12. A region in space contains a total positive charge that is distributedspherically such that the volume charge density is given byfor r ≤ R/2for R/2 ≤ r ≤ Rfor r≥ Rp(r) = 3ar/(2R)p(r) = a[1 (r/R)²]p(r) = 0Here a is a positive constant having units of C/m³ (a) Determine a in termsof Q and R (b) Using Gauss's law, derive an expression E as a function of rDo this separately for all regions. Express your answers in terms of the totalcharge Q.Be sure to check that your results agree on the boundaries of theregions. (c) What fraction of the total charge is contained within the regionr≤R/2(d) If an electron q' = -e with charge is oscillating back and forthabout r=0 (the center of the distribution) with an amplitude less than R/2show that the motion is simple harmonic. (Hint: Review the discussion ofsimple harmonic motion). If, and only if, the net force on the electron isproportional to its displacement from equilibrium, then the motion is simpleharmonic.) (e) What is the period of the motion in part (d)? (f) If theamplitude of the motion described in part (e) is greater than R/2 is themotion still simple harmonic? Why or why not?

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A region in space contains a total positive charge that is distributed spherically such that the volume charge density is given by for r ≤R/2 for R/2 ≤r≤ R for r > R p(r) = 3ar/(2R) p(r) = a[1 (r/R)²] p(r) = 0 Here a is a positive constant having units of C/m³ (a) Determine a in terms -