The shaded region is a quarter-annular disc of outer radius R and inner radius R/2 with uniform surface charge density σ, centred at O. Its potential at O is σR/(4nε₀). Find n.
Physics · Mixed Chapter
JEE Mains

Question

The shaded region is a quarter-annular disc of outer radius R and inner radius R/2 with uniform surface charge density σ, centred at O. Its potential at O is σR/(4nε₀). Find n.

G1 Physics · Accepted Answer

Step 1 — Geometry: The charged region is one quarter of an annulus, with inner radius R/2, outer radius R and angular extent π/2. Step 2 — Potential contribution of an area element: dq = σ(r dr dθ), and dV = dq/(4πε₀r) = σ dr dθ/(4πε₀). Step 3 — Integrate over the region: V = [σ/(4πε₀)]∫₀^(π/2)dθ ∫ from R/2 to R dr = [σ/(4πε₀)](π/2)(R/2) = σR/(16ε₀). Step 4 — Match the question: σR/(4nε₀) = σR/(16ε₀) ⇒ 4n = 16 ⇒ n = 4. Answer: 4.