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Answer: In Exercises 928, find the limit or explain why it does not exist

Thomas' Calculus Early Transcendentals | 12th Edition | ISBN: 9780321588760 | Authors: George B. Thomas ISBN: 9780321588760 156

Solution for problem 16 Chapter 2

Thomas' Calculus Early Transcendentals | 12th Edition

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Thomas' Calculus Early Transcendentals | 12th Edition | ISBN: 9780321588760 | Authors: George B. Thomas

Thomas' Calculus Early Transcendentals | 12th Edition

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Problem 16

In Exercises 928, find the limit or explain why it does not exist

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Short Assignment By 2/8/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Short Assignment By 2/8/2016 Due: 11:00am on Monday, February 8, 2016 To understand how points are awarded, read theading Policy for this assignment. Kirchhoff's Loop Rule Conceptual Question The circuit shown belowconsists of four different resistors and a battery. You don't know the strength of the battery or the value any of the four resistances. Part A Select the expressions that will be equal to the voltage of the battery in the circuit, whe, for example, is the potential drop across resistor A. Check all that apply. Hint 1. Kirchhoff's voltage rule for closed circuit loops Kirchhoff’s loop rule states that in any closed circuit loop, the voltage supplied by a battery must be used by the devices in the loop. Therefore, the voltage drop across all of the resistors in a single closed circuit loop must add up to the voltage of the battery. Carefully identify all of the closed loops in this circuit. ANSWER: Correct Short Assignment By 2/8/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Problem 21.58 Part A Find the magnitude of the current in the figure . ANSWER: = 0.889 Correct Part B Find the direction (clockwise or counterclockwise) of the current. ANSWER: clockwise counterclockwise Correct Score Summary: Your score on this assignment is 100%. You received 1.5 out of a possible total of 1.5 points. Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Short Assignment By 2/10/2016 Due: 11:00am on Wednesday, February 10, 2016 To understand how points are awarded, read therading Policy for this assignment. Capacitors in Parallel Learning Goal: To understand how to calculate capacitance, voltage, and charge for a parallel combination of capacitors. Frequently, several capacitors are connected together to form a collection of capacitors. We may be interested in determining the overall capacitance of such a collection. The simplest configuration to analyze involves capacitors connected in series or in parallel. More complicated setups can often (though not always!) be treated by combining the rules for these two cases. Consider the example of a parallel combination of capacitors: Three capacitors are connected to each other and to a battery as shown in the figure. The individual capacitances are , , and , and the battery's voltage is . Part A If the potential of plate 1 i, then, in equilibrium, what are the potentials of plates 3 and 6 Assume that the negative terminal of the battery is at zero potential. Hint 1. Electrostatic equilibrium When electrostatic equilibrium is reached, all objects connected by a conductor (by wires, for example) must have the same potential. Which plates on this diagram are at the same potential ANSWER: and and and and Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Correct Part B If the charge of the first capacitor (the one with capacitan) is , then what are the charges of the second and third capacitors Hint 1. Definition of capacitance Capacitance is given by , where is the charge of the capacitor and is the voltage across it. Hint 2. Voltages across the capacitors As established earlier, the voltage across each capacitor i. The voltage is always the same for capacitors connected in parallel. ANSWER: and and and and Correct Part C Suppose we consider the system of the three capacitors as a single "equivalent" capacitor. Given the charges of the three individual capacitors calculated in the previous part, find the total chargfor this equivalent capacitor. Express your answer in terms of and . ANSWER: = Correct Part D Using the value of , find the equivalent capacitance for this combination of capacitors. Express your answer in terms of . Hint 1. Using the definition of capacitance Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Use the general formula to find . The charge on the "equivalent" capacitor is , and the voltage across this capacitor is the voltage across the battery, . ANSWER: = Correct The formula for combining three capacitors in parallel is . How do you think this formula may be generalized