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Homework Physics #5
Spring 2009
*** This assignment will be covered on Prelim Exam #1. *** Read: Chapter 23, all (section 23.5 for conceptual content); Chapter 26, sections 26.2 and 26.5
Learning Goals: (Be sure you understand where and how each goal in each assignment applies to our homework, discussion, lecture, and lab activities.) * * * * * * * * * Use Kirchhoff's Rules to analyze electric circuits relate and determine electric currents and voltages. Recognize when it is possible or not possible to reduce networks of electrical resistances to simpler equivalent forms using series and parallel resistance combinations. Do this reduction where possible and useful. Determine properties of electric circuits (V, I, R, or P), and predict how those properties will change when components or connections are changed. Calculate electric potential as a function of position in space, either directly using the PointCharge Superposition Principle or directly from a known electric field. Use electric potential along with Conservation of Energy to relate speeds and positions of moving charged particles. Sketch graphs for electric potential and electric field as functions of position (without doing detailed numerical calculations), and show how such graphs are related. GENERAL SKILL: Set up and solve numerical problems with symbolic algebra before plugging in numbers in order to check the conceptual validity of reasoning and results. GENERAL SKILL: Show that your algebraic and numerical results have the correct units. GENERAL SKILL: Show that your algebraic and numerical results are physically reasonable. Chap. 23: Q's #Q23.2, 4-11, 15-21 E's & P's: #23.9, 13, 19, 21, 31, 45, 57, 79 Chap. 26: Q's #Q26.13-18, 22 E's & P's: #26.13, 15, 17, 79, 92 [answer: (5/6)R]
For extra practice:
To be worked and discussed before Prelim Exam #1, but not quizzed: #26.23 [Kirchoff's Circuit Rules] #26.50 [Electric Dryer] Please add: (d) Does your answer to (c) make physical sense? #23.82 [Alpha Particle Collision] Please add: GENERAL SKILL: First, solve this problem algebraically in order to check the conceptual validity of your result. Show that your algebraic result makes physical sense before inserting numbers. #23.22 [Pair of (+) Charges] Please add: (f) Show that the electric potential V(x) in part (c) gives the correct electric field E(x) on the x-axis, using the relationship Ex(x) = - dV(x)/dx. Show that the graph of V(x) vs. x correctly corresponds to the graph of E(x) vs. x. #23.72 & 73 [Solid Insulating Sphere of Charge] An extension of our discussion of the electric potential for a charged conducting sphere or shell. Follow the procedure used in lecture to find V(r) E(r). from Please add: (c) GENERAL SKILL: Show that your graph for V(r) correctly corresponds to your resulting algebraic expressions. [Assignment CONTINUES on next page]
1. [Current Divider] In this circuit the electric current through the battery and resistor R1 divides between resistors R2 and R3 . (a) Show that the currents I2 and I3 in resistors R2 and R3 are inversely proportional to those resistances.
+
R1
R2
R3
(b) Write an expression for the equivalent resistance of resistors R2 and R3 . GENERAL SKILLS: ("open-circuit") and when R3 ("short-circuit")? What are the currents through R1, R2, and R3 in each case? (c) GENERAL SKILLS: Why is each of the special limiting values of current in (b) physically reasonable? Why are the nicknames "short-circuit" and "open-circuit" appropriate for these two limiting cases? How could you achieve each of these limiting cases in the circuit without using resistor R3? You may use other wires if you wish. #2. [Irreducible Circuit] Here is a circuit called a "bridge network" discussed briefly in lecture. Your ultimate goal is to find answers to questions (c) and (d) on the handout sheet. To do this, work through questions (i)-(iv) below.
a
What values does this equivalent resistance take when R3
0
I
+
R1 = R I1
b
R2 = 2R R3 = R I2
c
[HINTS: There are no series or parallel combinations of I4 I5 I3 resistors to use to reduce the circuit to a simpler from. All we can do is apply Kirchhoff's Circuit Rules for R4 = 2R R5 = R currents and voltages. The 6 unknown currents through d the resistors and battery are labeled. We've guessed at likely directions for these currents (which may or may not be correct). To answer the questions on the lecture handout, we'll first need to find all these individual currents in terms of R and .] (i) We can reduce the number of independent unknown currents to 3 by writing currents I, I4, and I5 in terms of I1, I2, and I3. Do so, and mark your results on the diagram. (ii) Now we need 3 more equations in order to solve for I1, I2, and I3. Write voltage loop equations in terms of I1, I2, and I3 for these 3 circuit loops: (1) d-battery-a-b-d, (2) d-battery-a-c-d, (3) a-b-c-a. (iii) Use equations (1) and (2) to find I1 and I2 in terms of I3. Use these relations in equation (3) to find I3 in terms of R and . Then work back using equations (1) and (2) to find I1 and I2 each explicitly in terms of R and . Do any of these currents turn out to have negative values? If so, what does this mean physically? (iv) Use these results to answer lecture handout questions (c) and (d). What is the effective resistance of this circuit between points a and b?