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 < c, where r is the distance from the center of the shells? Explain your answer.
(e) What is the electric field vector E⃗ (r) for c < r < d?
(f) (Numeric). Compute the magnitude of the electric field vector |E⃗(r)| and its direction at the point P1, located a distance 0.25 m to the right of the center of the shells. [Ans: E-field magnitude|E⃗(r)|=2.9×103 N/C.]
(g) What is the electric field vector E⃗ (r) for d < r < a?
(h) (Numeric). Compute the magnitude of the electric field vector |E⃗(r)| and its direction at the point P2, located a distance 0.4 m to the right of the center of the shells. [Ans: E-field magnitude|E⃗(r)|=2.8×103 N/C.]
(i) What is the electric field vector E⃗ (r) for a < r < b?
(j) 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.