For radii R₁ ≤ r ≤ R₂, a spherical shell contains volume charge density ρ(r) = a/r; a point charge Q is at the common centre. The field is constant in this shell when a = Q/(√z πR₁²). Find z.
Physics · Mixed Chapter
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Question

For radii R₁ ≤ r ≤ R₂, a spherical shell contains volume charge density ρ(r) = a/r; a point charge Q is at the common centre. The field is constant in this shell when a = Q/(√z πR₁²). Find z.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — For R₁ ≤ r ≤ R₂, Gauss’s law depends on the charge enclosed within a sphere of radius r. Step 2 — The central charge is Q. The additional charge in the shell is q(shell) = ∫(R₁→r) ρ(r′)4πr′²dr′ = 4πa∫(R₁→r)r′dr′ = 2πa(r² − R₁²). Step 3 — Total enclosed charge q = Q + 2πa(r² − R₁²). Therefore E(r) = q/(4πε₀r²) = a/(2ε₀) + [Q − 2πaR₁²]/(4πε₀r²). Step 4 — For E(r) to be constant, the coefficient of 1/r² must vanish: Q − 2πaR₁² = 0 ⇒ a = Q/(2πR₁²). Step 5 — Given a = Q/(√z πR₁²), compariso…