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1 1 of 8 3/30/2010 2:18 PM Chapter 32 Homework Due: 8:00am on Tuesday, March 30, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View] 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. Hint A.2 Hint A.3 Hint A.4 Hint A.5 Hint A.6 Compare the brightness of bulbs A and B Compare the brightness of bulbs D and E Compare the brightness of bulbs C and D Compare the brightness of bulbs C and A Compare the brightness of bulbs D and A Rank the bulbs from brightest to dimmest. To rank items as equivalent, overlap them. 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.) Does bulb A get brighter, dimmer, or stay the same brightness? Hint B.1 Hint B.2 How to approach the problem Determine how the current changes Light bulb A gets brighter. Light bulb A gets dimmer. Light bulb A stays the same brightness. Conceptual Question 32.8 Rank in order, from largest to smallest, the powers to dissipated by the four resistors in the figure. Rank from largest to smallest. To rank items as equivalent, overlap them.

2 2 of 8 3/30/2010 2:18 PM View Problem 32.8 How much power is dissipated by the 12 resistor in the figure? Express your answer using two significant figures. = 1.9 How much power is dissipated by the 18 resistor in the figure? Express your answer using two significant figures. = 2.9 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 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 for various combinations of resistors.

3 3 of 8 3/30/2010 2:18 PM 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. 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 For the combination of resistors shown, find the equivalent resistance between points A and B. Hint C.1 Hint C.2 How to approach the question 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.

4 4 of 8 3/30/2010 2:18 PM Hint D.1 How to approach the question Hint D.2 Find for the "4-6-12" combination Hint D.3 Find for the top branch Hint D.4 Find for the bottom branch Express your answer in Ohms. = 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. Problem 32.4 What is the magnitude of the current in the 30 resistor in the figure? = What is the direction of the current in the 30 resistor in the figure? from right to left through the resistor from left to right through the resistor. 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).

5 5 of 8 3/30/2010 2:18 PM The junction rule describes the conservation of which quantity? Note that this rule applies only to circuits that are in a steady state. Hint A.1 At the junction current voltage resistance Apply the junction rule to the junction labeled with the number 1 (at the bottom of the resistor of resistance ). Hint B.1 Elements in series Answer in terms of given quantities, together with the meter readings and and the current. If you apply the juncion rule to the junction above flow enforces the same relationship among the currents when they separate as when they recombine., you should find that the ezpression you get is equivalent to what you just obtained for the junction labeled 1. Obviously the conservation of charge or current 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 Hint C.2 Hint C.3 Elements in series have same current Sign of voltage across resistors Voltage drop across ammeter Express the voltage drops in terms of,,, the given resistances, and any other given quantities. 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. 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. Problem What is the resistance in the figure? 20.0

6 6 of 8 3/30/2010 2:18 PM What is the emf of the battery in the figure? 60.0 Problem What is the current through the 10 resistor in the figure? Express your answer using two significant figures. = 0.12 Is the current from left to right or right to left? left to right right to left RC Circuit and Current Conceptual Question In the diagram below,the two resistors, and, are identical and the capacitor is initially uncharged with the switch open. How does the current through compare with the current through immediately after the switch is first closed? Hint A.1 Using Kirchhoff's junction rule for currents The current through the current through. How does the current through compare with the current through a very long time after the switch has been closed? Hint B.1 Hint B.2 Using Kirchhoff's junction rule for currents Current associated with a fully charged capacitor The current through the current through. Part C

7 7 of 8 3/30/2010 2:18 PM How does the current through compare with the current through immediately after the switch is opened (after being closed a very long time)? Hint C.1 Effect of a discharging capacitor The current through the current through. 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. What happens to the charge on the capacitor immediately after the switch is thrown? The electrons on the negative plate of the capacitor are held inside the capacitor by the positive charge on the other plate. Only the surface charge is held in the capacitor; the charge inside the metal plates flows through the resistor. The electrons on the negative plate immediately pass through the resistor and neutralize the charge on the positive plate. The electrons on the negative plate eventually pass through the resistor and neutralize the charge on the positive plate. What is the current that flows through the resistor immediately after the switch is thrown? Hint B.1 How to approach the problem Find the voltage across the resistor at. Then use Ohm's law to find the current through the resistor. Hint B.2 Find the voltage at What is the voltage across the capacitor at time? Express your answer in terms of any or all of the quantities,, and. = If you apply Kirchhoff's loop rule to this cicuit you will find that the magnitude of the voltage gain across the capacitor is equal to the magnitude of the voltage drop across the resistor. Express your answer in terms of any or all of the quantities,, and. = Note that since current is charge per time, the preceeding formula shows that the units of must be time. The combination of variables is called the time constant. It will occur frequently in problems involving a resistor and a capacitor. Problem A 30.0 capacitor initially charged to 30.0 is discharged through a 1.00 resistor. How long does it take to reduce the capacitor's charge to 9.00? 36.1 The capacitor in the figure begins to charge after the switch closes at. Problem 32.73

8 8 of 8 3/30/2010 2:18 PM What is a very long time after the switch has closed? What is in terms of,, and? Part C In this circuit, does or? Part D Find an expression for the current at time. Score Summary: Your score on this assignment is 99.9%. You received out of a possible total of 70 points.

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