A solid nonconducting sphere of radius R has charge density ρ(r) = ρ₀(1 − r/R), and dielectric constant K = 1. The energy stored in the region outside the sphere is U = πρ₀²R⁵/(9nε₀). Find n.
Physics · Mixed Chapter
JEE Mains

Question

A solid nonconducting sphere of radius R has charge density ρ(r) = ρ₀(1 − r/R), and dielectric constant K = 1. The energy stored in the region outside the sphere is U = πρ₀²R⁵/(9nε₀). Find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Charge density: ρ(r) = ρ₀(1 − r/R) for 0 ≤ r ≤ R. Step 2 — Total charge is Q = ∫(0→R) 4πr²ρ(r)dr = 4πρ₀∫(0→R)[r² − r³/R]dr = 4πρ₀(R³/3 − R³/4) = πρ₀R³/3. Step 3 — Outside the sphere, Gauss’s law gives E(r) = Q/(4πε₀r²), for r ≥ R. Step 4 — External electric-field energy density is u = (1/2)ε₀E². Integrate only over r ≥ R: U = ∫(R→∞) (1/2)ε₀E²(4πr²dr) = Q²/(8πε₀R). Step 5 — Insert Q = πρ₀R³/3: U = [π²ρ₀²R⁶/9]/(8πε₀R) = πρ₀²R⁵/(72ε₀). Step 6 — Given U = πρ₀²R⁵/(9nε₀). Thus 9n = 72 ⇒ n =…