to capacitors Capacitors in Series Learning Goal: To understand how to calculate capacitance, voltage, and charge for a combination of capacitors connected in series. Consider the combination of capacitors shown in the figure. Three capacitors are connected to each other in series, and then to the battery. The values of the capacitances are , , and , and the applied voltage is . Initially, all of the capacitors are completely discharged; after the battery is connected, the charge on plate 1 is . Part A What are the charges on plates 3 and 6 Hint 1. Charges on capacitors connected in series When the plates of two adjacent capacitors are connected, the sum of the charges on the two plates must remain zero, since the pair is isolated from the rest of the circuit; that is, and Q_4+Q_5=0, where \texttip{Q_{\it i}}{Q_i} is the charge on plate \texttip{i}{i}. Hint 2. The charges on a capacitor's plates When electrostatic equilibrium is reached, the charges on the two plates of a capacitor must have equal magnitude and opposite sign. Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... ANSWER: +Q and +Q -Q and -Q +Q and -Q -Q and +Q 0 and +Q 0 and -Q Correct Part B If the voltage across the first capacitor (the one with capacitance \texttip{C}{C}) is \texttip{\Delta V_{\rm 1}}{DeltaV_1}, then what are the voltages across the second and third capacitors Hint 1. Definition of capacitance The capacitance \texttip{C}{C} is given by \large{{\frac {Q}{\Delta V}}}, where \texttip{Q}{Q} is the charge of the capacitor and \texttip{\Delta }{Delta} \texttip{V}{V} is the voltage across it. Hint 2. Charges on the capacitors As established earlier, the absolute value of the charge on each plate is \texttip{Q}{Q}: It is the same for all three capacitors and thus for all six plates. ANSWER: 2\Delta V_1 and 3\Delta V_1 \large{\frac{1}{2} \Delta V_1} and \large{\frac{1}{3}\Delta V_1} \Delta V_1 and \Delta V_1 0 and \Delta V_1 Correct Part C Find the voltage \texttip{\Delta V_{\rm 1}}{DeltaV_1} across the first capacitor. Express your answer in terms of \texttip{\Delta V}{DeltaV}. Hint 1. How to analyze voltages According to the law of conservation of energy, the sum of the voltages across the capacitors must equal the voltage of the battery. Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... ANSWER: \texttip{\Delta V_{\rm 1}}{DeltaV_1} = \large{{\frac{6}{11}}{\Delta}V} Correct Part D Find the charge \texttip{Q}{Q} on the first capacitor. Express your answer in terms of \texttip{C}{C} and \texttip{\Delta V_{\rm 1}}{DeltaV_1}. ANSWER: \texttip{Q}{Q} = C{\Delta}V_{1} Correct Part E Using the value of \texttip{Q}{Q} just calculated, find the equivalent capacitance \texttip{C_{\rm eq}}{C_eq} for this combination of capacitors in series. Express your answer in terms of \texttip{C}{C}. Hint 1. Using the definition of capacitance The "equivalent" capacitor has the same charge as each of the individual capacitors: \texttip{Q}{Q}. Use the general formula \large{C={\frac {Q}{\Delta V}}} to find \texttip{C_{\rm eq}}{C_eq}. ANSWER: \texttip{C_{\rm eq}}{C_eq} = \large{\frac{6C}{11}} Correct The formula for combining three capacitors in series is \large{\frac{1}{C_{\rm series}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}}. How do you think this formula may be generalized to \texttip{n}{n} capacitors Equivalent Capacitance Consider the combination of capacitors shown in the diagram, where \texttip{C_{\rm 1}}{C_1} = 3.00 {\rm \rm \mu F} , \texttip{C_{\rm 2}}{C_2} = 11.0 {\rm \rm \mu F} , \texttip{C_{\rm 3}}{C_3} = 3.00 {\rm \rm \mu F} , and \texttip{C_{\rm 4}}{C_4} = 5.00 {\rm \rm \mu F} . Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Part A Find the equivalent capacitance \texttip{C_{\rm A}}{C_A} of the network of capacitors. Express your answer in microfarads. Hint 1. How to reduce the network of capacitors To find the equivalent capacitance of the given network of capacitors, it is most convenient to reduce the network in successive stages. First, replace the capacitors \texttip{C_{\rm 2}}{C_2}, \texttip{C_{\rm 3}}{C_3}, and \texttip{C_{\rm 4}}{C_4}, which are in parallel, with a single capacitor with an equivalent capacitance. By doing so, you will reduce the network to a series connection of two capacitors. At this point, you only need to find their equivalent capacitance. Hint 2. Find the capacitance equivalent to \texttip{C_{\rm 2}}{C_2}, \texttip{C_{\rm 3}}{C_3}, and \texttip{C_{\rm 4}}{C_4} Find the capacitance \texttip{C_{\rm 234}}{C_234} equivalent to the parallel connection of the capacitors \texttip{C_{\rm 2}}{C_2}, \texttip{C_{\rm 3}}{C_3}, and \texttip{C_{\rm 4}}{C_4}. Express your answer in microfarads. Hint 1. Find the capacitance equivalent to \texttip{C_{\rm 3}}{C_3} and \texttip{C_{\rm 4}}{C_4} Find the capacitance \texttip{C_{\rm 34}}{C_34} equivalent