22.66 ... CP CALC A region in space contains a total positive charge that is distributed spherically such that the volume charge density is given by
p(r) = α for r <= R/2
p(r) = 2α (1 -r /R) for R /2 <= r <=R
p(r)=0 For r >= R
Here a is a positive constant having units of C/m3. (a) Determine a in terms of Q and R. (b) Using Gauss’s 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. (c) What fraction of the total charge is contained within the region r … R/2? (d) If an electron with charge q¿ = -e is oscillat- ing back and forth about r = 0 (the center of the distribution) with an amplitude less than R/2, show that the motion is simple har- monic. (Hint: Review the discussion of simple harmonic motion in Section 14.2. If, and only if, the net force on the electron is propor- tional to its displacement from equilibrium, then the motion is sim- ple harmonic.) (e) What is the period of the motion in part (d)? (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?