? (Hint: batteries supply power when the current runs through them in the direction you would expect, and they use up power when the current runs through them in the opposite direction to what you would expect.)  Consult your group partners before you proceed. If possible, consult an instructor. V. Additional practice 1. A group is considering the following circuit. Three resistors are connected as shown in a circuit with a 12 volt battery. The resistances are: R A = 4  R B = 3  R C is of unknown value (larger than 0) A. Andy thinks R A has the most current going through it. A. J. says that it is impossible to determine unless the value of R C is known. Who do you agree with? Explain your reasoning B. Rank the resistors based on the potential difference across them, from largest to smallest. Justify your choices.

Tutorial: Analyzing Circuits Created by Alexandru Maries 6 (Adapted from BU) C. If R C is increased, what happens to the currents through the different elements of the circuit? i. The current through the battery… ii. The current through R A … iii. The current through R B … iv. The current through R C … D. If the current through the battery is 2.0 amps, what is the value of R C ? 2. For the circuit on the right, R 1 = 10.0Ω, R 2 = 8.0Ω, ε 1 = 10.0V and ε 2 = 12.0V. A. Find the currents going through each resistor. ( Hint: The current through R 1 can easily be found if you pick your loop carefully!) B. Double check that the other loop rules are satisfied. i. Left loop (R 1 , R 2 and ε 2 ): ii. Right loop (R 2 , ε 1 , ε 2 ):

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Tutorial: Analyzing Circuits Created by Alexandru Maries 7 (Adapted from BU) 3. Consider the circuit on the right where ε 1 = 6.0V, ε 2 = 10.0V, ε 3 = 20.0V, R 1 = 8 .0Ω , R 2 = 10. 0Ω , and R 3 = 15. 0Ω . A. Find the current going through each resistor. B. Suppose R 1 is a fuse which burns out due to a sudden surge of current, thus, it essentially becomes an open switch. How do the currents change after this? i. Current through R 1 ii. Current through R 2 iii. Current through R 3  Consult your group partners before you proceed. If possible, consult an instructor.