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A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by for「SRI2 Here α is a positive constant having units of C/m3 (a) Determine a in terms of Q and R (b) Using Gausss law, derive an expression for the magnitude of E as a function of r. Do this separately for all three regions. Express your answers in terms of the total charge Q. Be sure to check that your results agree on the boundaries of the regions. (Use the following as necessary: r, R, Q, and ε。.) (c) What fraction of the total charge is contained within the region rs R/2? (d) If an electron with charge q-e is oscillating back and forth about 0 (the center of the distribution) with an amplitude less than R/2, show that the motion is simple harmonic. (Hint: If, and only if, the net force on the electron is proportional to its displacement from equilibrium, then the motion is simple harmonic. Do this on paper. Your instructor may ask you to turn in this work.) (e) What is the period of the motion in part (d)? (Use the following as necessary: r, R, Q, e, eo, and me for the mass of an electron.) (f) If the amplitude of the motion described in part (e) is greater than R/2, is the motion still simple harmonic? Why or why not?
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