HW__9_P213_F08 - Physics 2213 Read: Homework #9 Fall 2008...

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Unformatted text preview: Physics 2213 Read: Homework #9 Fall 2008 Chapter 24, intro., sections 24.1, 24.2 Chapter 26, section 26.4 Chap. 24: Q's #Q24.2, 6, 7, 8; E's & P's: #24.2, 9, 13, 18, 19 Chap. 26: Q's #Q26.19, 21; E's & Ps: #26.38, 41, 43, 83 For study: To be prepared for your 2nd recitation class the week of October 27-31: #24.59 #24.66 [Capacitor Network] [Capacitor + Metal Slab] #24.60 #26.39 [Throw the Switch] [Discharging Capacitor] 1. [Global Atmospheric Capacitor] In problem #1 on HW #7, we found that there is an 5 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 drives a downward global electric current of roughly 1300 A, and the resistance of the atmosphere between the ground and ionosphere is roughly 230 . Let's now 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 and with 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 of 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? 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 r. (c) 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 determined values of global 5 surface charge (-6.1 x 10 C) and the electric potential difference between ionosphere and 5 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? (d) What is time constant of the global atmospheric electrical circuit based on our calculated values for its resistance and capacitance? How does 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 on HW #7? What process maintains the charge separation on the global atmospheric capacitor? [Assignment CONTINUES on next page] 2. [Charging Capacitor] The capacitor is initially uncharged, and the switch is open. Then at time t = 0, the switch is closed. + b a R c ! S C d Vbc(t) Switch closed 0 0 t (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. For the following questions: Vcd(t) = 10 V, R = 2000 , C = 400 F (F = 10 F) -6 (b) What is the current in the resistor just after the switch is closed? Please show your work. 0 0 t (c) What is the final charge on the capacitor after a long time? Please show your work. (d) At what time t is the voltage across the resistor equal in magnitude to the voltage across the capacitor? Please show your work. Vdb(t) 0 0 t [Assignment CONTINUES on next page] 2 3. [Charge Sheets, Superposition, & Parallel Conducting Plates] Shown below are three different arrangements (A-C) of large flat parallel sheets of charge with surface charge densities (charge/area) or /2 and five corresponding regions (1-5) labeled on each diagram. Where a charge density is marked as "0" there is no sheet of charge. (A) (B) (C) (a) Use the Superposition Principle along with what you know about the electric field on each side of a single large flat sheet of charge to determine the magnitude and direction of the electric field in each region (1-5) for each charge sheet arrangement (A-C). To do this, you should first find the magnitude and direction of the electric field vector in each region due to each charge sheet separately. Then you should add together the electric field vectors in each region due to the various charge sheets to get the net electric field vector there. Please illustrate your vector calculations neatly in the form of a table for each arrangement (A-C). (b) Which charge sheet arrangements have electric field configurations that correctly represent the expected electric fields in and around two flat conducting plates in electrostatic equilibrium occupying regions #2 and 4? Please explain your reasoning carefully. 3 ...
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