to the parallel connection of the capacitors \texttip{C_{\rm 3}}{C_3} and \texttip{C_{\rm 4}}{C_4}. Express your answer in microfarads. Hint 1. Two capacitors in parallel Consider two capacitors of capacitance \texttip{C_{\rm a}}{C_a} and \texttip{C_{\rm b}}{C_b} connected in parallel. They are equivalent to a capacitor with capacitance \texttip{C_{\rm eq}}{C_eq} given by C_{\rm eq}=C_{\rm a} + C_{\rm b}. ANSWER: \texttip{C_{\rm 34}}{C_34} = 8.00 \rm \mu F Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... ANSWER: \texttip{C_{\rm 234}}{C_234} = 19.0 \rm \mu F Hint 3. Two capacitors in series Consider two capacitors of capacitance \texttip{C_{\rm a}}{C_a} and \texttip{C_{\rm b}}{C_b} connected in series. They are equivalent to a capacitor of capacitance \texttip{C_{\rm eq}}{C_eq} that satisfies the following relation: \large{\frac{1}{C_{\rm eq}}=\frac{1}{C_{\rm a}}+\frac{1}{C_{\rm b}}}. ANSWER: \texttip{C_{\rm A}}{C_A} = 2.59 \rm \mu F Correct Part B Two capacitors of capacitance \texttip{C_{\rm 5}}{C_5} = 6.00 {\rm \rm \mu F} and \texttip{C_{\rm 6}}{C_6} = 3.00 {\rm \rm \mu F} are added to the network, as shown in the diagram. Find the equivalent capacitance \texttip{C_{\rm B}}{C_B} of the new network of capacitors. Express your answer in microfarads. Hint 1. How to reduce the extended network of capacitors To determine the equivalent capacitance of the extended network of capacitors, it is again convenient to reduce the network in successive stages. First, determine the equivalent capacitance of the series connection of the capacitors \texttip{C_{\rm 2}}{C_2} and \texttip{C_{\rm 6}}{C_6}. Then, combine it with the equivalent capacitance of the parallel connection of \texttip{C_{\rm 3}}{C_3}, \texttip{C_{\rm 4}}{C_4}, and \texttip{C_{\rm 5}}{C_5}, and replace the five capacitors with their equivalent capacitor. The resulting network will consist of two capacitors in series. At this point, you only need to find their equivalent capacitance. Hint 2. Find the equivalent capacitance of \texttip{C_{\rm 2}}{C_2}, \texttip{C_{\rm 3}}{C_3}, \texttip{C_{\rm 4}}{C_4}, \texttip{C_{\rm 5}}{C_5}, and \texttip{C_{\rm 6}}{C_6} Find the equivalent capacitance \texttip{C_{\rm 2-6}}{C_2-6} of the combination of capacitors \texttip{C_{\rm 2}}{C_2}, \texttip{C_{\rm 3}}{C_3}, \texttip{C_{\rm 4}}{C_4}, \texttip{C_{\rm 5}}{C_5}, and \texttip{C_{\rm 6}}{C_6}. Typesetting math: 56% Express your answer in microfarads. Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Hint 1. Find the equivalent capacitance of \texttip{C_{\rm 2}}{C_2} and \texttip{C_{\rm 6}}{C_6} Find the equivalent capacitance \texttip{C_{\rm 26}}{C_26} of the series connection of \texttip{C_{\rm 2}}{C_2} and \texttip{C_{\rm 6}}{C_6}. Express your answer in microfarads. Hint 1. Two capacitors in series Consider two capacitors of capacitance \texttip{C_{\rm a}}{C_a} and \texttip{C_{\rm b}}{C_b} connected in series. They are equivalent to a capacitor of capacitance \texttip{C_{\rm eq}}{C_eq} that satisfies the following relation: \large{\frac{1}{C_{\rm eq}}=\frac{1}{C_{\rm a}}+\frac{1}{C_{\rm b}}}. ANSWER: \texttip{C_{\rm 26}}{C_26} = 2.36 \rm \mu F Hint 2. Find the equivalent capacitance of \texttip{C_{\rm 3}}{C_3}, \texttip{C_{\rm 4}}{C_4} , and \texttip{C_{\rm 5}}{C_5} Find the equivalent capacitance \texttip{C_{\rm 345}}{C_345} of the parallel connection of \texttip{C_{\rm 3}}{C_3}, \texttip{C_{\rm 4}}{C_4} , and \texttip{C_{\rm 5}}{C_5} Express your answer in microfarads. Hint 1. Three capacitors in parallel Consider three capacitors of capacitance \texttip{C_{\rm a}}{C_a}, \texttip{C_{\rm b}}{C_b}, and \texttip{C_{\rm c}}{C_c} connected in parallel. They are equivalent to a capacitor with capacitance \texttip{C_{\rm eq}}{C_eq} given by C_{\rm eq}=C_{\rm a} + C_{\rm b} + C_{\rm c}. ANSWER: \texttip{C_{\rm 345}}{C_345} = 14.0 \rm \mu F ANSWER: \texttip{C_{\rm 2-6}}{C_2-6} = 16.4 \rm \mu F Hint 3. Two capacitors in series Consider two capacitors of capacitance \texttip{C_{\rm a}}{C_a} and \texttip{C_{\rm b}}{C_b} connected in series. They are equivalent to a capacitor of capacitance \texttip{C_{\rm eq}}{C_eq} that satisfies the following relation: \large{\frac{1}{C_{\rm eq}}=\frac{1}{C_{\rm a}}+\frac{1}{C_{\rm b}}}. Typesetting math: 56% Short Assignment By 2/10/2016 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... ANSWER: \texttip{C_{\rm B}}{C_B} = 2.54 \rm \mu F Correct Score Summary: Your score on this assignment is 91.7%. You received 2.75 out of a possible total of 3 points. Typesetting math: 56%

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Textbook: Thomas' Calculus Early Transcendentals
Edition: 12
Author: George B. Thomas
ISBN: 9780321588760

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Answer: In Exercises 928, find the limit or explain why it does not exist