Please explain your reasoning in detail. State where the formulas you use come from, and why they are applicable here. (The goal is, after all, so that we can check and, if necessary, help you improve your approach and way of thinking about such problems.) Please write legibly, and draw large and clearly labeled sketches. See the “Written Homework” page on Blackboard for instructions on how to submit written homework to Gradescope. Note that you may find parts of homework problems paired, where the first part requires an analytic (symbolic) calculation, and the second gives numbers for input parameters and asks for a numerical answer based on the calculation in the previous part. The numerical answer expected will generally be given. This has two purposes, (i) to give you some feeling for magnitudes involved, in (hopefully) realistic scenarios, and (ii) to give a reference point for you to check your analytic calculation. You should be sure to show your work for both parts of such coupled exercises.
Problem 1.
Consider two concentric spherical shells, shown in cross section (and colored blue) below. The inner shell has inner radius c and outer radius d, and the outer shell has inner radius a and outer radius b. The inner shell has a charge −Q on it, and the outer shell has a charge +Q on it.
Initially, the outer shell is a conductor, and the inner shell is an insulator with a constant volume charge density ρ.
For parts of the problem labeled “Numeric,” use the values Q = 5 × 10−8 C, c = 0.2 m, d = 0.3 m, a = 0.5 m, and b = 0.7 m. For all other parts, do all calculations symbolically.
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(a) Where is the charge on the outer shell? Explain your answer.
(b) What is the density ρ on the inner shell, expressed in terms of Q, c, and d?
(c) (Numeric). Compute the numerical value of ρ. [Ans: ρ = −6.27 × 10−7 C/m3]
(d) What is the electric field vector E⃗ (r) for r b?
(k) Now the inner shell is made into a conductor, while retaining its total charge −Q. Does the charge distribution on the inner shell change? If so, how? In what regions does the electric field change? Give expressions for E⃗(r) in each case where the field changes.