A hemispherical bowl with inner radius R and outer radius 2R has volume charge density ρ(r) = ρ₀r/R. The electric field at point O is nρ₀R/(4ε₀). Find n.

Question

G1 Physics · Accepted Answer

Step 1: Treat the charged hemispherical bowl as a stack of infinitesimal hemispherical shells. For an element between radii r and r + dr, use ρ(r)=ρ₀r/R and integrate its axial electric-field contribution at O over the hemisphere. Step 2: Performing the angular integral first and then the radial integral from R to 2R gives E = 3ρ₀R/(4ε₀). Step 3: Comparing with E = nρ₀R/(4ε₀) gives n = 3.