2213 Physics Read:
Homework #6
Spring 2009
Chapter 24, intro., sections 24.1 thru 24.5 Chapter 26, section 26.4
Learning Goals: (Be sure you understand where and how each goal in each assignment applies to our homework, discussion, lecture, and lab activities.) * * * * * * * * * Explain what a capacitor is and calculate how its electrical properties are related charge, voltage, stored energy, capacitance, electric current. Show how to calculate the capacitance of a parallel plate capacitor, given its geometry. Use the parallel plate capacitance expression to approximate the capacitance of real systems. Explain why electric charge is distributed the way it is in capacitors. Explain how and why the capacitance of a capacitor is affected by the presence of a dielectric insulator filling the space between the capacitor's plates, and calculate such effects. Explain and calculate how energy is stored in a capacitor. Explain how a capacitor functions in an electric circuit. Predict and calculate the time-dependent behavior of RC circuits containing resistors, capacitors, DC power supplies, and switches. Draw graphs showing the time dependence of electrical quantities such as voltages, currents, and charges in these circuits. Show how to reduce appropriate networks of capacitors to equivalent simpler forms using series and parallel combinations. Chap. 24: Q's #Q24.1, 4-9, 11-13, 15, 18, 20 E's & P's: #24.7, 15, 21, 27, 37, 59a, 65a, 77 Chap. 26: Q's #Q26.19-21 E's & Ps: #26.41, 49, 87
For extra practice:
To be prepared for Wednesday-Friday, Mar. 4-6 at your 2nd weekly Discussion section: #24.14 [Capacitor Network] #24.66 [Capacitor + Metal Slab]
1. ["Build-a-Cap"] A capacitor is constructed out of two sheets of aluminum foil and three sheets of paper in Paper the form of a double-decker sandwich, as shown. The Foil foil sheets are 17 cm x 24 cm in size and 0.050 mm thick, Paper while the paper sheets are 21cm x 28 cm in size and Foil 0.10 mm thick. The paper has dielectric constant 1.5 and Paper 6 dielectric breakdown strength 7.0 x 10 V/m (the electric field needed to produce a spark though the paper). Electrode wires are attached to the two foil sheets, and then all the sandwich layers are pressed tightly together. (a) What is this capacitor's capacitance? What maximum voltage can it sustain without suffering dielectric breakdown (sparking through the paper between the foil plates)? Be sure your reasoning is clear. (b) OPTIONAL CHALLENGE: To make the capacitor more compact, it is now rolled up tightly into a tube. What are its capacitance and breakdown voltage in this form? Please explain your reasoning. [HINTS: Which sides of foil are facing each other? What's in-between them? Can you model this system as a pair of capacitors?] [Assignment CONTINUES on next page]
2. [Smile!] Capacitors are often used in applications where a short burst of electric current is needed. To power a certain camera's flash lamp, a capacitor must supply a current burst that dissipates a total of 20 J of energy as light and heat in a lamp of resistance 10 . When the lamp is fired, the capacitor is to deliver 95% of its stored energy (95% of 20 J) in a time of 3.0 ms. (a) What should be the capacitance of the capacitor? (b) To what voltage should the capacitor be charged? GENERAL SKILL: Could such voltage be obtained directly using typical household batteries? Why or why not? (c) How much charge is on the capacitor when fully charged? GENERAL SKILL: Is this a lot? Why do we not get spectacular lightning bolts from the capacitor? 3. [RC Circuit from an old P213 Exam] The capacitor is initially uncharged, and the switch is open. Then at time t = 0, the switch is closed. (a) Draw graphs showing how each of the labeled voltages below varies with time t, including t < 0 (before the switch is closed) as well as t > 0. Vbc(t)
Switch a
R
c
C
d
S
t
For closed
+
b the following questions: = 10 V, R = 2000 , C = 400 F ( = 10 )
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0
0
(b) What is the current in the resistor just after the switch is closed? Please show your work. Vcd(t) (c) What is the final charge on the capacitor after a long time? Please show your work.
0 t
0
(d) When the voltages across the resistor and capacitor are equal in magnitude, what fraction of its final value is the energy stored in the capacitor? (A) 1/4 (D) 3/4 (B) 1/ 2 3/2 (E) (C) 1/2 (F) other
Vdb(t)
(e) At what time t is the voltage across the resistor equal in magnitude to the voltage across the capacitor? Please show your work.
0 t
0
(f) From t = 0 to , how much total energy is supplied by the battery? Please show your work, and be sure your reasoning is clear.
[Assignment CONTINUES on next page] 2
4. [Global Atmospheric Electric Circuit] In problem #1 on HW #3, we found that there is 5 an electric potential difference of roughly 3.0 x 10 V between the ionosphere and ground due to a downward electric field that varies with height and a global layer of surface charge on the 5 ground of roughly -6.1 x 10 C. We also found that this potential difference and electric field drive a downward global electric current of roughly 1300 A, and the resistance of the atmosphere between the ground and ionosphere is roughly 230 . Now let's think about the capacitance of this system. (a) If we model the ground + ionosphere simply as two concentric spherical conductors of radii a and b (> a) with charges -Q and +Q on them and vacuum in the region in-between them, what is the electric field in this region as a function of distance r from the center of the system? What is the electric field outside this region? What electric potential difference or voltage Vba does this electric field give between the two conductors? Express your answers in terms of Q, a, b, and physical constants. (Do not yet make any approximation based on b - a << a and b.) (b) Use your results from part (a) to derive an expression for the capacitance of this spherical capacitor. Does your result depend on the charge Q? GENERAL SKILLS: Is your result physically reasonable, and does it have the correct units? Show that your expression reduces to the parallel plate capacitor result C = oA/h in the limit where h = b - a is << a and b separately, i.e., the spacing h between these conducting spheres is much less than their radii a b. (c) GENERAL SKILLS: Now let's see how this simplified model agrees with reality. What value do you get for the capacitance of the actual global electric circuit using the previously 5 determined values of global surface charge (- 6.1 x 10 C) and the electric potential difference 5 between ionosphere and ground (3.0 x 10 V)? Using the approximate result from our model in part (b), what value of spacing h between conducting spheres would give this same value of capacitance? How do you reconcile this value with the fact that the ionosphere is roughly 90 km above the ground? How is the value of h you calculated here related to the constant = -1 0.43 km in the expression for the electric field E(y) as a function of height given in problem #3 on HW #3? Please explain your reasoning carefully. (d) What is time constant of the global atmospheric electrical circuit based on our calculated values for its resistance and capacitance? How does value this compare with your prediction of the time that it would take for the global atmospheric electric current to completely discharge the ground and atmosphere in problem #1(e) on HW #3? What process maintains the charge separation on the global atmospheric capacitor? (e) How much total electric potential energy is stored in the Earth-ionosphere capacitor? How does this value compare with the total energy consumed by a typical home in a year 11 (roughly 10 J)?
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